THz generation based on photoconductivity is a resonant process in which a femtosecond optical pulse is absorbed through interband transitions in a semiconductor to produce charge carriers. These carriers are subsequently accelerated in either an externally applied dc electric field or a built-in electric field in the depletion or accumulation region of the semiconductor. A transient current is thus formed, which in turn emits a THz electromagnetic transient that can propagate either on a transmission line or in free space (Smith et al., 1988).
基于光电导的太赫兹产生是一种共振过程，其中飞秒光脉冲通过半导体中的带间跃迁被吸收以产生电荷载流子。这些载流子随后在外部施加的直流电场或半导体耗尽或积累区域的内置电场中加速。这样就形成了一个瞬态电流，进而发射出一个 THz 电磁瞬变，该瞬态可以在传输线上或自由空间中传播 （ Smith et al., 1988 ）。

For this purpose, the semiconductor can either be incorporated into an antenna or transmission line structure, or it can radiate directly. In the former case, an external electric field is applied across a gap formed by electrodes, which is excited by the optical pulse (Auston, 1983; Auston et al., 1984; Fattinger and Grischkowsky, 1989). The optical pulse is often arranged at normal incidence and the bias field is parallel to the photoconductor surface. The bias field can also be provided by the built-in electric field near the surface of a semiconductor wafer. A depletion or accumulation region is formed in a doped semiconductor as a result of Fermi level pinning. In this case the emitter is excited by an optical pulse at an oblique angle in order to couple the emission into the free space (Hu et al., 1990). The output coupling efficiency is zero for normal incidence. Studies have also shown that magnetic fields can enhance the radiation output coupling efficiency by altering the direction of the current through the Lorentz force (Shan et al., 2001). These studies not only provide conditions to optimize the THz emission, but also means to investigate ultrafast carrier dynamics in semiconductors in magnetic fields (Shan and Heinz, 2004).
为此，半导体可以集成到天线或传输线结构中，也可以直接辐射。在前一种情况下，外部电场施加在电极形成的间隙上，该间隙由光脉冲 （ Auston, 1983 ; Auston et al., 1984 ; Fattinger and Grischkowsky, 1989 ）。光脉冲通常以法向入射排列，偏置场平行于光电导体表面。偏置场也可以由半导体晶片表面附近的内置电场提供。由于费米能级固定，在掺杂半导体中形成耗尽或积累区。在这种情况下，发射器被斜角的光脉冲激发，以便将发射耦合到自由空间 （ Hu et al., 1990 ） 中。对于正常入射，输出耦合效率为零。研究还表明，磁场可以通过改变电流通过洛伦兹力 （ ） 的方向来提高辐射输出耦合效率 Shan et al., 2001 。这些研究不仅为优化太赫兹发射提供了条件，而且为研究磁场中半导体中的超快载流子动力学提供了手段 （ Shan and Heinz, 2004 ）。

In the far field the emitted THz electric field is proportional to the first time derivative of the transient current. The current transient is limited by the duration of the optical excitation pulse and the carrier scattering time, as well as the recombination lifetime of the semiconductor and the time that it takes the carriers to drift out of the active emitter area. Therefore, commonly used semiconductors for THz generation are defect-rich to reduce the fall time of the transient current. Examples include low-temperature grown or ion-implanted GaAs and silicon (McIntosh et al., 1995; Shan and Heinz, 2004; Krotkus and Coutaz, 2005; Mikulics et al., 2006). Following the pioneering work of Auston (1983), Grischkowsky (1993), and their coworkers, researchers optimized ultrafast photoconductive switches in the past two decades to permit generation and field-resolved detection of electromagnetic transients up to $\sim 5\mathrm{THz}$. Such a bandwidth, while impressive, actually reflects the finite response time of photoconductive materials rather than the ideal bandwidth that could be obtained from current state-of-the-art mode-locked laser pulses. For instance, a 10-fs transform-limited optical pulse (with a bandwidth of $\sim 50\mathrm{THz}$) should in principle permit generation and detection of electromagnetic transients up to $\sim 50\mathrm{THz}$. In this regime, however, the comparatively slow response of the carriers in available photoconductive media significantly degrades the high-frequency performance. A complete understanding of the frequency response of the emission process can yield insight into a material’s carrier dynamics. Upon excitation of a 10-fs optical pulse, the transient photocurrent rises rapidly with a rise time of 10 fs followed by a ballistic acceleration before the onset of the momentum relaxation processes and the carrier recombination processes. The 10-fs rise part of the transient current provides the highest spectral components of the emitted THz radiation, but the spectral bandwidth of the emission is significantly below $\sim 50\mathrm{THz}$. The spectral bandwidth is determined by the main contributions to the transient current, i.e., the subsequent much slower processes. A careful analysis of the THz emission has, for example, allowed the investigation of high-field transport on the fs time scale in compound semiconductors such as GaAs and InP (Leitenstorfer et al., 1999, 2000). In contrast, optical rectification, as described in the next section, can potentially generate THz emission with a bandwidth limited only by the duration of the optical excitation pulse. Under the assumption of perfect phase matching and a second-order nonlinearity of the emitter independent of frequencies in the region of interest, the emitted THz electric field in the far field is proportional to the second time derivative of the nonlinear polarization which follows the intensity envelope of the excitation pulse.
在远场中，发射的太赫兹电场与瞬态电流的第一次导数成正比。电流瞬变受光激发脉冲持续时间和载流子散射时间、半导体的复合寿命和载流子从有源发射器区域漂移所需的时间的限制。因此，用于产生太赫兹的常用半导体具有丰富的缺陷，以减少瞬态电流的下降时间。示例包括低温生长或离子注入的 GaAs 和硅 （ McIntosh et al., 1995 ; Shan and Heinz, 2004 ; Krotkus and Coutaz, 2005 ; Mikulics et al., 2006 ）。继 、 Grischkowsky (1993) 及其同事的 Auston (1983) 开创性工作之后，研究人员在过去二十年中优化了超快光电导开关，以允许产生和场分辨检测高达 $\sim 5\mathrm{THz}$ 的电磁瞬变。这样的带宽虽然令人印象深刻，但实际上反映了光电导材料的有限响应时间，而不是从当前最先进的锁模激光脉冲中获得的理想带宽。例如，10 fs 变换限制的光脉冲（带宽为 $\sim 50\mathrm{THz}$ ）原则上应允许产生和检测高达 $\sim 50\mathrm{THz}$ 的电磁瞬变。然而，在这种状态下，载流子在可用光电导介质中相对较慢的响应显着降低了高频性能。全面了解发射过程的频率响应可以深入了解材料的载流子动力学。 在 10 fs 光脉冲激发时，瞬态光电流以 10 fs 的上升时间迅速上升，随后在动量弛豫过程和载流子复合过程开始之前进行弹道加速。瞬态电流的 10 fs 上升部分提供了发射的 THz 辐射的最高光谱分量，但发射的光谱带宽明显低于 $\sim 50\mathrm{THz}$ 。频谱带宽由对瞬态电流的主要贡献决定，即随后的慢得多的过程。例如，对太赫兹发射的仔细分析允许研究化合物半导体（如 GaAs 和 InP）在飞秒时间尺度上的高场传输 （ Leitenstorfer et al., 1999 ， 2000 ）。相比之下，如下一节所述的光校正可能会产生太赫兹发射，其带宽仅受光激发脉冲的持续时间限制。在完美相位匹配和发射器的二阶非线性与感兴趣区域内的频率无关的假设下，远场中发射的太赫兹电场与非线性极化的二阶时间导数成正比，该偏振跟随激发脉冲的强度包络。

With respect to the strength of the THz emission, a linear dependence of the THz electric field on the dc bias (Darrow et al., 1991; Reimann, 2007) has been observed. At low excitation fluence, the THz field also varies linearly with fluence; however, high excitation fluence often leads to saturation of the THz emission. There are two main reasons for saturation: (i) the resultant high charge densities effectively screen the bias electric field; and (ii) the electric-field of the emitted radiation acts back and further decreases the net bias field (Benicewicz et al., 1994; Kim and Citrin, 2006). Photoconductive antennas and coplanar transmission lines with a small gap (tens of microns) are often used with a femtosecond oscillator source that delivers optical pulses of energy on the order of ${10}^{-9}\mathrm{J}/\mathrm{pulse}$. A bias electric field of $\sim {10}^6\mathrm{V}/\mathrm{m}$ can be applied and a typical THz pulse energy of $\sim {10}^{-13}\mathrm{J}$ (and of peak power of ${10}^{-5}\mathrm{W}$) can be achieved. With an amplified femtosecond laser source that delivers pulses of energy on the order of ${10}^{-3}\mathrm{J}/\mathrm{pulse}$, to avoid saturation, large-aperture structures ( $\sim \mathrm{mm}$ gap size) or bare semiconductor wafers are often used. THz emission with a peak electric field up to $150\mathrm{kV}/\mathrm{cm}$, corresponding to an energy of ${10}^{-7}\mathrm{J}/\mathrm{pulse}$ and a peak power of ${10}^5\mathrm{W}$, has been reported (You et al., 1993). Details can be found in Sakai (2005) and Cheville (2008).
就太赫兹发射的强度而言，太赫兹电场对直流偏置的线性依赖性 （ Darrow et al., 1991 ; Reimann, 2007 ） 已被观察到。在低激发磁通量下，太赫兹场也随磁通量线性变化;然而，高激发磁通量通常会导致太赫兹发射饱和。饱和有两个主要原因：（i） 由此产生的高电荷密度有效地屏蔽了偏置电场;（ii） 发射辐射的电场反作用并进一步减小净偏置场 （ Benicewicz et al., 1994 ; Kim and Citrin, 2006 ）。具有小间隙（几十微米）的光电导天线和共面传输线通常与飞秒振荡器源一起使用，该振荡器源提供能量量级的光脉冲 ${10}^{-9}\mathrm{J}/\mathrm{pulse}$ 。可以施加偏置电场 ， $\sim {10}^6\mathrm{V}/\mathrm{m}$ 并且可以实现典型的太赫兹脉冲能量 $\sim {10}^{-13}\mathrm{J}$ （和峰值功率 ${10}^{-5}\mathrm{W}$ ）。对于放大的飞秒激光源，其能量脉冲为 ${10}^{-3}\mathrm{J}/\mathrm{pulse}$ ，为避免饱和，通常使用大孔径结构（ $\sim \mathrm{mm}$ 间隙大小）或裸露的半导体晶片。据报道，峰值电场高达 $150\mathrm{kV}/\mathrm{cm}$ 的 THz 发射对应于 的 ${10}^{-7}\mathrm{J}/\mathrm{pulse}$ 能量和 ${10}^5\mathrm{W}$ 的峰值功率 （ You et al., 1993 ）。有关详细信息，请参阅 Sakai (2005) 和 Cheville (2008) 。

An alternative method to generate THz radiation is to rely on nonresonant nonlinear optical processes such as optical rectification. Optical rectification is a second-order nonlinear process in which a dc or low-frequency polarization is developed when an intense laser beam propagates through a non-centro-symmetric crystal. It can be viewed as difference-frequency generation between the frequency components within the band of an optical excitation pulse. In contrast to photoconductivity, it is a nonresonant process and can therefore withstand higher excitation fluences and, importantly, generate THz emission with a bandwidth limited only by that of the optical excitation pulse.
产生太赫兹辐射的另一种方法是依靠非谐振非线性光学过程，例如光学校正。光学整流是一种二阶非线性过程，当强激光束穿过非中心对称晶体时，会产生直流或低频偏振。它可以看作是光激发脉冲频带内频率分量之间的差频产生。与光电导性相比，它是一个非谐振过程，因此可以承受更高的激发通量，重要的是，它产生的太赫兹发射的带宽仅受光激发脉冲的带宽限制。

In choosing appropriate nonlinear crystals for THz generation, several factors need to be considered: [for more details, see Reimann (2007)]
在为太赫兹生成选择合适的非线性晶体时，需要考虑几个因素：[有关更多详细信息，请参阅 Reimann (2007) ]

Based on the above requirements, standard choices for nonlinear crystals for optical rectification using a Ti:sapphire laser are ZnTe for the 0–3 THz range, GaP for 2–7 THz, and GaSe for 8–40 THz (Wu and Zhang, 1997; Kaindl et al., 1999; Huber et al., 2000). There have also been considerable efforts in the development of organic materials for optical rectification. Examples include 4-dimethylamino- $N$-methyl-4-stilbazolium-tosylate (DAST) (Katayama et al., 2010), $2N$-a-(methylbenzylamino)-5-nitropyridine (NBANP) crystals, and dye-doped polymers (Zheng et al., 2007). Organic materials often possess larger nonlinearities than inorganic materials, but may exhibit relatively low damage threshold or a lack of photostability.
基于上述要求，使用 Ti：sapphire 激光器进行光学校正的非线性晶体的标准选择是 0-3 THz 范围的 ZnTe、2-7 THz 的 GaP 和 8-40 THz 的 GaSe （ Wu and Zhang, 1997 ; Kaindl et al., 1999 ; Huber et al., 2000 ）。在开发用于光学校正的有机材料方面也付出了相当大的努力。例子包括 4-二甲氨基- $N$ -甲基-4-二唑-甲苯磺酸酯 （DAST） （ Katayama et al., 2010 ）、 $2N$ -a-（甲基苄基氨基）-5-硝基吡啶 （NBANP） 晶体和染料掺杂聚合物 （ Zheng et al., 2007 ）。有机材料通常比无机材料具有更大的非线性，但可能表现出相对较低的损伤阈值或缺乏光稳定性。

Plasmas have recently been recognized as another attractive nonlinear medium for THz generation. The lack of absorbing phonon modes in plasma permits the generation of broadband radiation without spectral gaps. Spectral components up to 75 THz have been observed and center frequencies up to 30 THz have been predicted for an excitation pulse of 50 fs duration (Kim et al., 2008). Plasmas have also been shown to generate intense THz radiation, with energies exceeding $>5\mu \mathrm{J}$ per pulse (Kim et al., 2008). In these experiments a focused fundamental pulse at 800 nm is mixed with its second harmonic at 400 nm in a gas (Cook and Hochstrasser, 2000). The fundamental beam ionizes the gas in its focus, which then acts as the nonlinear medium. Several groups have worked on understanding and optimizing this process. Researchers examined the dependence of the THz emission on the parameters, such as the phase and polarization of the optical excitation (Xie et al., 2006; Houard et al., 2008), the optical pulse duration (Bartel et al., 2005), the external dc bias (Thomson et al., 2007), and the type of gas (Chen, Yamaguchi et al., 2007; Lu et al., 2008). However, the underlying microscopic mechanism of the process is still a subject of debate (Thomson et al., 2007; Karpowicz et al., 2009).
等离子体最近被认为是另一种有吸引力的太赫兹产生非线性介质。等离子体中缺乏吸收声子模式，可以产生没有光谱间隙的宽带辐射。已观察到高达 75 THz 的频谱分量，并且预测了持续时间为 50 fs 的激发脉冲高达 30 THz 的中心频率 （ Kim et al., 2008 ）。等离子体还被证明会产生强烈的太赫兹辐射，其能量超过 $>5\mu \mathrm{J}$ 每个脉冲 （ Kim et al., 2008 ）。在这些实验中，800 nm 处的聚焦基波脉冲与气体 （ ） 中 400 nm 处的二次谐波混合 Cook and Hochstrasser, 2000 。基波束将其聚焦中的气体电离，然后充当非线性介质。几个小组已经致力于了解和优化这一过程。研究人员研究了太赫兹发射对参数的依赖性，例如光学激发的相位和偏振 （ Xie et al., 2006 ; Houard et al., 2008 ）、光脉冲持续时间 （ Bartel et al., 2005 ）、外部直流偏置 （ Thomson et al., 2007 ） 和气体类型 （ Chen, Yamaguchi et al., 2007 ; Lu et al., 2008 ）。然而，该过程的潜在微观机制仍然是一个争论的话题 （ Thomson et al., 2007 ; Karpowicz et al., 2009 ）。

In most studies THz radiation is used to probe the material response without perturbation, as expected for typical conditions of a peak electric-field amplitude up to a few $\mathrm{kV}/\mathrm{cm}$ and a pulse energy of $\sim {10}^{-10}\mathrm{J}$. THz pulses of high energy are thus not required. It is, however, of great interest to be able to study the response of charge carriers and low-energy excitations in the nonlinear regime, thus requiring THz pulses of high energy as a pump. The advent of amplified Ti:sapphire systems supplying near-infrared pulses in the mJ regime and novel THz generation schemes have yielded THz radiation with energies in the range of $\mu \mathrm{J}/\mathrm{pulse}$. Such a pulse energy is comparable to that of subpicosecond THz pulses generated by free-electron lasers (Carr et al., 2002).
在大多数研究中，太赫兹辐射用于探测无扰动的材料响应，正如在峰值电场振幅高达几 $\mathrm{kV}/\mathrm{cm}$ 和脉冲能量 的典型 $\sim {10}^{-10}\mathrm{J}$ 条件下所预期的那样。因此不需要高能量的 THz 脉冲。然而，能够研究非线性状态下载流子和低能量激发的响应具有极大的意义，因此需要高能量的太赫兹脉冲作为泵。在 mJ 范围内提供近红外脉冲的放大钛蓝宝石系统的出现和新颖的太赫兹产生方案产生了能量在 $\mu \mathrm{J}/\mathrm{pulse}$ .这种脉冲能量与自由电子激光器产生的亚皮秒 THz 脉冲 （ Carr et al., 2002 ） 相当。

An obvious approach to scale up the THz emission based on nonlinearity is to use higher fluences of the optical excitation pulse. This approach is often limited by two-photon absorption in the nonlinear material. The photoinduced charge carriers screen a substantial part of the generated THz radiation, thereby limiting the conversion efficiency. To circumvent this problem, one can either enlarge the pump beam to lower the excitation pulse fluence while maintaining the total energy or use wide band gap materials. For instance, it has been shown that THz pulses of $1.5\mu \mathrm{J}$ energy at a center frequency of 0.6 THz can be generated with an optical pulse of $48\mathrm{mJ}/\mathrm{pulse}$ in a ZnTe crystal of 75 mm size (Blanchard et al., 2007). THz pulses of energy up to $30\mu \mathrm{J}/\mathrm{pulse}$ at a center frequency of 0.6 THz were also generated using a 28 mJ optical pulse (Stepanov et al., 2008) from magnesium-doped lithium niobate ( $\mathrm{Mg}\mathrm{\text{:}}\mathrm{Li}{\mathrm{NbO}}_3$). This material has a high damage threshold and a large nonlinear susceptibility (Hebling, Yeh, Hoffman, Bartal, and Nelson, 2008; Hebling, Yeh, Hoffman, and Nelson, 2008), but significant phonon absorption above 1 THz (Bartal et al., 2007).
基于非线性放大太赫兹发射的一种明显方法是使用更高的光激发脉冲通量。这种方法通常受到非线性材料中双光子吸收的限制。光诱导电荷载流子屏蔽了大部分产生的太赫兹辐射，从而限制了转换效率。为了规避这个问题，可以扩大泵浦光束以降低激发脉冲通量，同时保持总能量，或者使用宽带隙材料。例如，已经表明，中心频率为 0.6 THz 的 $1.5\mu \mathrm{J}$ THz 能量脉冲可以用 75 mm 尺寸的 ZnTe 晶体 （ ） 中的光脉冲 $48\mathrm{mJ}/\mathrm{pulse}$ 产生 Blanchard et al., 2007 。使用来自镁掺杂铌酸锂 （ Stepanov et al., 2008 ） 的 28 mJ 光脉冲 （ ） 也产生了高达 $30\mu \mathrm{J}/\mathrm{pulse}$ 0.6 THz 中心频率的 THz 能量脉冲 $\mathrm{Mg}\mathrm{\text{:}}\mathrm{Li}{\mathrm{NbO}}_3$ 。这种材料具有较高的损伤阈值和较大的非线性磁化率 （ Hebling, Yeh, Hoffman, Bartal, and Nelson, 2008 ; Hebling, Yeh, Hoffman, and Nelson, 2008 ），但 1 THz （ ） 以上有显著的声子吸收 Bartal et al., 2007 。

The criterion to drive nonlinear processes with THz radiation is not, however, the pulse energy, but rather the pump intensity or its electric-field strength. Long-wavelength beams like the ones created in the aforementioned studies suffer from reduced focusability. Thus, it is difficult to reach peak amplitudes beyond $100\mathrm{kV}/\mathrm{cm}$ (Loffler et al., 2005). The situation becomes easier for higher-frequency beams. Pulses with a spectrum from 1 to 7 THz created in laser-induced plasma have been shown to reach peak amplitudes of $400\mathrm{kV}/\mathrm{cm}$, despite a pulse energy of just 30 nJ (Bartel et al., 2005).
然而，用太赫兹辐射驱动非线性过程的标准不是脉冲能量，而是泵浦强度或其电场强度。像上述研究中产生的长波长光束的聚焦能力降低。因此，很难达到超过 （ Loffler et al., 2005 ） 的 $100\mathrm{kV}/\mathrm{cm}$ 峰值振幅。对于高频光束，这种情况变得更容易。在激光诱导等离子体中产生的光谱范围为 1 至 7 THz 的脉冲已被证明可以达到 $400\mathrm{kV}/\mathrm{cm}$ 的峰值振幅，尽管脉冲能量仅为 30 nJ （ Bartel et al., 2005 ）。

With photoconductive switches, one can scale up the THz emission by increasing both the bias field and the optical excitation fluence. The screening effects by the photoexcited carriers can be significantly reduced by using a large-aperture structure with a gap size of up to a few mm. The excitation power and the bias voltage are scaled up to maintain the magnitude of the excitation fluence and bias field.
使用光电导开关，可以通过增加偏置场和光激发通量来放大太赫兹发射。通过使用间隙尺寸高达几毫米的大孔径结构，可以显著降低光激发载流子的屏蔽效应。激发功率和偏置电压按比例放大，以保持激励通量和偏置场的大小。

On the other hand, for applications such as imaging and steady-state spectroscopy, high average THz power is desirable. While the ultrafast laser-based techniques can generate THz radiation of an average power up to ${10}^{-4}\mathrm{W}$; 20 W has been reported for free-electron lasers (Carr et al., 2002). Recent developments in quantum-cascade lasers (QCL) and high-frequency electronics have demonstrated the possibility of providing cw THz radiation up to $\sim 100\mathrm{mW}$ (Williams, 2006, 2007). The development of quantum-cascade lasers has seen rapid progress since the first prototype was fabricated in 1994 (Faist et al., 1994). They can now offer reliable cw or pulsed mode operation in the midinfrared at room temperature (Williams, 2007; Liu et al., 2010). Scaling down of the emission to THz frequencies is however technically challenging (Scalari et al., 2009). The first QCL that emitted light at THz frequencies was only demonstrated in 2002 (Kohler et al., 2002). THz QCLs still require cryogenic working temperatures and commercially available models are scarce. We are currently not aware of any applications in spectroscopy. Some groups have, however, used THz spectroscopy to study lasing and charge carrier dynamics in QCLs. We briefly treat some of these studies in Sec. IV B 1.
另一方面，对于成像和稳态光谱等应用，高平均 THz 功率是可取的。虽然基于超快激光的技术可以产生平均功率高达 ${10}^{-4}\mathrm{W}$ ;据报道，自由电子激光器的功率为 20 W （ Carr et al., 2002 ）。量子级联激光器 （QCL） 和高频电子学的最新发展证明了提供高达 $\sim 100\mathrm{mW}$ （ Williams, 2006 ， 2007 ） 的连续太赫兹辐射的可能性。自 1994 年制造第一个原型以来，量子级联激光器的发展取得了快速进展 （ Faist et al., 1994 ）。它们现在可以在室温 （ Williams, 2007 ; Liu et al., 2010 ）。然而，将发射缩小到 THz 频率在技术上具有挑战性 （ Scalari et al., 2009 ）。第一个以 THz 频率发射光的 QCL 仅在 2002 年才得到展示 （ Kohler et al., 2002 ）。太赫兹 QCL 仍然需要低温，并且市售型号稀缺。我们目前不知道光谱学有任何应用。然而，一些小组使用太赫兹光谱来研究 QCL 中的激光和电荷载流子动力学。我们在 Sec 中简要介绍了其中一些研究。 IV B 1

The linear response of material can be described by its complex dielectric function $ϵ(\nu )$ as a function of the frequency $\nu$. The complex dielectric function is directly related to the complex refractive index $n(\nu )$ by $ϵ(\nu )=n^2(\nu )$, and to the optical conductivity $\sigma (\nu )$ by $\sigma (\nu )=-2\pi i\nu {ϵ}_0[ϵ(\nu )-1]$, where ${ϵ}_0$ is the vacuum permittivity. The goal of a THz-TDS experiment is usually to extract this material response. In this section, we discuss the general procedure for doing so.
材料的线性响应可以用其复介电函数 $ϵ(\nu )$ 来描述，它是频率 $\nu$ 的函数。复介电函数与复折射率 $n(\nu )$ 直接相关 ， $ϵ(\nu )=n^2(\nu )$ 与光导 $\sigma (\nu )$ 率直接相关 $\sigma (\nu )=-2\pi i\nu {ϵ}_0[ϵ(\nu )-1]$ ，其中 ${ϵ}_0$ 是真空介电常数。THz-TDS 实验的目标通常是提取这种材料响应。在本节中，我们将讨论执行此作的一般过程。

The material response can be extracted through the Fourier transforms of the measured electric-field waveforms for the sample $E(\nu )$ and the reference $E_{\mathrm{ref}}(\nu )$. The transmission $t(\nu )$ or reflection $r(\nu )$ of the electric field [ $=E(\nu )/E_{\mathrm{ref}}(\nu )$] can be related to the parameters of the sample. In the general case of a multilayered planar system of $N$ layers, a transfer matrix analysis can be applied (Furman, 1996). For simplicity, we assume the permeability of the materials to be 1. The characteristic matrix for normal incidence at $m$th layer of thickness $d_m$ and dielectric function ${ϵ}_m$ is given by
材料响应可以通过样品 $E(\nu )$ 和参考 $E_{\mathrm{ref}}(\nu )$ 的测量电场波形的傅里叶变换来提取。电场 [ $=E(\nu )/E_{\mathrm{ref}}(\nu )$ ] $r(\nu )$ 的透射 $t(\nu )$ 或反射可能与样品的参数有关。在多层平面 $N$ 层系统的一般情况下，可以应用传递矩阵分析 （ Furman, 1996 ）。为简单起见，我们假设材料的渗透率为 1。厚度 $d_m$ 和介电函数 ${ϵ}_m$ 法 $m$ 向入射的特征矩阵由下式给出

where ${\beta }_m=2\pi \nu d_m\sqrt{{ϵ}_m}/c$ is the phase delay associated with propagation inside the $m$th layer and $c$ is the speed of light in vacuum. The characteristic matrix of the entire multilayer system is given by the product of individual layer matrices $z=z_N{z}_{N-1}\cdots z_1$. The transmission and reflection coefficient of the system are determined by the matrix elements ( $i$, $j=1$, 2) of the characteristic matrix of the entire system $z$:
其中 ${\beta }_m=2\pi \nu d_m\sqrt{{ϵ}_m}/c$ 是与 $m$ th 层内传播相关的相位延迟， $c$ 是真空中的光速。整个多层系统的特征矩阵由各个层矩阵 的乘积给出。 $z=z_N{z}_{N-1}\cdots z_1$ 系统的透射系数和反射系数由整个系统的 $z$ 特征矩阵的矩阵元素 （ $i$ ， $j=1$ ， 2） 决定：

In case of a slab of homogeneous material of thickness $d$ and dielectric function ${ϵ}_2(\nu )$ with media 1 and 3 on its two sides, a common experimental geometry, one retrieves the following more familiar expression for the transmission coefficient:
对于厚度 $d$ 和介电函数 ${ϵ}_2(\nu )$ 的均质材料板，其两侧有介质 1 和 3，这是一个常见的实验几何，可以检索以下更熟悉的传输系数表达式：

Here $t_{ij}=2\sqrt{{ϵ}_i}/(\sqrt{{ϵ}_i}+\sqrt{{ϵ}_j})$ and $r_{ij}=(\sqrt{{ϵ}_i}-\sqrt{{ϵ}_j})/(\sqrt{{ϵ}_i}+\sqrt{{ϵ}_j})$ are the Fresnel transmission and reflection coefficients for normal incidence to an interface from medium $i$ to $j$ and $E_{\mathrm{inc}}(\nu )$ is the incident field. The task is then to solve Eq. (2) using experimentally determined $E(\nu )$ and $E_{\mathrm{inc}}(\nu )$ and knowledge of the sample thickness to obtain ${ϵ}_2(\nu )$. Generally, this will involve numerical calculations: Eq. (2) can be solved either using an iterative approach or piecewise in frequency to minimize phase and amplitude errors (Duvillaret et al., 1996).
这里 $t_{ij}=2\sqrt{{ϵ}_i}/(\sqrt{{ϵ}_i}+\sqrt{{ϵ}_j})$ 和 $r_{ij}=(\sqrt{{ϵ}_i}-\sqrt{{ϵ}_j})/(\sqrt{{ϵ}_i}+\sqrt{{ϵ}_j})$ 是从介质 $i$ 到 $j$ 和 的界面法向入射的菲涅耳透射系数和反射系数， $E_{\mathrm{inc}}(\nu )$ 是入射场。然后，任务是使用实验确定 $E(\nu )$ 的方程 和 $E_{\mathrm{inc}}(\nu )$ 样品厚度的知识来求解 (2) 方程，得到 ${ϵ}_2(\nu )$ 。通常，这将涉及数值计算： (2) 方程可以使用迭代方法求解，也可以按频率分段求解，以最小化相位和幅度误差 （ Duvillaret et al., 1996 ）。

In the limit of a thin-film sample of thickness $d\ll {\lambda }_{\mathrm{THz}}$ (where ${\lambda }_{\mathrm{THz}}$ is the wavelength of THz radiation), an analytical solution for the film dielectric function $ϵ(\nu )$ is possible. Here we take the THz transmission through the film on a substrate as $E(\nu )$ and the THz radiation transmitted through the bare substrate as $E_{\mathrm{ref}}(\nu )$. One can carry out a Taylor expansion of the exponential terms in Eq. (2) to obtain
在厚度 $d\ll {\lambda }_{\mathrm{THz}}$ 为薄膜样品的极限（其中 ${\lambda }_{\mathrm{THz}}$ 是太赫兹辐射的波长）中，可以得到薄膜介电函数 $ϵ(\nu )$ 的解析解。这里我们取通过衬底上薄膜的太赫兹透射率为 $E(\nu )$ ，通过裸衬底透射的太赫兹辐射为 $E_{\mathrm{ref}}(\nu )$ 。可以对方程中的指数项进行泰勒展开， (2) 得到

where ${ϵ}_{\mathrm{sub}}$ is the substrate dielectric function. Equation (3) can be readily solved analytically to obtain $ϵ(\nu )$.
其中 ${ϵ}_{\mathrm{sub}}$ 是衬底介电函数。方程 (3) 可以很容易地通过解析求解得到 $ϵ(\nu )$ 。

We note that in deriving the characteristic matrix described in Eq. (1), we assume that the THz beam can be represented as a superposition of plane waves of frequency $\nu$ propagating along a fixed direction (normal to the sample in this case) and the beam shape is not altered by the sample. These assumptions are usually valid. In case of significant diffraction, the spatial profile of the THz pulse needs to be measured and a spatial Fourier transform needs to be used in addition to the temporal Fourier transform above.
我们注意到，在推导出方程 (1) 中描述的特征矩阵时，我们假设太赫兹光束可以表示为沿固定方向（在这种情况下垂直于样品）传播的频率 $\nu$ 平面波的叠加，并且光束形状不会因样品而改变。这些假设通常是有效的。在显著衍射的情况下，需要测量太赫兹脉冲的空间分布，并且除了上述时间傅里叶变换外，还需要使用空间傅里叶变换。

The measurement technique and the analysis method described here are very general. They can be applied to material systems ranging from dielectrics to semiconductors to metals, in gas, liquid, and solid forms. They can be applied to a homogeneous sample or a composite. In the next Sec. II D 2 we describe how to extract the dielectric properties of constituents of a composite based on the effective medium theory (EMT). The THz TDS technique and the analysis method described above, however, have restrictions. We list some of the major ones below.
这里描述的测量技术和分析方法非常通用。它们可以应用于从电介质到半导体再到金属的材料系统，以气体、液体和固体形式存在。它们可以应用于均质样品或复合材料。在下一节中， II D 2 我们将介绍如何基于有效介质理论 （EMT） 提取复合材料成分的介电特性。然而，太赫兹 TDS 技术和上述分析方法存在限制。我们在下面列出了一些主要的。

In Sec. II D 1 we considered homogeneous media. The analysis is also applicable to inhomogenous materials with inhomogeneities on a length scale much smaller than the wavelength of the THz radiation. These systems include nanocomposites and nanoporous materials. An effective dielectric function $ϵ(\nu )$ can be extracted from the THz measurements. In order to obtain the properties of the constituents, an effective medium theory is usually applied.
在第 Sec. II D 1 中，我们考虑了同质介质。该分析也适用于长度尺度上不均匀性远小于太赫兹辐射波长的不均匀材料。这些系统包括纳米复合材料和纳米多孔材料。可以从太赫兹测量中提取有效的介电函数 $ϵ(\nu )$ 。为了获得成分的性质，通常应用有效的介质理论。

There are many EMTs in Choy (1999). Maxwell-Garnett (MG) (Garnett, 1904) and Bruggemann (1935) are the two most widely used ones. For particles made of a material of dielectric function ${ϵ}_p$ imbedded in a medium of dielectric function ${ϵ}_m$, the effective dielectric function of the composite $ϵ$ is the solution of the following self-consistent equations:
中 Choy (1999) 有许多 EMT。Maxwell-Garnett （MG） （ Garnett, 1904 ） 是 Bruggemann (1935) 两个使用最广泛的。对于由嵌入介电介质 ${ϵ}_m$ 中的介电功能 ${ϵ}_p$ 材料制成的粒子，复合材料 $ϵ$ 的有效介电功能是以下自洽方程的解：

Here $f$ is the filling factor in volume and $\kappa$ is the geometric factor related to depolarization ( $\kappa =2$ for spheres) (Garnett, 1904). In contrast to the Bruggemann approximation, in Maxwell-Garnett EMT the role of the particles and their surroundings is not symmetric and is applicable only for dilute composites. The general consensus is that Bruggemann formula is more appropriate for composites consisting of two or more components at high filling factors. However, it is well known that the Bruggemann effective medium theory does not reproduce some important material properties, such as plasmon resonances in percolated conductors (Riikonen et al., 2005).
这是 $f$ 体积中的填充因子， $\kappa$ 是与去极化相关的几何因子（ $\kappa =2$ 对于球体）（ Garnett, 1904 ）。与 Bruggemann 近似相反，在 Maxwell-Garnett EMT 中，颗粒及其周围环境的作用是不对称的，仅适用于稀复合材料。普遍的共识是 Bruggemann 公式更适合于由两个或多个组分组成的高填充因子复合材料。然而，众所周知，Bruggemann 有效介质理论并没有再现一些重要的材料特性，例如渗流导体中的等离激元共振 （ Riikonen et al., 2005 ）。

Now we turn our discussion to the analysis of typical optical pump and THz probe measurements. In these experiments the electric-field waveform $E(t)$ of THz pulses transmitted through an unexcited sample and the pump-induced change in the THz waveform $\Delta E(t)$ are recorded. The latter is done by using a lock-in technique with modulation of the optical pump beam for each fixed delay time $\tau$ between the optical pump and THz probe pulse. In such experiments, we are interested in retrieving the photoinduced change $\Delta ϵ(\nu )$ in the complex dielectric function $ϵ(\nu )$ as a function of the frequency $\nu$. This can be achieved using the Fourier transform of the two waveforms $E(\nu )$ and $\Delta E(\nu )$ following a similar procedure as in Sec. II D 1, where
现在，我们将讨论转向典型光泵和太赫兹探头测量的分析。在这些实验中，记录了通过未激发样品传输的太赫兹脉冲 $E(t)$ 的电场波形和太赫兹波形 $\Delta E(t)$ 中泵浦引起的变化。后者是通过使用锁相技术完成的，该技术对光泵浦和 THz 探头脉冲之间的每个固定延迟时间 $\tau$ 进行光泵光束调制。在这样的实验中，我们感兴趣的是检索复电介质函数 $ϵ(\nu )$ 中的光诱导变化 $\Delta ϵ(\nu )$ 作为频率 $\nu$ 的函数。这可以使用两个波形的傅里叶变换来实现， $E(\nu )$ 并 $\Delta E(\nu )$ 遵循与秒类似的过程， II D 1 其中

Note that, for now, we consider the simple case of a quasistatic limit, in which the photoinduced change in the material properties occurs on a much longer time scale than the THz pulse duration. Such a quasistatic condition can be satisfied in a material with a long carrier lifetime and probed several picoseconds after pump excitation when the fast hot-carrier effects have abated.
请注意，现在我们考虑准静态极限的简单情况，其中材料属性的光诱导变化发生在比太赫兹脉冲持续时间长得多的时间尺度上。这种准静态条件可以在具有长载流子寿命的材料中得到满足，并在泵激发后快速热载流子效应减弱时探测数皮秒。

We first consider the analysis for a homogeneously excited slab of material in vacuum of thickness $d$ [Fig. 2(a)] In the limit of a weak perturbation to the THz response, a linear relation between the pump-induced response and the ratio of the Fourier transform of the waveforms can be found, ignoring multiple reflections (Knoesel et al., 2004)
我们首先考虑在厚度 $d$ 为 [Fig. 2(a) ] 在对 THz 响应的弱扰动极限下，可以找到泵浦诱导响应与波形傅里叶变换之比之间的线性关系，忽略多重反射 （ Knoesel et al., 2004 ）

This expression includes propagation of the THz probe through the sample, as well as interfacial reflection losses at the boundary of the sample.
该表达式包括太赫兹探针在样品中的传播，以及样品边界处的界面反射损耗。

FIG. 2  图 2

A useful approximation can also be applied when only a very thin region near the surface of a sample is excited [Fig. 2(b)]. This situation occurs when the pump light is strongly absorbed by a material, such as for above-band-gap photoexcitation of a semiconductor. It is convenient to approximate the excited region by a homogeneous region of width $l_0$, the penetration depth of the optical excitation in the sample. If the excitation density is sufficiently small, the phase accumulated by a THz wave propagating through the excited region will be small (i.e., $\sqrt{(ϵ+\Delta ϵ)}2\pi \nu l_0/c\ll 1$). The exponential terms in the transmission function of the photoexcited region given by Eq. (6) can again be approximated by Taylor expansions. We thus obtain the following expression for the average change in the dielectric function inside the photoexcited region:
当仅激发样品表面附近的非常薄的区域时，也可以应用有用的近似值 [图 D]。 2(b) ]。当泵浦光被材料强烈吸收时，例如半导体的带上光激发，就会出现这种情况。通过宽度 $l_0$ 的均匀区域 ，样品中光学激发的穿透深度来近似激发区是很方便的。如果激发密度足够小，则通过激发区传播的太赫兹波累积的相位将很小（即）。 $\sqrt{(ϵ+\Delta ϵ)}2\pi \nu l_0/c\ll 1$ 由方程给出的光激发区传输函数中的指数项 (6) 可以再次用泰勒展开近似。因此，我们获得了光激发区内介电函数的平均变化的以下表达式：

Note that Eq. (7) follows directly from Eq. (6) under the neglect of the reflection contribution, which is not altered by photoexcitation for this case. Equation (7) applies to a broad range of THz pump-probe experiments where the condition of $\sqrt{(ϵ+\Delta ϵ)}2\pi \nu l_0/c\ll 1$ is fulfilled [see, e.g., (Hendry, Koeberg, O’Regan, and Bonn (2006) and Laarhoven et al. (2008)].
请注意，方程直接 (7) 来自忽略反射贡献 (6) 的方程，在这种情况下，光激发不会改变反射贡献。该方程 (7) 适用于满足 条件 $\sqrt{(ϵ+\Delta ϵ)}2\pi \nu l_0/c\ll 1$ 的广泛太赫兹泵浦探针实验 [参见，例如 （ Hendry, Koeberg, O’Regan, and Bonn (2006) 和 Laarhoven et al. (2008) ]。

The analysis, however, is complicated when the penetration depth $l_0$ of the optical excitation in the material of interest is comparable to the wavelength of THz radiation. A numerical analysis is then required in which we divide the sample into many thin, homogeneous slabs, with different excitation densities. Similarly problematic is the effect of inhomogeneous excitation along the direction perpendicular to the propagation direction of the THz probe: finite difference time domain simulations are required (Kindt and Schmuttenmaer, 1999).
然而，当目标材料中光学激发的穿透深度 $l_0$ 与太赫兹辐射的波长相当时，分析就很复杂了。然后需要进行数值分析，将样品分成许多具有不同激发密度的薄而均匀的板片。同样存在问题，即沿垂直于太赫兹探头传播方向的方向的非均匀激励效应：需要有限差分时域仿真 （ Kindt and Schmuttenmaer, 1999 ）。

The discussion above applies to quasistatic changes in the material response. When the change in the material response occurs on a time scale comparable to or shorter than that of the THz pulse, one can no longer use the method described above for data analysis. This regime is typically encountered when probing immediately after optical excitation or in systems with very fast carrier recombination times. In these cases a complete two-dimensional scan of the pump-induced THz electric-field waveforms $\Delta E(t,\tau )$ at each optical pump-THz delay time $\tau$ must be considered within an overall model of the material response, since the time-dependent frequency-domain material response function discussed above has no general physical meaning. In the limit of weak material response to the pump beam, the measurement can be considered as a particular form of four-wave mixing. More details can be found in Nemec et al. (2002).
上述讨论适用于材料响应中的准静态变化。当材料响应的变化发生在与太赫兹脉冲相当或更短的时间尺度上时，就不能再使用上述方法进行数据分析。这种状态通常在光学激发后立即进行探测时或在载流子复合时间非常快的系统中遇到。在这些情况下，必须在材料响应的整体模型中考虑每个光泵浦-THz 延迟时间 $\Delta E(t,\tau )$ $\tau$ 的泵浦感应太赫兹电场波形的完整二维扫描，因为上面讨论的瞬态频域材料响应函数没有一般的物理意义。在材料对泵波束的响应较弱的极限下，可以将测量视为四波混合的一种特殊形式。有关更多详细信息，请参阅 Nemec et al. (2002) 。

In bulk solids, the THz response of free carriers, whether from dopants or photoexcitation, can often be described with by a Drude response function. The result can be derived from several descriptions of charge motion within a solid, including the semiclassical Boltzmann transport equation and the Kubo-Greenwood analysis (Kubo, 1957; Hickey et al., 2001). The model, as discussed below, involves just a single relaxation parameter. Despite its simplicity, it has been verified for charge transport over a spectral range from MHz to the optical frequencies (Dressel and Scheffler, 2006) in various materials.
在散装固体中，自由载流子的太赫兹响应，无论是来自掺杂剂还是光激发，通常都可以用 Drude 响应函数来描述。结果可以从固体内电荷运动的几种描述中得出，包括半经典的玻尔兹曼输运方程和 Kubo-Greenwood 分析 （ Kubo, 1957 ; Hickey et al., 2001 ）。如下所述，该模型只涉及一个松弛参数。尽管它很简单，但它已经在各种材料中验证了从 MHz 到光频率 （ Dressel and Scheffler, 2006 ） 的光谱范围内的电荷传输。

Heuristically, the Drude response can be derived from a simple semiclassical model of charge transport. In this picture, the charges are accelerated under the external electric field $E(t)$ according to an equation of motion of
从启发式角度讲，Drude 响应可以从一个简单的电荷传输半经典模型推导出来。在这张图片中，电荷在外部电场 $E(t)$ 下根据

where $r$ is the ensemble average of the displacement of the charge carriers, $\gamma$ denotes the damping rate or scattering rate, $e$ is the electronic charge, and $m$ is the effective mass of the charge carrier. For a given carrier density $N$ the dielectric response function ${ϵ}_D$ to an alternating external electric field at angular frequency $\omega (=2\pi \nu )$ can be solved from Eq. (8). This yields the well-known Drude response:
其中 $r$ 是电荷载流子位移的集合平均值， $\gamma$ 表示阻尼率或散射率， $e$ 是电子电荷， $m$ 是电荷载流子的有效质量。对于给定的载流子密度 $N$ ，在角频率 $\omega (=2\pi \nu )$ 下对交变外部电场的介电响应函数 ${ϵ}_D$ 可以由方程求解。 (8) 这产生了众所周知的 Drude 响应：

where ${ϵ}_{\infty }$ is the background dielectric constant.
其中 ${ϵ}_{\infty }$ 是背景介电常数。

The Drude response is characterized by a Lorentzian resonance centered at zero frequency with a linewidth $\gamma$, which is the charge carrier scattering rate; the inverse of the scattering time $\tau$. The amplitude of the dielectric response is determined by the plasma frequency ${\omega }_p$, defined as
Drude 响应的特征是以零频率为中心的洛伦兹谐振，线宽 $\gamma$ 为 ，即电荷载流子散射率;散射时间 $\tau$ 的倒数 。介电响应的振幅由等离子体频率 ${\omega }_p$ 决定，定义为

with $N$ being the charge density and ${ϵ}_0$ the permittivity of free space. The real part of ${ϵ}_D$ crosses the zero point at the screened plasma frequency ${\omega }_{sp}={\omega }_p/\sqrt{{ϵ}_{\infty }}$. By plotting the inverse of the dielectric response $1/{ϵ}_D$, also known as the dielectric loss function, we find a peak centered at ${\omega }_{sp}$.
是 $N$ 电荷密度和 ${ϵ}_0$ 自由空间的介电常数。的 ${ϵ}_D$ 实部在筛选的等离子体频率 ${\omega }_{sp}={\omega }_p/\sqrt{{ϵ}_{\infty }}$ . 上穿过零点。通过绘制介电响应 $1/{ϵ}_D$ （也称为介电损耗函数） 的逆函数，我们找到一个以 为中心的 ${\omega }_{sp}$ 峰值。

We can also write the Drude response in terms of the complex conductivity of the material
我们还可以根据材料的复杂电导率来写 Drude 响应

Here the parameter ${\sigma }_{\mathrm{dc}}=N{e}^2\tau /m$ gives the value of the dc conductivity of the material within the Drude model.
这里的参数 ${\sigma }_{\mathrm{dc}}=N{e}^2\tau /m$ 给出了 Drude 模型中材料的直流电导率值。

Depending on the relative value of the scattering rate and the THz frequency window, the observed Drude response will vary. Figure 3 shows the predicted dielectric function and complex conductivity for the case of a scattering rate within the THz frequency window and for a far higher scattering rate. When the scattering rate is within the THz probe window, both the real and imaginary parts of the response are finite and dispersive. In this case, we can determine both of the parameters in the Drude model, i.e., the plasma frequency and the scattering rate. If the carrier mass is known, this analysis then provides the carrier concentration $N$ from the plasma frequency and the carrier mobility $\mu =e/m\gamma$ from the scattering rate. When the scattering rate far exceeds the available THz frequency range, then the analysis can only yield the dc conductivity ${\sigma }_{\mathrm{dc}}$. In this limit, the real part of the conductivity is finite, while the imaginary part is negligible. This corresponds to a purely imaginary dielectric function, with a trivial $1/\omega$ frequency dependence. For this case, one consequently cannot independently determine the carrier concentration and scattering time. In many experiments involving photogenerated carriers, the excitation conditions provide information on the carrier concentration. The THz data within the Drude model are then seen to provide the carrier mobility (or scattering rate, assuming existing knowledge of the carrier effective mass).
根据散射速率和 THz 频率窗口的相对值，观察到的 Drude 响应会有所不同。图 3 显示了在 THz 频率窗口内散射率和更高散射率的情况下预测的介电函数和复电导率。当散射率在太赫兹探针窗口内时，响应的实部和虚部都是有限且色散的。在这种情况下，我们可以确定 Drude 模型中的两个参数，即等离子体频率和散射速率。如果载流子质量已知，则此分析会提供等离子体频率的载流子浓度 $N$ 和散射速率的载流子迁移率 $\mu =e/m\gamma$ 。当散射率远超过可用的太赫兹频率范围时，分析只能产生直流电导率 ${\sigma }_{\mathrm{dc}}$ 。在此极限下，电导率的实部是有限的，而虚部可以忽略不计。这对应于一个纯虚的介电函数，具有微不足道 $1/\omega$ 的频率依赖性。因此，对于这种情况，我们无法独立确定载流子浓度和散射时间。在许多涉及光生载流子的实验中，激发条件提供了有关载流子浓度的信息。然后，可以看到 Drude 模型中的太赫兹数据提供了载流子迁移率（或散射率，假设对载流子有效质量的现有了解）。

FIG. 3  图 3

Charge carriers in many semiconductors exhibit scattering times of a few hundred femtoseconds, corresponding to linewidths of the Drude peak of around 1 THz. Such a response is therefore readily characterized by THz pulses generated with either photoconductive antennas or ZnTe crystals, which allow one to determine the real and imaginary components of the dielectric response over a broad spectral window exceeding 1 THz. Hence, when the Drude model is applicable, THz spectroscopy can be used to simultaneously characterize the two key parameters associated with electrical conductivity, the plasma frequency and carrier scattering rate. Values for the carrier mobility $\mu$ and the charge concentration $N$ then follow immediately if the effective mass of the carriers is known.
许多半导体中的电荷载流子表现出几百飞秒的散射时间，对应于大约 1 THz 的 Drude 峰的线宽。因此，这种响应很容易通过用光电导天线或 ZnTe 晶体产生的太赫兹脉冲来表征，这使得人们可以在超过 1 THz 的宽光谱窗口内确定介电响应的实部和虚部。因此，当 Drude 模型适用时，太赫兹光谱可用于同时表征与电导率相关的两个关键参数， 等离子体频率和载流子散射速率。如果载流子的有效质量已知，则载流子迁移率 $\mu$ 和电荷浓度 $N$ 的值会立即随之而来。

In addition to the free-carrier response described above, a second distinct type of THz response frequently observed is that with a resonance at a finite frequency. The THz spectral regime covers many fundamental excitations including rotations, vibrations, and low-lying electronic transitions in molecules and collective modes in condensed matter such as phonons, plasma, magnons, and energy gaps associated with superconductivity (Averitt and Taylor, 2002; Kaindl et al., 2006) as well as intraexcitonic transitions for excitons (Kira et al., 2004; Koch et al., 2006). A response with a resonance at finite frequencies, as will be shown in the remainder of this review, also occurs for a number of other situations, including the effective medium-type response (Garnett, 1904; Stroud, 1975; Choy, 1999), the plasmon-type response (Nienhuys and Sundstrom, 2005a), the Drude-Smith response (Smith, 2001), and the response associated with “dispersive” transport (Dyre and Schroder, 2000).
除了上述的空载流子响应外，经常观察到的第二种不同类型的太赫兹响应是在有限频率下发生谐振的响应。太赫兹光谱范围涵盖了许多基本激发，包括分子的旋转、振动和分子和凝聚态中的低位电子跃迁，以及凝聚态物质（如声子、等离子体、磁振子和与超导相关的能隙）中的低位电子跃迁 （ Averitt and Taylor, 2002 ; Kaindl et al., 2006 ） 以及激子的激子内跃迁 （ Kira et al., 2004 ; Koch et al., 2006 ）。如本文的其余部分所示，在有限频率下产生共振的响应也发生在许多其他情况下，包括有效的中等类型响应 （ Garnett, 1904 ; Stroud, 1975 ; Choy, 1999 ）、等离激元型响应 （ Nienhuys and Sundstrom, 2005a ）、Drude-Smith 响应 （ Smith, 2001 ） 和与“色散”传输相关的响应 （ Dyre and Schroder, 2000 ）。

Here we present the simplest description of response at a finite frequency, the Lorentz oscillator (given by the scattering rate). This is a direct generalization of the Drude model including a response a a finite frequency and follows from the generalization of Eq. (8) to include a restoring force. The dielectric function thus can be described by
在这里，我们提出了有限频率下响应的最简单描述，即洛伦兹振荡器（由散射率给出）。这是 Drude 模型的直接推广，包括有限频率的响应，并遵循方程的推广 (8) ，包括恢复力。因此，介电函数可以通过以下方式描述

where $A$ is the amplitude, ${\omega }_0$ denotes the resonance frequency, and $\gamma$ defines the width of the resonance. Figure 4 shows the corresponding response as measured over a finite spectral range for different parameters. Applications of this treatment will be discussed below.
其中 $A$ 是振幅， ${\omega }_0$ 表示谐振频率，并 $\gamma$ 定义谐振的宽度。图 4 显示了在有限光谱范围内针对不同参数测得的相应响应。这种处理的应用将在下面讨论。

FIG. 4  图 4

THz-TDS allows one to measure both the amplitude and phase (or, equivalently, both the real and imaginary parts) of the material response over a wide range of frequencies. Frequently, the mobility of electrons exceeds that of the holes, and the data can be described by the Drude model with one carrier type. Hence the two key conduction parameters, the plasma frequency and scattering rate, (Beard et al., 2000; Hendry, Wang, Shan et al., 2004; Hendry, Koeberg, Pijpers, and Bonn, 2007; ), can be determined by simultaneously fitting both the real and imaginary parts of the frequency dependence of the conductivity.
THz-TDS 允许在很宽的频率范围内测量材料响应的振幅和相位（或者等效地测量实部和虚部）。电子的迁移率通常超过空穴的迁移率，数据可以用一种载流子类型的 Drude 模型来描述。因此，两个关键的传导参数，等离子体频率和散射率 （ Beard et al., 2000 ; Hendry, Wang, Shan et al., 2004 ; Hendry, Koeberg, Pijpers, and Bonn, 2007 ;），可以通过同时拟合电导率的频率依赖性的实部和虚部来确定。

In practice, however, the finite range of probe frequencies, determined by the bandwidth of the spectrometer, can set limitations. For example, for a Drude model conductor, if the charge scattering rate lies well outside the probe frequency range, the phase of the conductivity is approximately zero across the spectrum, and one can obtain reasonable fits to experimental data with a wide range of Drude parameters (Hendry, Wang, Shan et al., 2004). This limitation can be overcome by determining, in addition, the density of absorbed photons in the sample. This allows a method of independently determining the plasma frequency (assuming that the carrier mass is known), so that the THz spectrum can be fitted by the Drude model with only one adjustable parameter: the scattering rate. A similar approach is useful for samples with more than one photoexcitation species contributing to the THz spectrum, helping to separate carrier mobility from density.
然而，在实践中，由光谱仪带宽决定的有限探测频率范围可能会产生限制。例如，对于 Drude 模型导体，如果电荷散射率远超出探针频率范围，则电导率的相位在整个光谱中大约为零，并且可以使用广泛的 Drude 参数 （ Hendry, Wang, Shan et al., 2004 ） 获得与实验数据的合理拟合。此外，还可以通过确定样品中吸收光子的密度来克服这一限制。这允许一种独立确定等离子体频率的方法（假设载流子质量已知），因此 Drude 模型可以使用一个可调参数来拟合太赫兹光谱：散射率。类似的方法对于具有多个光激发物质对太赫兹光谱有贡献的样品很有用，有助于将载流子迁移率与密度分开。

This ability of THz spectroscopy to independently determine both scattering rates and plasma frequencies has proven particularly useful in recent work that assessed the efficiency of impact ionization processes in the low-band gap semiconductors PbS and PbSe (Pijpers et al., 2009). Impact ionization refers to a mechanism in which a photon possessing an energy of several times the band gap is able to convert its excess energy into the generation of more than one electron-hole pair. The terahertz response was measured following optical excitation at photon energies ranging from the infrared (close to band gap) to the ultraviolet. By verifying the uniformity of the scattering rate for all pump wavelengths and fluences, one is assured that the plasma frequency, and hence the signal amplitude, is directly proportional to the charge carrier density. The number density of absorbed excitation photons can be determined separately. Combining this quantity with the photogenerated charge carrier density inferred from terahertz measurements, one can directly determine the efficiency of impact ionization, measured in electron-hole pairs per photon. The results for bulk PbS ( $E_{\mathrm{gap}}=0.42\mathrm{eV}$) are shown in Fig. 6 as a function of the photon energy of the excitation. It is apparent from this figure that the impact ionization efficiency is rather low (it takes a photon energy of $\sim 7E_{\mathrm{gap}}$ to produce two, rather than one, electron-hole pairs). However, compared to the efficiency of this process in quantum dots (QDs), the impact ionization efficiency is larger in bulk materials than in QDs for a given photon energy. This is in contrast to previous expectations that quantum confinement would lead to an enhancement of impact ionization.
太赫兹光谱法独立确定散射速率和等离子体频率的这种能力在最近的工作中被证明特别有用，这些工作评估了低带隙半导体 PbS 和 PbSe （ Pijpers et al., 2009 ） 中冲击电离过程的效率。撞击电离是指一种机制，其中具有数倍带隙能量的光子能够将其多余的能量转化为产生多个电子-空穴对。太赫兹响应是在光子能量范围从红外（接近带隙）到紫外线的光激发下测量的。通过验证所有泵浦波长和磁通量的散射速率的均匀性，可以确保等离子体频率以及信号振幅与电荷载流子密度成正比。吸收激发光子的数密度可以单独确定。将这个量与从太赫兹测量中推断出的光生电荷载流子密度相结合，可以直接确定撞击电离的效率，以每个光子的电子-空穴对数为单位。块体 PbS （ $E_{\mathrm{gap}}=0.42\mathrm{eV}$ ） 的结果如图 2 所示。 6 作为激发光子能量的函数。从该图中可以明显看出，冲击电离效率相当低（产生两个而不是一个电子-空穴对需要光子能量 $\sim 7E_{\mathrm{gap}}$ ）。然而，与量子点 （QD） 中该过程的效率相比，对于给定的光子能量，块状材料中的冲击电离效率大于 QD。这与之前的预期形成鲜明对比，即量子限制会导致冲击电离的增强。

FIG. 6  图 6

It is interesting to compare the THz results to device measurements, shown as the solid sloping line in Fig. 6 (Smith and Dutton, 1958). Contrary to the THz measurements, device measurements to determine impact ionization efficiencies rely on photocurrent measurements (Brinza et al., 2005), which require charges to move over large distances on long time scales. As a result, recombination losses such as Auger recombination and trapping at surface defects can introduce uncertainties in the assessment of inherent carrier multiplication efficiencies in devices. In THz studies, the carrier density is determined picoseconds after photoexcitation, thus excluding the influence of all but the fastest trapping and recombination processes.
将太赫兹结果与器件测量值进行比较是很有趣的，如图 1 中的实线斜线所示。 6 （ Smith and Dutton, 1958 ）。与太赫兹测量相反，用于确定冲击电离效率的器件测量依赖于光电流测量 （ Brinza et al., 2005 ），这需要电荷在长时间尺度上长距离移动。因此，俄歇复合和表面缺陷捕获等复合损失会给器件中固有载流子倍增效率的评估带来不确定性。在太赫兹研究中，载流子密度是在光激发后皮秒确定的，因此排除了除最快的捕获和复合过程之外的所有影响。

Although the Drude model has been successfully applied in many instances where the frequency-dependent conduction was determined, there are several examples in which a description of the electron response requires more refined models.
尽管 Drude 模型已成功应用于许多确定频率相关传导的情况，但有几个例子表明，电子响应的描述需要更精细的模型。

For instance, the conductivity of $n$-type and $p$-type doped silicon samples at low charge densities departs from the Drude form in the THz spectral range. The data can, however, be fit by the Cole-Davidson model (Jeon and Grischkowsky, 1997; Jeon and Grischkowsky, 1998):
例如，在低电荷密度下， $n$ 型和 $p$ 型掺杂硅样品的电导率在太赫兹光谱范围内偏离了 Drude 形式。但是，数据可以通过 Cole-Davidson 模型 （ Jeon and Grischkowsky, 1997 ; Jeon and Grischkowsky, 1998 ）：

where the parameter $\beta \ne 1$ describes the deviation from the Drude behavior and can be considered a measure of the departure from a carrier response characterized by a single scattering time (see Fig. 7). For low doping densities, the data could be described with $\beta \approx 0.8$. Only at charge densities above ${10}^{17}{\mathrm{cm}}^{-3}$, a fit with $\beta =1$ was obtained, hence recovering the Drude model.
其中，该参数 $\beta \ne 1$ 描述了与 Drude 行为的偏差，可以被视为与以单个散射时间为特征的载流子响应的偏差的量度（见图 D）。 7 ）。对于低掺杂密度，数据可以用 $\beta \approx 0.8$ 来描述。仅在 电荷密度高于 ${10}^{17}{\mathrm{cm}}^{-3}$ 时，才获得了 拟合 $\beta =1$ ，因此恢复了 Drude 模型。

FIG. 7.  图 7.

A second example where the Drude model should be handled with care is for very high charge densities. In these regimes scattering events between electrons and holes can no longer be neglected, and the scattering time and hence the mobility decrease with increasing charge density. This effect is most pronounced in materials with a small (real) dielectric constant ${ϵ}^{\prime }$ since charge carriers are less effectively screened from each other. For instance, in silicon (small ${ϵ}^{\prime }$) a hundredfold higher charge density from ${10}^{21}$ to ${10}^{23}{\mathrm{m}}^{-3}$ leads to a reduction of the scattering time by roughly the same amount, while in titanium dioxide (large ${ϵ}^{\prime }$) the same increase has barely any effect on $\tau$ (Hendry, Koeberg, Pijpers, and Bonn, 2007). We note that, even when electron-hole interactions become important, the dispersion of the conductivity may still appear Drude-like; the Drude parameters will be modified, however.
应该小心处理 Drude 模型的第二个例子是非常高的电荷密度。在这些状态下，电子和空穴之间的散射事件不能再被忽视，散射时间和迁移率会随着电荷密度的增加而降低。这种效应在介电常数 ${ϵ}^{\prime }$ 较小的（实际）材料中最为明显，因为电荷载流子之间的屏蔽效果较差。例如，在硅（小 ${ϵ}^{\prime }$ ）中，从 ${10}^{21}$ to ${10}^{23}{\mathrm{m}}^{-3}$ 的电荷密度提高 100 倍导致散射时间减少大致相同的量，而在二氧化钛（大 ${ϵ}^{\prime }$ ）中，相同的增加对 $\tau$ （ ） 几乎没有任何影响 Hendry, Koeberg, Pijpers, and Bonn, 2007 。我们注意到，即使电子-空穴相互作用变得重要，电导率的色散仍然可能看起来类似 Drude;但是，Drude 参数将被修改。

Another limitation in the applicability of the Drude model can be found in the limit of very strong electron-phonon interactions. For relatively weak electron-phonon interactions, one can consider electrons to exhibit their band mass, but to scatter by emission of absorption of a phonon. As the electron-phonon interaction becomes strong, it becomes appropriate to consider a local deformation of the lattice around a charge carrier. The free electron approximation basically assumes a rigid lattice. In reality the electric field exerted by a charge carrier displaces the lattice ions in its vicinity, forming a polarization field around it. The charge carrier and its accompanying lattice deformation result in the formation of a quasiparticle, the polaron (Devreese and Peeters, 1984; Klingshirn, 2006). The degree of coupling scales with the lattice polarizability and is measured by the Fröhlich constant $\alpha$, which characterizes the electron-phonon coupling energy in terms of the phonon energy (Frohlich et al., 1950). The potential well created by the polarization field hinders the movement of the polaron since it has to “drag” the field along. This results in an increased effective mass of the carriers, termed polaron mass. For sufficiently weak coupling ( $\alpha \ll 6$), the wave function of the carrier remains extended (large polaron) and coupling merely increases the polaron mass, but leaves its transport Drude-like in nature, as THz-TDS measurements on sapphire have shown (Shan et al., 2003). For $\alpha \gg 6$, the potential well is deep enough to localize the carrier wave function, i.e., trap the charge (small polaron formation). In this limit transport can only occur via tunneling or hopping and the conductivity is not expected to follow Drude behavior.
Drude 模型适用性的另一个限制可以在非常强的电子-声子相互作用的限制中找到。对于相对较弱的电子-声子相互作用，可以认为电子表现出它们的能带质量，但通过声子的吸收发射来散射。随着电子-声子相互作用变得很强，考虑电荷载流子周围晶格的局部变形就变得合适了。自由电子近似基本上假设一个刚性晶格。实际上，电荷载流子施加的电场会使附近的晶格离子发生位移，在其周围形成极化场。电荷载流子及其伴随的晶格变形导致准粒子的形成，即极化子 （ Devreese and Peeters, 1984 ; Klingshirn, 2006 ）。耦合程度与晶格极化率成比例，由 Fröhlich 常数 测量，该常数 $\alpha$ 根据声子能量 （ ） 来表征电子-声子耦合能量 Frohlich et al., 1950 。极化场产生的势阱阻碍了极化子的运动，因为它必须“拖动”磁场。这导致载流子的有效质量增加，称为极化子质量。对于足够弱的耦合 （ $\alpha \ll 6$ ），载流子的波函数保持延伸（大极化子），耦合仅增加了极化子质量，但其传输性质类似 Drude，正如蓝宝石上的 THz-TDS 测量所示 （ Shan et al., 2003 ）。对于 $\alpha \gg 6$ ，势阱足够深，可以定位载波函数，即捕获电荷（小极化子形成）。在这个限制下，运输只能通过隧道或跳跃进行，并且电导率不会遵循 Drude 行为。

Hence, charge carrier transport can in principle be divided into two idealized cases: coherent, bandlike transport in delocalized states when intersite coupling is large and incoherent hopping transport from one localized state to another, characterized by low intersite coupling. In addition to different types of response (Drude-like versus non-Drude-like), a second way to distinguish the two types of response is through the temperature dependence of the charge carrier mobility. While the mobility is generally thermally activated for hopping processes, i.e., it increases with temperature, it generally decreases with temperature in bandlike transport due to increased carrier scattering with phonons at elevated temperatures. In many complex materials this distinction might be an oversimplification since several types of charge transport mechanisms can occur on different time and length scales in a single material. This is particularly true for organic semiconductors where one often has to distinguish between intramolecular and intermolecular transport, both of which are strongly influenced by the sample morphology. Given the surge in recent applications of THz TDS on these systems, we will elaborate on these issues in more detail.
因此，电荷载流子传输原则上可以分为两种理想化的情况：当位点间耦合很大时，离域状态下的相干带状传输，以及从一个局部状态到另一个定位状态的非相干跳跃传输，其特征是低位点间耦合。除了不同类型的响应（类 Drude 与非类 Drude）之外，区分这两种响应的第二种方法是通过载流子迁移率的温度依赖性。虽然迁移率通常被热激活用于跳跃过程，即它随温度的增加而增加，但在带状传输中，由于在高温下声子的载流子散射增加，它通常会随着温度的增加而降低。在许多复杂材料中，这种区别可能过于简单化，因为在单一材料中，几种类型的电荷传输机制可能发生在不同的时间和长度尺度上。对于有机半导体来说尤其如此，因为人们经常必须区分分子内和分子间传递，这两者都受到样品形态的强烈影响。鉴于最近太赫兹 TDS 在这些系统上的应用激增，我们将更详细地详细说明这些问题。

The organic semiconductors are commonly divided into two classes: (i) small molecules that are typically deposited as crystalline material by evaporation and (ii) semiconducting organic polymers that are processed in solution. Small-molecule semiconductors like acenes are typically characterized by an extended $\pi$ system along the molecular backbone on which charge carriers are delocalized. Under optimized process conditions they can assemble into crystals, bonded by intermolecular van der Waals forces. The key to efficient intermolecular electronic coupling lies in an optimum overlap between the $\pi$ orbitals of adjacent molecules. Several THz studies have been performed on single crystals of pentacene (Thorsmolle et al., 2004) and functionalized pentacene (Hegmann et al., 2002), as well as on thin films of both materials (Ostroverkhova et al., 2005). The photoinduced conductivity is real and essentially independent of frequency in the THz range Fig. 8. This can in principle be described by the Drude model with scattering times of only a few femtoseconds. Such a low scattering time would correspond to a carrier mean free path on the order of the intermolecular distance, indicating that charge carriers are likely to be localized on the molecule.
有机半导体通常分为两类：（i） 通常通过蒸发沉积为晶体材料的小分子和 （ii） 在溶液中加工的半导体有机聚合物。像乙烷这样的小分子半导体通常以沿分子骨架的扩展 $\pi$ 系统为特征，在该系统上电荷载流子离域。在优化的工艺条件下，它们可以组装成晶体，通过分子间范德华力键合。高效分子间电子耦合的关键在于相邻分子的 $\pi$ 轨道之间的最佳重叠。已经对五苯 （ Thorsmolle et al., 2004 ） 和功能化五苯 （ Hegmann et al., 2002 ） 的单晶以及两种材料的薄膜 （ ） 进行了多项太赫兹研究 Ostroverkhova et al., 2005 。光诱导电导率是实数，基本上与太赫兹范围内的频率无关 图 8 。原则上，这可以用 Drude 模型来描述，散射时间只有几飞秒。如此低的散射时间对应于分子间距离量级的载流子平均自由程，表明电荷载流子可能位于分子上。

FIG. 8.  图 8.

For single crystals, the photoconductivity drops within 2 ps and subsequently decays on longer time scales following a power-law decay $\sim t^{-\beta }$ with $\beta =0.5$. This was originally interpreted as the onset of dispersive (e.g., hopping) transport found in disordered media, pointing to localized charge carriers. While material dispersion can manifest itself in a time dependence (after photoexcitation) of the dc conductivity, in THz-TDS material polarization is normally expressed in the frequency dependence of the conductivity, while population dynamics will generally present themselves as a function of time after photoexcitation (see Sec. II F). Moreover, the mobility of all measured samples, single crystals and thin films, was found to decrease with increasing temperature, which would indicate bandlike transport.
对于单晶，光电导率下降到 2 ps 以内，随后在幂律衰变后在更长的时间尺度上衰减 $\sim t^{-\beta }$ 。 $\beta =0.5$ 这最初被解释为在无序介质中发现的分散（例如跳跃）传输的开始，指向局部电荷载流子。虽然材料色散可以表现为直流电导率的时间依赖性（光激发后），但在 THz-TDS 中，材料极化通常表示为电导率的频率依赖性，而种群动力学通常会表现为光激发后时间的函数（见第 Sec. II F ）。此外，发现所有测量样品、单晶和薄膜的迁移率都随着温度的升高而降低，这表明存在带状传输。

A possible explanation for this apparent contradiction was offered by Troisi and Orlandi (2006). Since the rate-limiting transport step is the intermolecular charge transfer, which in turn depends crucially on the distance between molecules, a so-called nonlocal (Peierls) electron-phonon coupling effect was proposed. In essence, it predicts a time-dependent modulation of the intermolecular coupling by the molecular vibrations of the molecules in the crystal that is sufficiently strong to localize carriers on the molecule and yet yields a bandlike temperature dependence of the mobility.
对这个明显的矛盾提供了一个可能的解释。 Troisi and Orlandi (2006) 由于限速传输步骤是分子间电荷转移，而这反过来又主要取决于分子之间的距离，因此提出了一种所谓的非局部 （Peierls） 电子-声子耦合效应。从本质上讲，它预测了晶体中分子的分子振动对分子间耦合的时间依赖性调制，这种调制的强度足以将载流子定位在分子上，但会产生迁移率的带状温度依赖性。

This was directly verified in a terahertz experiment on pentacene where on charge injection a pronounced absorption peak at 1.1 THz was observed (Laarhoven et al., 2008). This peak was attributed to a charge-induced activation of a soft vibrational mode at that frequency that couples strongly to the charge carrier dynamics. In other words, the intermolecular distance is modulated at a frequency of 1.1 THz and so is the intermolecular electronic coupling.
这在五苯的太赫兹实验中得到了直接验证，其中在充注时观察到 1.1 THz 的明显吸收峰 （ Laarhoven et al., 2008 ）。该峰值归因于该频率下软振动模式的电荷诱导激活，该模式与电荷载流子动力学强烈耦合。换句话说，分子间距离以 1.1 THz 的频率调制，分子间电子耦合也是如此。

Another class of organic semiconductors can be found in semiconducting polymers, which are characterized by delocalized $\pi$-electron systems along the polymer backbone. The electronic properties of polymer thin films are generally inferior compared to their small-molecule counterparts. The morphology lacks the long-range order that characterizes polycrystalline thin films and especially single crystals of small molecules. The electronic coupling between polymers is hence comparably weak, resulting in a bottleneck for carrier transport, which leads to lower macroscopic charge carrier mobilities. The intrachain mobility, however, is anticipated to be high, since charge carriers are, at least in absence of torsional disorder originating from kinks, expected to be delocalized on the conjugated backbone. Techniques that predominantly probe charge transport on the molecule, i.e., are only sensitive to charge carrier displacements on the nanometer scale, are therefore desirable in order to gain access to this regime. This problem can be translated into time by stating that such a technique ought to be sensitive to charge carrier movements on picosecond time scales, i.e., operating at GHz or THz frequencies. Indeed, microwave measurements at 34 GHz on ladder-type poly(p-phenylenes) revealed that charge transport along the backbone that is only obstructed by the chain ends would result in mobilities as high as $600{\mathrm{cm}}^2/\mathrm{Vs}$ (Prins et al., 2006). THz TDS can therefore be expected to deliver similar results and with it the time-resolved evolution of photoexcited species.
另一类有机半导体可以在半导体聚合物中找到，其特征是沿聚合物主链的离域 $\pi$ 电子系统。与小分子薄膜相比，聚合物薄膜的电子特性通常较差。形态缺乏表征多晶薄膜，尤其是小分子单晶的长程有序。因此，聚合物之间的电子耦合相对较弱，导致载流子传输成为瓶颈，从而导致宏观载流子迁移率降低。然而，预计链内迁移率会很高，因为至少在没有由扭结引起的扭转无序的情况下，电荷载流子预计会在共轭主链上离域。因此，为了获得这种状态，需要主要探测分子上电荷传输的技术，即仅对纳米级的电荷载流子位移敏感。这个问题可以通过说明这种技术应该对皮秒时间尺度上的电荷载流子运动敏感，即在 GHz 或 THz 频率下运行，从而将这个问题转化为时间。事实上，在 34 GHz 下对梯形聚（对苯乙烯）进行的微波测量表明，沿主链的电荷传输仅被链端阻碍，将导致迁移率高达 $600{\mathrm{cm}}^2/\mathrm{Vs}$ （ Prins et al., 2006 ）。因此，可以预期太赫兹 TDS 将提供类似的结果，并随之提供光激发物质的时间分辨进化。

A typical photoconductivity spectrum for semiconducting polymers, in this case from a MEH-PPV thin film, is shown in Fig. 9. The existence of a real part of the conductivity indicates the presence of free charge carriers. However, contrary to the Drude model, the real part decreases with decreasing frequency, approaching very low values close to zero frequency. This indicates that the long-range transport at dc frequencies is poor. The imaginary part differs from the Drude response as well, being negative and increasing with frequency. This type of behavior is characteristic of localized charge carriers in disordered systems. In these types of systems lacking long-range order, a highly corrugated potential energy landscape exists for carrier motion. Even along the backbone of one polymer strand, the mobility will be hindered by interruption of the $\pi$-electron continuum caused, for instance, by chemical or torsional defects. At increasingly high frequencies of the terahertz field, the mobility will be probed over increasingly short length scales. Hence, at high frequencies, the carrier response is probed in regions of high mobility.
半导体聚合物的典型光电导光谱，在本例中来自 MEH-PPV 薄膜，如图 1 所示。 9 。电导率的实部的存在表明存在自由电荷载流子。然而，与 Drude 模型相反，实部随着频率的降低而减小，接近接近零频率的非常低的值。这表明直流频率下的长距离传输很差。虚部也与 Drude 响应不同，它是负数并且随着频率的增加而增加。这种行为是无序系统中局部电荷载流子的特征。在这些缺乏长程有序的系统中，载流子运动存在高度波纹的势能景观。即使沿着一条聚合物链的主链，迁移率也会受到电子连续体中断 $\pi$ 的阻碍，例如，由化学或扭转缺陷引起的中断。在太赫兹场的频率越来越高时，将在越来越短的长度尺度上探测迁移率。因此，在高频下，在高迁移率区域探测载波响应。

FIG. 9.  图 9.

The complex conductivity is thus characteristic of dispersive free charge transport in a disordered medium, where localization caused by the disorder in the material structure causes non-Drude behavior. Indeed, the observed THz conductivity spectrum could be reproduced very well using a model by Grozema et al. (2002) to describe this frequency dependence . This model simulates the conductivity of a charge along a polymer chain. It is based on the tight-binding approximation combined with static torsional disorder deviations from planar alignment of the chain determining the effective conjugation length of the polymer chain.
因此，复电导率是无序介质中色散自由电荷传输的特征，其中由材料结构中的无序引起的定位会导致非 Drude 行为。事实上，使用模型可以很好地再现观察到的太赫兹电导率谱 来 Grozema et al. (2002) 描述这种频率依赖性 。此模型模拟了沿聚合物链的电荷的电导率。它基于紧密结合近似与静态扭转无序与链平面对齐的偏差，确定了聚合物链的有效共轭长度。

From these initial terahertz conductivity measurements on semiconducting polymers, it is apparent that there is an important effect due to the morphology of the polymer chains. It was subsequently shown that interactions between polymer chains also affect the photoconductivity of these materials: in a dropcast film of semiconducting polymers, the close proximity of nearest-neighbor chains facilitates exciton dissociation on subpicosecond time scales by allowing the resulting electron and hole charges to escape on separate chains and permitting significant real conductivity on much longer time scales. In solution, where polymer chains interact to a much lesser extent, significantly fewer charges are photogenerated, but the increased freedom of the polymer chains results in a larger conjugation length and an exciton with greater spatial extent and corresponding larger average polarizability than in the film sample (Hendry et al., 2005). More systematic studies of morphology issues, combining THz results with thin-film transistor device measurements, have revealed the influence of conjugation effects, molecular weight and film deposition conditions on the charge carrier mobility (Esenturk, Kline et al., 2008; Esenturk, Melinger, and Heilweil, 2008).
从半导体聚合物的这些初始太赫兹电导率测量中可以明显看出，由于聚合物链的形态，存在重要影响。随后表明，聚合物链之间的相互作用也会影响这些材料的光电导率：在半导体聚合物的滴铸薄膜中，最近相邻链的紧密接近通过允许产生的电子和空穴电荷在单独的链上逸出并允许在更长的时间尺度上显着的实电导率来促进亚皮秒时间尺度上的激子解离。在溶液中，聚合物链的相互作用程度要小得多，光生电荷明显减少，但聚合物链自由度的增加导致更大的共轭长度和具有更大空间范围的激子，以及相应的比薄膜样品更大的平均极化率 （ Hendry et al., 2005 ）。将太赫兹结果与薄膜晶体管器件测量相结合，对形态问题进行更系统的研究，揭示了共轭效应、分子量和薄膜沉积条件对电荷载流子迁移率 （ Esenturk, Kline et al., 2008 ; Esenturk, Melinger, and Heilweil, 2008 ）。

Apart from pristine polymers, films blended with fullerene derivatives [usually phenyl-C61-butyric acid methyl ester (PCBM)] have been investigated as well (Hendry, Schins et al., 2004; Ai et al., 2006; Cunningham and Hayden, 2008; Nemec et al., 2008; Parkinson et al., 2008; Nemec, Kuzel, and Sundstrom, 2009). The inclusion of PCBM is primarily intended for the application in organic solar cells; they provide a means to dissociate excitons after photoexcitation by acting as electron scavengers. Accordingly, free-carrier yields between a few and 30 times the yield in pristine films have been reported. Additionally, in blends a large fraction of free carriers is still present after nanoseconds, corroborating efficient spatial electron-hole separation (Parkinson et al., 2008). Two studies on poly(3-hexylthiophene) (P3HT)-PCBM blends found that photoexcitation below the band gap with 800 nm wavelength yielded comparable quantum efficiencies for charge generation and similar decay dynamics as compared to excitation at or above the band gap (Cunningham and Hayden, 2008; Parkinson et al., 2008). The origin for this behavior is still unclear. It was suggested that a charge-transfer state that is present in blends but absent in pristine polymers might act as an intermediate state for free-carrier generation.
除了原始聚合物外，还研究了与富勒烯衍生物 [通常是苯基-C61-丁酸甲酯 （PCBM）] 混合的薄膜 （ Hendry, Schins et al., 2004 ; Ai et al., 2006 ; Cunningham and Hayden, 2008 ; Nemec et al., 2008 ; Parkinson et al., 2008 ; Nemec, Kuzel, and Sundstrom, 2009 ）。PCBM 的加入主要用于有机太阳能电池中的应用;它们提供了一种在光激发后通过充当电子清除剂来解离激子的方法。因此，据报道，自由载流子的产量是原始薄膜产量的几到三十倍。此外，在混合物中，纳秒后仍然存在大量自由载流子，证实了有效的空间电子-空穴分离 （ Parkinson et al., 2008 ）。对聚（3-己基噻吩） （P3HT）-PCBM 混合物的两项研究发现，与带隙或带隙以上的激发相比，800 nm 波长的带隙以下的光激发产生了相当的电荷产生量子效率和类似的衰变动力学 （ Cunningham and Hayden, 2008 ; Parkinson et al., 2008 ）。这种行为的起源尚不清楚。有人提出，存在于共混物中但在原始聚合物中不存在的电荷转移态可能充当自由载流子生成的中间态。

Recent work has employed Monte Carlo simulations to explain measurements on fullerene blends (Nemec, Nienhuys et al., 2009). The application of Monte Carlo simulations seems a promising, albeit complex, route to take in order to interpret terahertz spectra in complex material systems such as semiconducting polymers. These simulations can be adapted to the specific morphological conditions and include different types of dynamics. The results hence offer important microscopic insights that more general and phenomenological models simply cannot provide.
最近的工作采用了蒙特卡洛模拟来解释富勒烯共混物 （ Nemec, Nienhuys et al., 2009 ） 的测量。蒙特卡洛仿真的应用似乎是一条很有前途但很复杂的方法，可以解释复杂材料系统（如半导体聚合物）中的太赫兹光谱。这些模拟可以适应特定的形态条件，并包括不同类型的动力学。因此，结果提供了更一般和现象学模型根本无法提供的重要微观见解。

It is apparent that the spectral signatures associated with conductivities of delocalized and localized charge carriers are different in the terahertz regime—and their differentiation thus possible using THz spectroscopy. This feature is particularly helpful for the characterization of materials that fall into the intermediate regime between large and small polarons. Measurements on these materials using other techniques are often inconclusive about the nature of charge transport. As an example, in titanium dioxide terahertz measurements revealed that, despite a large extracted Fröhlich constant of six, charge carriers were still delocalized (Hendry, Wang, Shan et al., 2004), contradicting previous reports that claimed the presence of small polarons. These examples demonstrate that THz TDS provides a unique tool to investigate photoconductivity—following the response to photoexcitation over a large frequency window. These insights, combined with complementary techniques that are sensitive to longer time and length scales, make a valuable contribution to developing an overall picture of photoconductivity in a variety of materials. In the following section, we discuss another appealing feature of THz TDS: the possibility to determine photoconductivity at different times after photoexcitation, with very high time resolution.
很明显，在太赫兹范围内，与离域和局域电荷载流子的电导率相关的光谱特征是不同的，因此可以使用太赫兹光谱法进行区分。此功能对于表征属于大极化子和小极化子之间的中间状态的材料特别有用。使用其他技术对这些材料进行测量通常无法确定电荷传输的性质。例如，在二氧化钛中，太赫兹测量表明，尽管提取的 Fröhlich 常数很大，为 6，但电荷载流子仍然离域 （ Hendry, Wang, Shan et al., 2004 ），这与之前声称存在小极化子的报告相矛盾。这些例子表明，太赫兹 TDS 提供了一种独特的工具来研究光电导性——跟踪大频率窗口上对光激发的响应。这些见解与对较长时间和长度尺度敏感的互补技术相结合，为开发各种材料中光电导性的整体图景做出了宝贵贡献。在下一节中，我们将讨论太赫兹 TDS 的另一个吸引人的特点：可以以非常高的时间分辨率确定光激发后不同时间的光电导率。

Unlike other ultrafast techniques such as luminescence and transient absorption, THz TDS is applicable to the study of carrier dynamics in indirect-gap semiconductors (Koch et al., 2006). Moreover, other ultrafast techniques are usually sensitive to either the sum or the product of the electron and hole population densities and do not measure the conductivity directly, while methods which are sensitive to the diffusive motion of carriers, such as transient grating or four-wave mixing, are not able to determine the conductivity on very fast time scales (Othonos, 1998; Chemla and Shah, 2001; Axt and Kuhn, 2004; Cundiff, 2008). THz TDS is therefore an extremely useful tool for determining material conductivities on very fast time scales.
与发光和瞬态吸收等其他超快技术不同，太赫兹 TDS 适用于间接间隙半导体中的载流子动力学研究 （ Koch et al., 2006 ）。此外，其他超快技术通常对电子和空穴群密度的总和或乘积敏感，并且不直接测量电导率，而对载流子扩散运动敏感的方法，例如瞬态光栅或四波混合，无法在非常快的时间尺度上确定电导率（ Othonos, 1998 ; Chemla and Shah, 2001 ; Axt and Kuhn, 2004 ; Cundiff, 2008 ）。因此，太赫兹 TDS 是在非常快的时间尺度上确定材料电导率的非常有用的工具。

The first THz-TDS measurements were performed on impurity-doped semiconductors (Vanexter and Grischkowsky, 1990; Katzenellenbogen and Grischkowsky, 1992). The emergence of amplified Ti:sapphire systems in the 1990s (Backus et al., 1998) eventually provided the possibility to readily create charge carriers by optical means. Not only did this offer greater flexibility in generating different charge carrier densities, but it also permitted the study of systems driven out of equilibrium by optical excitation processes and to monitor the evolution (Elsaesser and Woerner, 1999; Shah, 1999). This is particularly useful for the determination of the mobility in systems that possess a high density of trap states or fast electron-hole recombination channels, for instance. Traditional techniques like time-of-flight or photocurrent spectroscopy measure the dc current after photoexcitation (Brinza et al., 2005). Naturally, the carriers have to travel significant distances to reach the electrodes—too long for some materials to collect them all. THz TDS can take conductivity “snapshots” before trapping or recombination mechanisms set in. These are not influenced by processes occurring after distances much larger than the carrier mean free path which is typically less than 100 nm. Mobilities obtained with THz-TDS measurements can therefore be considered as an upper limit for semiconductors and semiconductor devices. From the observed dynamics, trapping processes occurring on picosecond time scales can be identified (Jepsen et al., 2001; Messner et al., 2001). Especially in materials that possess a low degree of crystallinity or porous materials, the conduction pathway can be significantly obstructed, resulting in the localization of charge carriers. Terahertz measurements on such materials will be discussed in Sec. IV A. Dynamics in the sub-ps regime are within reach as well. Probe pulses in the few-THz range allow under certain circumstances time resolutions on the order of 100s of fs. This permits for instance the study of intraband dynamics of charge carriers driven out of their equilibrium. With state-of-the-art lasers generating midinfrared probe pulses the probe spectrum not only covers carrier responses but vibrational modes of the lattice as well and the time resolution can be as high as 10 fs (Huber et al., 2000; Sell, Scheu et al., 2008). This gives access to the dynamics of quasiparticle correlations like the coupling between charge carriers and phonons, as will be shown below.
第一次 THz-TDS 测量是在掺杂杂质的半导体 （ Vanexter and Grischkowsky, 1990 ; Katzenellenbogen and Grischkowsky, 1992 ）。1990 年代 （ Backus et al., 1998 ） 放大的 Ti：sapphire 系统的出现最终提供了通过光学手段轻松创建电荷载流子的可能性。这不仅在产生不同的电荷载流子密度方面提供了更大的灵活性，而且还允许研究被光学激发过程驱使出平衡的系统，并监测演变 （ Elsaesser and Woerner, 1999 ; Shah, 1999 ）。例如，这对于确定具有高密度陷阱态或快速电子-空穴复合通道的系统中的迁移率特别有用。飞行时间或光电流光谱等传统技术测量光激发后的直流电流 （ Brinza et al., 2005 ）。当然，载体必须行驶很长的距离才能到达电极——对于某些材料来说，时间太长了，无法将它们全部收集起来。太赫兹 TDS 可以在捕获或复合机制开始之前拍摄电导率“快照”。这些不受远大于载流子平均自由程（通常小于 100 nm）的距离后发生的过程的影响。因此，通过 THz-TDS 测量获得的迁移率可以被视为半导体和半导体器件的上限。从观察到的动力学中，可以识别出发生在皮秒时间尺度上的俘获过程 （ Jepsen et al., 2001 ; Messner et al., 2001 ）。特别是在结晶度低的材料或多孔材料中，传导路径可能会受到严重阻碍，从而导致电荷载流子的局域化。此类材料的太赫兹测量将在 Sec. IV A .sub-ps 制度中的动态也是触手可及的。 在某些情况下，几太赫兹范围内的探测脉冲允许时间分辨率达到 100 fs 的数量级。例如，这允许研究被赶出平衡的电荷载流子的带内动力学。借助产生中红外探针脉冲的先进激光器，探针光谱不仅涵盖载流子响应，还涵盖晶格的振动模式，时间分辨率可高达 10 fs （ Huber et al., 2000 ; Sell, Scheu et al., 2008 ）。这可以访问准粒子相关性的动力学，例如载流子和声子之间的耦合，如下所示。

a. Intraband dynamics The first THz studies of intraband relaxation dynamics were carried out in model semiconducting materials like GaAs. While THz transmission studies of GaAs were carried out as far back as in the early 1990s (Greene et al., 1991), the first truly ultrafast THz study of the relaxation processes immediately following photoexcitation was carried out by Beard et al. (2000). In this pioneering work, they showed that by deconvolving the detector response function one can obtain information about material conduction on time scales down to $\sim 200\mathrm{fs}$ after photoexcitation, while accurately determining the carrier dynamics and mobilities.
一个。带内动力学 带内弛豫动力学的首次太赫兹研究是在 GaAs 等模型半导体材料中进行的。虽然早在 1990 年代初就进行了 GaAs 的太赫兹透射研究 （ Greene et al., 1991 ），但光激发后立即弛豫过程的第一次真正超快太赫兹研究是由 Beard et al. (2000) 进行的。在这项开创性的工作中，他们表明，通过对探测器响应函数进行反卷积，可以获得到光激发 $\sim 200\mathrm{fs}$ 后的时间尺度上的材料传导信息，同时准确确定载流子动力学和迁移率。

The photoexcitation energies used by Beard et al. (2000) were high enough to excite carriers into adjacent valleys in the band structure of GaAs (labeled $L$ and $X$ valleys). Since the conduction electrons in either the $L$ or $X$ valley have a much lower mobility than electrons in the main valley ( $\Gamma$), the measured dynamics reflected the electrons returning to the $\Gamma$ valley. While the conductivity on long time scales was found to be well described by the Drude model, with good agreement with literature values for the Drude parameters, they uncovered a contribution to the frequency-dependent conductivity on time scales (from 0.2 to 3 ps) after photoexcitation that had not been previously observed. This was essentially described by a time-dependent scattering rate (see Fig. 10), which was attributed to the influence of phonon dynamics on intraband free-carrier absorption. At early times phonons can be either absorbed or emitted to assist the THz photoabsorption [inset (a)], whereas at long delay times, after the carriers have relaxed to the bottom of the conduction band, their energy is not great enough to allow emission of a phonon to accompany photoabsorption [inset (b)], corresponding to a reduced scattering rate as shown in Fig. 10. This was also predicted earlier by Vengurlekar and Jha (1991).
使用的 Beard et al. (2000) 光激发能足够高，可以将载流子激发到 GaAs 能带结构中的相邻谷（标记 $L$ 和 $X$ 谷）中。由于 或 $L$ $X$ 谷中的传导电子比主谷 （ ） 中的电子具有低得多的迁移率 $\Gamma$ ，因此测得的动力学反映了返回 $\Gamma$ 谷谷的电子。虽然 Drude 模型很好地描述了长时间尺度上的电导率，并且与 Drude 参数的文献值非常一致，但他们发现了光激发后在时间尺度（从 0.2 到 3 ps）上对频率依赖性电导率的贡献，这是以前没有观察到的。这基本上是由随时间变化的散射率来描述的（见图 1）。 10 ），这归因于声子动力学对带内自由载流子吸收的影响。在早期，声子可以被吸收或发射以辅助太赫兹光吸收 [插图 （a）]，而在较长的延迟时间内，在载流子松弛到导带底部后，它们的能量不足以让声子的发射伴随着光吸收 [插图 （b）]，对应于降低的散射率，如图 1 所示。 10 。早些时候也预测到 Vengurlekar and Jha (1991) 了这一点。

FIG. 10.  图 10.

In this example hot carriers received the required kinetic energy to scatter into the $L$ or $X$ valley by the excess energy of the optical pump. It is also possible to induce intervalley scattering through the application of an electric field. Charge carriers then get accelerated until their kinetic energy is sufficient to scatter into another valley. THz measurements on a GaAs wafer that was subjected to a dc field showed a decrease in conductivity for increasing electric fields and for field strengths above $3\mathrm{kV}/\mathrm{cm}$ (Zhou et al., 2008). The decrease was attributed to an increasing amount of charge carriers that have scattered into the L valley and hence exhibit a lower mobility.
在这个例子中，热载流子接收到所需的动能，通过光泵的过量能量散射到 $L$ OR $X$ 谷中。也可以通过施加电场来诱导间隔散射。然后，电荷载流子会加速，直到它们的动能足以散射到另一个谷中。在受到直流场作用的 GaAs 晶片上的太赫兹测量表明，随着电场的增加和场强高于 $3\mathrm{kV}/\mathrm{cm}$ （ Zhou et al., 2008 ），电导率会降低。减少归因于分散到 L 谷中的载流子数量的增加，因此表现出较低的迁移率。

The recent development of high-energy single-cycle pulses in the low-THz regime as mentioned in Sec. II A 3 has provided a unique tool to study such hot-carrier effects with high time resolution. Currently, electric fields of several hundred $\mathrm{kV}/\mathrm{cm}$ for pulses with a spectral coverage from 0.1 to 3 THz have been realized (Blanchard et al., 2007; Hebling, Yeh, Hoffman, Bartal, and Nelson, 2008). Viewed in the time domain such frequencies correspond to electric fields that retain their polarity for up to several hundred femtoseconds—long enough to accelerate charges to very high velocities. An obvious application is thus the study of nonlinearities in the drift velocity of charge carriers and their impact on the system with sub-ps time resolution—something that is not feasible with electrically exerted fields.
正如 Sec. II A 3 中提到的低 THz 范围内的高能单周期脉冲的最新发展为研究这种具有高时间分辨率的热载流子效应提供了一种独特的工具。目前，已经实现了数百 $\mathrm{kV}/\mathrm{cm}$ 种光谱覆盖范围为 0.1 至 3 THz 的脉冲电场 （ Blanchard et al., 2007 ; Hebling, Yeh, Hoffman, Bartal, and Nelson, 2008 ）。从时域中看，这些频率对应于保持其极性长达几百飞秒的电场——足够长的时间将电荷加速到非常高的速度。因此，一个明显的应用是研究电荷载流子漂移速度的非线性及其对亚 ps 级时间分辨率系统的影响，这在电场中是不可行的。

GaAs served again to demonstrate this approach (Su et al., 2009). Single-cycle THz pulses with peak fields up to $200\mathrm{kV}/\mathrm{cm}$ were generated in a large-aperture ZnTe crystal (Blanchard et al., 2007). A single THz pulse was used to simultaneously perturb and probe carriers, following an optical pump pulse that photoinjected charge carriers in the material. The observed dynamics occurred within the bandwidth of the THz-pump pulse or, in other words, on time scales on the order of the pulse duration. Thus, the pump field was modulated as well during propagation. Analysis of the electric-field modulation in the time domain was then applied to infer the underlying processes.
GaAs 再次用于演示这种方法 （ Su et al., 2009 ）。在大孔径 ZnTe 晶体 （ Blanchard et al., 2007 ） 中产生峰值场高达 $200\mathrm{kV}/\mathrm{cm}$ （ ） 的单周期太赫兹脉冲。使用单个 THz 脉冲同时扰动和探测载流子，跟随光泵脉冲在材料中光注入载流子。观察到的动力学发生在太赫兹泵脉冲的带宽内，或者换句话说，发生在脉冲持续时间量级的时间尺度上。因此，泵场在传播过程中也受到调制。然后应用时域中的电场调制分析来推断基本过程。

Figure 11 shows the responses with and without optically generated charge carriers for two different THz field strengths: (a) low field, $4\mathrm{kV}/\mathrm{cm}$, and (b) high field, $173\mathrm{kV}/\mathrm{cm}$. Note that the transmission of the high-field beam is significantly enhanced as compared to the low-field case, suggesting a bleaching of the charge carrier absorption at higher fields. Additionally, the trailing part of the waveform features almost no change in amplitude or phase compared to the unexcited case at high THz fields. This particular behavior originates from the fact that several processes occurring within the pump pulse duration influence the overall conductivity and hence the time-dependent THz transmission: electrons are accelerated in the electric field to high velocities (which already lowers their mobility and hence bleaches the absorption) until they reach the threshold energy to scatter into the $L$ valley and thus exhibit a lower mobility. This is counterbalanced by backscattering into the $\Gamma$ valley. A model incorporating these and other parameters was implemented to reproduce the time-domain data. The calculated relative $\Gamma$ valley population in time is drawn as a solid line. The time it takes the electrons to be scattered back into the $\Gamma$ valley was calculated to be about 3 ps. This is comparable to the value of 1.9 ps found by THz pump THz probe measurements on n-doped GaAs (Hoffmann et al., 2009b). Note that a similar study has dealt with intervalley scattering in n-doped InGaAs thin films using the same experimental setup (Razzari et al., 2009).
图 11 显示了两种不同太赫兹场强下有和没有光生成电荷载流子的响应：（a） 低场 $4\mathrm{kV}/\mathrm{cm}$ 和 （b） 高场 $173\mathrm{kV}/\mathrm{cm}$ 。请注意，与低场情况相比，高场光束的透射率显著增强，这表明载流子吸收在较高场下会漂白。此外，与高太赫兹场下的未激发情况相比，波形的尾随部分在幅度或相位上几乎没有变化。这种特殊行为源于这样一个事实，即在泵脉冲持续时间内发生的几个过程会影响整体电导率，从而影响与时间相关的太赫兹传输：电子在电场中加速到高速（这已经降低了它们的迁移率，从而漂白了吸收），直到它们达到散射到 $L$ 谷中的阈值能量，从而表现出较低的迁移率。这可以通过反向散射到山谷来 $\Gamma$ 抵消。实施了一个包含这些参数和其他参数的模型来重现时域数据。计算出的相对 $\Gamma$ 山谷人口在时间上绘制为实线。电子散射回 $\Gamma$ 谷值所需的时间计算约为 3 ps。这与太赫兹泵浦太赫兹探针在 n 掺杂 GaAs 上测量发现的 1.9 ps 值相当 Hoffmann et al., 2009b （ ）。请注意，类似的研究已经使用相同的实验装置 （ Razzari et al., 2009 ） 处理了 n 掺杂 InGaAs 薄膜中的间隔散射。

FIG. 11  图 11

Another semiconductor compound on which the effects of high-field low-THz excitation have been studied is InSb (Hoffmann et al., 2009a; Hoffmann et al., 2009b). It possesses one of the highest electron mobility and saturation velocities of all known semiconductors. Even though this enables charge carriers to accelerate to very high velocities in an electric field, intervalley scattering is usually less pronounced—the energetic separation to the next valley of about 1 eV is too large (Chin et al., 1992). However, the band gap of only 170 meV at room temperature is very low. The combination of a high charge carrier saturation velocity and a low band gap facilitates the occurrence of impact ionization: here an electron in the conduction band gains sufficient energy to reach the ionization threshold of an electron in the valence band and excite it into the conduction band. For dc fields this effect can already be observed for field strengths of several hundred $\mathrm{V}/\mathrm{cm}$. The experiments were carried out on $n$-doped InSb¹ in a THz pump-THz probe scheme with the pump reaching field strengths of $100\mathrm{kV}/\mathrm{cm}$. Both beams were generated by optical rectification in lithium niobate (Hebling, Yeh, Hoffman, Bartal, and Nelson, 2008). Figure 12(a) shows the spectrally averaged dynamics of the absorption for a sample temperature of 200 K. The initial dip in the absorption is due to the common reduction of carrier mobility of hot electrons. Not only does the higher carrier velocity increase the phonon scattering rate, but also the effective mass: strongly nonparabolic bands imply a larger effective mass for higher-energy carriers. Both phenomena lower the mobility and thus the absorption. The subsequent absorption rise is caused by both the generation of new carriers through impact ionization and a mobility increase due to carrier cooling. The conductivity evolution shown in Fig. 12(b) reveals another interesting aspect: Apart from the rising Drude part on the low-frequency side a pronounced resonance appears at 1.2 THz. This peak was assigned to lattice vibrations: Hot charge carriers in InSb relax through the emission of optical phonons which in turn decay into acoustic modes. The sum and difference frequencies of these decay channels lie in the THz range. In particular, the difference frequency between longitudinal optical (LO) and longitudinal acoustic modes at the zone boundary can be found at 1.2 THz. The peak appearance can thus be attributed to a rising phonon population after energy transfer from hot electrons.
另一种研究过高场低 THz 激发影响的半导体化合物是 InSb （ Hoffmann et al., 2009a ; Hoffmann et al., 2009b ）。它拥有所有已知半导体中最高的电子迁移率和饱和速度之一。尽管这使得电荷载流子能够在电场中加速到非常高的速度，但间隔散射通常不太明显——到下一个约 1 eV 谷的能量分离太大了 （ Chin et al., 1992 ）。然而，在室温下只有 170 meV 的带隙非常低。高电荷载流子饱和速度和低带隙的结合促进了冲击电离的发生：在这里，导带中的电子获得了足够的能量来达到价带中电子的电离阈值，并将其激发到导带中。对于直流场，已经可以观察到几百 $\mathrm{V}/\mathrm{cm}$ 场强的这种效应。实验是在太赫兹泵浦-太赫兹探针方案中对 $n$ 掺杂的 InSb ¹ 进行的，泵浦达到 的场强。 $100\mathrm{kV}/\mathrm{cm}$ 两束光都是通过铌酸锂 （ ） 中的光学校正产生的 Hebling, Yeh, Hoffman, Bartal, and Nelson, 2008 。图 12(a) 显示了样品温度为 200 K 时的吸收光谱平均动力学。吸收的初始下降是由于热电子的载流子迁移率普遍降低。较高的载流子速度不仅会增加声子散射速率，还会增加有效质量：强非抛物线带意味着高能量载流子的有效质量更大。这两种现象都会降低迁移率，从而降低吸收率。 随后的吸收增加是由通过冲击电离产生新载流子和由于载流子冷却引起的迁移率增加引起的。电导率演变如图 2 所示。 12(b) 揭示了另一个有趣的方面：除了低频侧上升的 Drude 部分外，在 1.2 THz 处出现了明显的共振。这个峰值被分配给晶格振动：InSb 中的热电荷载流子通过光声子的发射来弛豫，而光声子又衰减到声学模式。这些衰减通道的和频率和差值频率在 THz 范围内。特别是，在区域边界处，纵向光学 （LO） 和纵向声学模式之间的差异频率可以在 1.2 THz 处找到。因此，峰值外观可归因于从热电子转移能量后声子数量增加。

FIG. 12  图 12

b. Dynamics of carrier-lattice coupling It is worth remembering that the collective oscillatory response of a charge carrier plasma to an alternating electric field is governed by many-body correlations: mobile carriers that are displaced from the equilibrium and their associated polarization field exert a restoring force on their surroundings and vice versa. A fundamental question concerns the time scale on which such many-body correlations form after charge carriers are injected in the conduction band—whether it is, for instance, instantaneous.
b.载流子-晶格耦合的动力学值得记住的是，电荷载流子等离子体对交变电场的集体振荡响应受多体相关性控制：脱离平衡的移动载流子及其相关的极化场对其周围环境施加恢复力，反之亦然。一个基本问题涉及电荷载流子注入导带后形成这种多体相关性的时间尺度——例如，它是瞬时的。

Huber et al. (2001) addressed this issue in one of the first publications that took advantage of the high time resolution and large bandwidth of mid-infrared (MIR) probes generated by 10 fs oscillator pulses in GaSe. The material of choice was GaAs. In a later publication by the same group the measurements were extended to InP, a semiconductor with a higher Fröhlich constant compared to GaAs (Huber, Kubler et al., 2005). The dynamics showed a similar evolution as in GaAs but spectral signatures that are characteristic for coupling between plasmon and phonon were more pronounced.
Huber et al. (2001) 在最早的出版物之一中解决了这个问题，该出版物利用了 GaŠ 中 10 fs 振荡器脉冲生成的中红外 （MIR） 探针的高时间分辨率和大带宽。选择的材料是 GaAs。在同一小组后来的出版物中，测量结果扩展到 InP，这是一种与 GaAs （ ） 相比具有更高 Fröhlich 常数的半导体 Huber, Kubler et al., 2005 。动力学显示出与 GaAs 相似的演变，但等离激元和声子之间耦合的特征光谱特征更为明显。

Figure 13(a) shows the temporal buildup of the plasma correlation in InP for increasing pump-probe delays. One should note that longitudinal modes of oscillation, such as bulk plasmons and longitudinal optical phonons, manifest themselves as nodes in the dielectric function (Kittel, 1995; Hasselbeck et al., 2002). For this reason, the inverse dielectric function, also known as the dielectric loss function $1/ϵ$, is plotted. For the unexcited material, the dielectric loss function features only one peak due to the longitudinal optical phonon. This resonance broadens and shifts to higher frequencies once electron-hole pairs have been injected. Within about 100 fs, the bandwidth narrows and eventually a peak appears at the plasma frequency of 14.4 THz. This behavior highlights the evolution from an uncorrelated electron-hole gas to a collective plasmon excitation that is mediated by many-body correlations, as shown in Fig. 13(b). The buildup time decreases with the charge carrier density; it scales with the inverse of the plasma frequency [see Fig. 13(c)]. The emergence of the bulk plasmon peak is accompanied by the appearance of a second smaller peak below the transverse optical phonon. This behavior is a direct consequence of the coupling between plasmon and phonon that forms a mixed mode. It causes a splitting of the normally linear plasmon dispersion into two branches: the upper mode $L_{+}$ and the lower mode $L_{-}$. Mapping the resonance frequencies as a function of charge density sketches the dispersion of the two branches [Fig. 13(d)]. The disappearance of the bare phonon resonance that is visible before carrier injection provides another indication for the existence of a mixed mode: its spectral weight shifts into the coupled modes once the plasmon-phonon coupling has been established.
该图 13(a) 显示了 InP 中等离子体相关性的时间累积，以增加泵浦探针延迟。应该注意的是，纵向振荡模式，如体等离激元和纵向光声子，在介电函数 （ Kittel, 1995 ; Hasselbeck et al., 2002 ）。因此，绘制了逆介电函数，也称为 dielectric loss function $1/ϵ$ 。对于未激发的材料，由于纵向光声子，介电损耗函数只有一个峰值。一旦注入电子-空穴对，这种谐振就会扩大并转移到更高的频率。在大约 100 fs 内，带宽变窄，最终在 14.4 THz 的等离子体频率处出现峰值。这种行为突出了从不相关的电子空穴气体到由多体相关性介导的集体等离激元激发的演变，如图 2 所示。 13(b) 。堆积时间随着电荷载流子密度的增加而减少;它与等离子体频率的倒数成比例 [见图 . 13(c) ]。体等离子体峰的出现伴随着横向光声子下方第二个较小峰的出现。这种行为是等离激元和声子之间耦合形成混合模式的直接结果。它导致通常线性的等离激元色散分裂成两个分支： 上模 $L_{+}$ 和 下模 $L_{-}$ 。将谐振频率映射为电荷密度的函数，勾勒出两个分支的色散 [图 D. 13(d) ]。 在载流子注入之前可见的裸声子共振的消失为混合模式的存在提供了另一个迹象：一旦等离激元-声子耦合建立，其频谱权重就会转移到耦合模式。

FIG. 13.  图 13.

The question of how the internal motion of a polaron within its potential well is affected at high electric fields was recently discussed in an experiment on GaAs by Gaal et al. (2007). To illustrate the physical mechanisms, the self-induced potential for a single polaron was calculated and is shown in Fig. 14 for low (a) and high (b) electric fields. A weak electric field merely induces a drift motion of the polaron (a). The electron stays at the center of the potential and the polaron response is determined by the center-of-mass motion of the entire quasiparticle. At high electric fields the electron gets appreciably displaced from the potential minimum [Fig. 14(b), filled circle). When the polaron reaches the saturation velocity, i.e., its energy equals the optical phonon energy, it impulsively transfers its energy to the lattice by emitting an optical phonon. The coherent oscillations of the emitted phonon appear as a stern wave of the moving electron. At a certain threshold, the induced oscillatory polarization changes are strong enough to modulate the lattice potential and alter the electron motion. The electron now oscillates within the potential with the optical phonon frequency [Fig. 14(b), open circles) on top of the polaron drift motion. The experiment succeeded in probing this wiggling motion by detecting the radiation emitted from the moving charge.
最近在 GaAs 的实验中讨论了极化子在其势阱内的内部运动在高电场下如何受到影响的问题 Gaal et al. (2007) 。为了说明物理机制，计算了单个极化子的自感应势，如图 1 所示。 14 适用于低 （A） 和高 （B） 电场。弱电场仅引起极化子 （a） 的漂移运动。电子保持在电位的中心，极化子响应由整个准粒子的质心运动决定。在高电场下，电子明显偏离了电位最小值 [图 D]。 14(b) ，实心圆圈）。当极化子达到饱和速度时，即其能量等于光声子能量时，它通过发射光声子脉冲将其能量转移到晶格上。发射的声子的相干振荡表现为移动电子的严厉波。在某个阈值下，诱导的振荡极化变化足以调节晶格电位并改变电子运动。电子现在随着光学声子频率在电位内振荡 [图 . 14(b) ，空心圆圈）的 Ζ S S Μ 的 Ζ S Η S Θ Θ Η S 该实验通过检测移动电荷发出的辐射成功地探测了这种摆动运动。

FIG. 14  图 14

In the actual experiment a low-THz beam with a field strength of $20\mathrm{kV}/\mathrm{cm}$ provided the electric field [Fig. 14(c) top) and a MIR beam with a center frequency of 17 THz served as the probe. The combination of both beams for a certain pump-probe delay $\tau$ is also shown. The time-integrated transmission should normally be zero, which is the case for low electric fields. For fields $>10\mathrm{kV}/\mathrm{cm}$ however a significant change in transmission occurs, as indicated for a pump-probe delay of 77 fs. Figure 14(d) shows that the magnitude and sign of the transmission change depends on the pump-probe delay. More precisely, it is modulated with a period equaling the LO phonon period (indicated as a dashed line). Apparently, the probe beam transmission is influenced by the radiation emitted from the wiggling charge, modulating it between absorption and gain.
在实际实验中，场强为 的 $20\mathrm{kV}/\mathrm{cm}$ 低 THz 光束提供了电场 [图 D]。 14(c) 上图）和中心频率为 17 THz 的 MIR 光束用作探头。还显示了特定泵浦探针延迟 $\tau$ 的两个光束的组合。时分传输通常应为零，对于低电场来说就是这种情况。然而，对于磁场 $>10\mathrm{kV}/\mathrm{cm}$ ，传输会发生显著变化，如 77 fs 的泵浦探测延迟所示。图 14(d) 显示传输变化的幅度和符号取决于泵浦探测延迟。更准确地说，它是用等于 LO 声子周期的周期（用虚线表示）调制的。显然，探针束传输受到摆动电荷发出的辐射的影响，在吸收和增益之间对其进行调制。

In this study the MIR beam was used as a probe that coherently mixes with radiation emitted from the system and thus gets modulated—similar to well-known coherent wave mixing techniques in the visible or infrared region. This probe scheme is thus somewhat different from the cases treated before in which the probe essentially interacts with an incoherent entity like an electron-hole gas. This wave mixing approach has been used in many cases to study coherent polarization dynamics in semiconductors (Chemla and Shah, 2001; Axt and Kuhn, 2004).
在这项研究中，MIR 光束被用作探针，它与系统发出的辐射相干混合，从而被调制——类似于可见光或红外区域中众所周知的相干波混合技术。因此，这种探针方案与之前讨论的情况有些不同，在这种情况下，探针基本上与电子空穴气体等非相干实体相互作用。这种波混合方法已在许多情况下用于研究半导体中的相干极化动力学 （ Chemla and Shah, 2001 ; Axt and Kuhn, 2004 ）。

It seems reasonable to address the influence of charge-lattice interactions in organic semiconductors as well. Compared to their inorganic counterparts they usually possess a multitude of vibrational modes, both intramolecular and intermolecular. And as it turns out, charge carriers in organic semiconductors can be of very polaronic nature and polaron binding energies² of 100 meV or more are not uncommon. This strong coupling suggests that injected charge carriers significantly alter the potential landscape of their host molecules—something that should be visible spectroscopically.
解决有机半导体中电荷-晶格相互作用的影响似乎也是合理的。与无机物相比，它们通常具有多种振动模式，包括分子内和分子间。事实证明，有机半导体中的电荷载流子可能具有非常极化的性质，并且 100 meV 或更高的极化子结合能 ² 并不少见。这种强耦合表明，注入的电荷载流子会显著改变其宿主分子的潜在景观——这应该在光谱学上可见。

Koeberg et al. (2007) investigated the ultrafast response of intramolecular vibrations after photoexcitation in rubrene, a small-molecule organic semiconductor that possesses one of the largest mobility values. The static absorption and dispersion spectrum is shown in Fig. 15(a). The featured resonances are intramolecular vibrations; whereas, intermolecular modes are restricted to lower frequencies ( $<6\mathrm{THz}$). The photomodulated spectrum for different pump-probe delays in the first picosecond is shown in Fig. 15(b). Analysis shows that it contains photoinduced modulations of the vibrational resonances carried on a nonresonant background absorption that originates from the spectrally broad free-carrier response. A close-up of the resonance around 15.5 THz in Fig. 15(c) reveals more details: The photomodulation of the mode lasts merely for 700 fs and its resonance frequency increasingly shifts in time to the blue [Fig. 15(d)]. The mechanisms responsible for this behavior are basically related to charge injection and redistribution which in turn alters the vibrational landscape. Both direct excitation of the vibronic states of the rubrene chromophores and their subsequent relaxation via internal conversion affects the vibrational potentials and frequencies. The magnitude of the observed resonance shift can thus be seen as a measure for the electron-phonon coupling strength.
Koeberg et al. (2007) 研究了 Rubrene 中光激发后分子内振动的超快响应，Rubrene 是一种具有最大迁移率值的小分子有机半导体。静态吸收和色散光谱如图 1 所示。 15(a) 。特色共振是分子内振动;而分子间模式仅限于较低频率 （ $<6\mathrm{THz}$ ）。第一皮秒内不同泵浦探针延迟的光调制光谱如图 1 所示。 15(b) 。分析表明，它包含对振动共振的光诱导调制，这种共振是在源自光谱宽自由载流子响应的非共振背景吸收上进行的。图 15.5 THz 附近谐振的特写。 15(c) 揭示了更多细节：该模式的光调制仅持续 700 fs，其共振频率随着时间的推移逐渐变为蓝色 [图 D]。 15(d) ]。导致这种行为的机制基本上与电荷注入和再分配有关，这反过来又改变了振动景观。红发色团的振动状态的直接激发和随后通过内部转换的弛豫都会影响振动电位和频率。因此，观察到的共振偏移的大小可以看作是电子-声子耦合强度的度量。

FIG. 15.  图 15.

Many of the following experiments were performed on stacked GaAs quantum well samples rather than bulk materials and hence on two-dimensional electron-hole gases. There are several reasons for that: Epitaxially grown, these layers are virtually strain-free and of high purity. Quantum confinement effects increase band gap and the exciton binding energy of originally 4.2 meV and can be tuned by changing the well thickness. The degeneracy of the valence band at $k=0$ is lifted and hence heavy and light hole levels are split. The band gap can be tuned to 1.55 eV to match the output energy of Ti:sapphire lasers which means that resonant exciton or continuum excitation is possible without the use of optical frequency conversion processes. Furthermore, since the quantum well (QW) thickness of around 10 nm is much smaller than the pump and probe wavelengths in use, propagation effects in the medium can be neglected. The concepts inferred from the studies of these samples can nevertheless be transferred to the three-dimensional bulk case.
以下许多实验是在堆叠的 GaAs 量子阱样品上进行的，而不是在块状材料上进行的，因此是在二维电子空穴气体上进行的。这有几个原因： 外延生长，这些层几乎无应力且纯度高。量子限制效应增加了带隙和激子结合能，最初为 4.2 meV，可以通过改变阱厚度来调整。价带 at $k=0$ 的简并性被提升，因此重空穴和轻空穴水平被分开。带隙可以调谐到 1.55 eV 以匹配 Ti：sapphire 激光器的输出能量，这意味着无需使用光学频率转换工艺即可实现谐振激子或连续激发。此外，由于大约 10 nm 的量子阱 （QW） 厚度远小于使用的泵浦和探针波长，因此可以忽略介质中的传播效应。然而，从这些样品的研究中推断出的概念可以转移到三维散装情况下。

The first detailed study on the dynamics of the formation and ionization of excitons was performed by Kaindl et al. (2003) on GaAs quantum wells. Later works by this group incorporated a more elaborate analysis in order to retrieve precise quantitative figures on exciton and plasma densities (Kaindl et al., 2009). The quantum wells had a thickness of 14 nm which led to a $1s\mathrm{\text{-}}2p$ transition energy of 7 meV ( $\sim 2\mathrm{THz}$)—and this “exciton fingerprint” being perfectly accessible by the probe THz spectrum that spanned from 2 to 12 meV. The optical pump spectrum centered around 800 nm was spectrally narrowed to about 1 meV in order to selectively excite into either exciton or continuum states. Two scenarios were investigated:
对激子形成和电离动力学的首次详细研究是通过 GaAs 量子阱进行的 Kaindl et al. (2003) 。该小组后来的工作结合了更详细的分析，以检索激子和等离子体密度的精确定量数据 （ Kaindl et al., 2009 ）。量子阱的厚度为 14 nm，导致过渡 $1s\mathrm{\text{-}}2p$ 能为 7 meV （ $\sim 2\mathrm{THz}$ ），并且这种“激子指纹”可以被跨越 2 到 12 meV 的探针太赫兹光谱完全接近。以 800 nm 为中心的光泵光谱被光谱缩小到大约 1 meV，以便选择性激发到激子或连续状态。调查了两种情况：

Regarding (1), directly after continuum excitation the spectrum is as expected dominated by the Drude response of free carriers. Surprisingly, however, already 40% of the electron-hole pairs are bound quasi-instantaneously (within the experimental time resolution of about 1 ps), indicated by a pronounced absorption peak at 7 meV [Fig. 16(a)]. Time-resolved photoluminescence measurements show no emission at these early time delays which led to the conclusion that these excitons occupy high-energy states with momenta $k>0$ and, as a result, are dark. This fast initial generation of excitons is followed by a slower condensation process on $\sim 100\mathrm{ps}$ time scales that leads to 90% of the electrons and holes being converted into excitons at 1 ns after photoexcitation. This process is accompanied by a rise of the exciton luminescence occurring on comparable time scales. Relaxation during the slow part involves at first the emission of optical and subsequently of acoustic phonons. The origin for the quasi-instantaneous exciton formation is as yet not understood in detail. Condensation facilitated by intricate energy and momentum exchange events between carriers, rather than a phonon bath, provides an explanation that is consistent with the observed time scale.
关于（1），在连续激发之后，光谱正如预期的那样由自由载流子的 Drude 响应主导。然而，令人惊讶的是，已经有 40% 的电子-空穴对准瞬时结合（在大约 1 ps 的实验时间分辨率内），由 7 meV 处的明显吸收峰表示 [图 . 16(a) ]。时间分辨光致发光测量显示，在这些早期时间延迟处没有发射，这导致得出结论，这些激子占据了动量的高能态 $k>0$ ，因此是暗的。这种快速的初始激子生成之后是时间尺度上 $\sim 100\mathrm{ps}$ 较慢的凝聚过程，导致 90% 的电子和空穴在光激发后 1 ns 转化为激子。这个过程伴随着在可比时间尺度上发生的激子发光的上升。慢速部分的弛豫首先涉及光的发射，然后是声声子的发射。准瞬时激子形成的起源尚未详细了解。载流子之间错综复杂的能量和动量交换事件（而不是声子浴）促进了凝结，这提供了一个与观察到的时间尺度一致的解释。

FIG. 16  图 16

Regarding (2), resonant interband excitation into the heavy-hole state initially creates a coherent interband polarization that dephases via phonon and carrier scattering within 5 ps into an incoherent exciton population (Lovenich et al., 2002). Accordingly, the anticipated THz exciton signature appears. At a sample temperature of 10 K the exciton fraction stays practically constant for all time delays, indicating that the thermal energy of the lattice is not sufficiently high to ionize a significant fraction of excitons. At higher temperatures, excitons get ionized on a time scale of several hundred picoseconds until they reach their temperature-dependent quasiequilibrium exciton concentration. As an example, the THz spectra for different pump-probe delays at $T=40\mathrm{K}$ are plotted in Fig. 16(b). At that temperature, the charge carrier plasma reaches its final exciton fraction of 60% after 400 ps. Above 50 K the ionization rate increases rapidly since now the ionization is facilitated through the absorption of LO phonons. At 80 K the quasiequilibrium exciton fraction of 30% is already established after 30 ps.
关于 （2），共振带间激发进入重空穴状态最初会产生一个相干的带间极化，该极化通过 5 ps 内的声子和载波散射分相为非相干激子群 （ Lovenich et al., 2002 ）。因此，出现了预期的 THz 激子特征。在 10 K 的样品温度下，激子分数在所有时间延迟内几乎保持恒定，这表明晶格的热能不够高，无法电离很大一部分激子。在较高温度下，激子在几百皮秒的时间尺度上被电离，直到达到与温度相关的准平衡激子浓度。例如，不同泵浦探针延迟的太赫兹频谱 $T=40\mathrm{K}$ 如图 2 所示。 16(b) 。在该温度下，电荷载流子等离子体在 400 ps 后达到其 60% 的最终激子分数。高于 50 K 时，电离速率迅速增加，因为现在通过吸收 LO 声子促进了电离。在 80 K 时，30% 的准平衡激子分数在 30 ps 后已经建立。

The mentioned experiments were performed at relatively low charge carrier densities of around ${10}^{10}{\mathrm{cm}}^{-2}$. An interesting change in the THz response occurs with increasing excitation fluence (Huber, Kaindl et al., 2005). For a GaAs QW sample held at a temperature of 6 K, optically exciting at the exciton line results in the formation of excitons of densities up to $5\times {10}^{10}{\mathrm{cm}}^{-2}$. When the excitation fluence results in densities exceeding this number, however, absorption starts to build up at low frequencies, indicative for the appearance of a conducting phase. The spectrum starts to deviate from being purely excitonic and the addition of a Drude component is required in order to describe it. Higher charge densities amplify this effect until beyond a critical density of $2\times {10}^{11}{\mathrm{cm}}^{-2}$, where all exciton signatures have vanished and the spectrum can be fully accounted for by the Drude model. This so-called excitonic Mott transition from an insulating phase at low charge carrier densities to a conducting one at high densities is driven by the increasing influence of Coulomb interactions between carriers. These many-body effects are usually negligible in a dilute electron-hole gas but become relevant at high charge densities when interexciton distances become comparable to the exciton Bohr radius. In that case, the electric field exerted by adjacent charge carriers is increasingly screening the attractive force between electrons and holes, thereby lowering the binding energy of excitons.
上述实验是在相对较低的电荷载流子密度下进行的，约为 ${10}^{10}{\mathrm{cm}}^{-2}$ 。太赫兹响应的一个有趣变化是随着激发能量通量 （ Huber, Kaindl et al., 2005 ） 的增加而发生的。对于在 6 K 温度下保持的 GaAs QW 样品，激子线处的光学激发导致形成密度高达 $5\times {10}^{10}{\mathrm{cm}}^{-2}$ 的激子。然而，当激发磁通量导致密度超过这个数字时，吸收开始在低频处积累，这表明出现了导电相。频谱开始偏离纯粹的激子，需要添加 Drude 分量才能描述它。较高的电荷密度会放大这种效应，直到超过临界密度 $2\times {10}^{11}{\mathrm{cm}}^{-2}$ ，此时所有激子特征都消失了，并且光谱可以完全由 Drude 模型解释。这种所谓的激子莫特从低电荷载流子密度的绝缘相到高密度的导电相的转变是由载流子之间库仑相互作用的影响越来越大所驱动的。这些多体效应在稀电子空穴气体中通常可以忽略不计，但当激子间距离与激子玻尔半径相当时，这些多体效应在高电荷密度下变得相关。在这种情况下，相邻电荷载流子施加的电场越来越多地屏蔽了电子和空穴之间的吸引力，从而降低了激子的结合能。

This lowering of the exciton binding energy can also be observed directly in the experiment: In the low-density case, in which the fraction of bound and unbound species in (quasi)equilibrium is essentially governed by the sample temperature, the THz spectrum simply reflects the relative contributions from excitons and plasma, but these individual contributions remain unchanged. During the Mott transition, however, the excitonic fine structure as seen in the THz spectrum is modified as well, reflecting the density-dependent influence of many-body correlations: The $1s\mathrm{\text{-}}2p$ transition energy decreases with increasing carrier density. This is a direct consequence of a lower exciton binding energy. This effect can also be observed in luminescence and absorption measurements as a blueshift of the exciton line towards the electronic band gap. Moreover, the $1s\mathrm{\text{-}}2p$ linewidth increases with charge density. The dephasing time of the polarization induced between the $1s$ and $2p$ states is reduced due to the growing perturbation from adjacent charge carriers.
这种激子结合能的降低也可以在实验中直接观察到：在低密度情况下，（准）平衡中结合和未结合物质的比例基本上由样品温度控制，太赫兹光谱仅反映激子和等离子体的相对贡献，但这些单独的贡献保持不变。然而，在莫特跃迁期间，太赫兹光谱中观察到的激子精细结构也被修改，反映了多体相关性的密度依赖性影响： $1s\mathrm{\text{-}}2p$ 跃迁能随着载流子密度的增加而降低。这是激子结合能较低的直接结果。这种效应也可以在发光和吸收测量中观察到，即激子线向电子带隙的蓝移。此外， $1s\mathrm{\text{-}}2p$ 线宽随着电荷密度的增加而增加。由于相邻电荷载流子的扰动增加， $1s$ 和 $2p$ 态之间感应的极化的去相时间缩短。

Similar observations have been made in a study on the bulk material zincoxide (ZnO), where at low temperatures excitons are readily formed (Hendry, Koeberg, and Bonn, 2007). The exciton binding energy of ZnO is 60 meV and hence several times higher than GaAs. The first exciton transition lies at 8 THz and can thus not be detected resonantly in the ZnTe probe window. However, its existence can be sensed off-resonantly by tracing its influence on the phase of the transmitted THz pulse, i.e., on the imaginary part of the conductivity (see Fig. 4). At low excitation densities, where the average exciton-exciton distance is much larger than the Bohr radius in ZnO, initially free charges are formed, as evidenced by the Drude-like behavior of the conductivity at short times after excitation. After $\sim 50\mathrm{ps}$, the imaginary component of the conductivity switches sign: from positive to negative. An analysis of the terahertz spectra reveals that this sign change originates from the conversion of free charges into excitons. This sign change can be understood by noting that the “resonance frequency” shifts from zero (Drude response) to finite frequency (exciton resonances). Indeed, at sufficiently long times the terahertz response can be described completely by the exciton response. For ZnO, at low temperatures the majority of the initially excited electron-hole pairs forms excitons.

The exciton formation dynamics can be characterized by an exponential decay time of 20 ps which is associated with the disappearance of the Drude response. This relatively slow evolution of conducting plasma into an exciton gas was explained by acoustic phonon emission being the rate determining step for exciton formation. This scenario contradicts the hypothesis that excitons are formed in the excited state, whereupon exciton cooling takes place (”hot exciton cascade”). In the hot exciton cascade picture, rapid emission of optical phonons by photocarriers leads to the formation of hot excitons, which subsequently cool to the emissive ( $K=0$) state by slow acoustic phonon emission [Fig. 17(a)]. However, emission of optical phonons by photocarriers in semiconductors typically occurs on subpicosecond time scales and one would expect disappearance of the Drude response on this time scale if the hot exciton cascade hypothesis were correct.

FIG. 17.

The formation of “cold” excitons from carriers, with acoustic phonon emission being the rate-limiting step, is corroborated by the temperature dependence of exciton formation. The temperature dependence of the exciton formation process is in good agreement with the temperature dependence expected for acoustic phonon scattering; the much stronger dependence expected for longitudinal optical phonons (expected in the hot exciton cascade picture) is not borne out experimentally.

At significantly higher excitation densities, where the interexciton distance is appreciably less than the Bohr radius, the photoconductivity reflects that of a Drude system with very high scattering rates. These high scattering rates can be attributed to carrier-carrier interactions. For these high densities, the electron-hole gas does not decay into an exciton state, but rather into an electron-hole plasma characterized by a density corresponding to an average carrier-carrier distance of roughly the Bohr radius. Apparently, above the Mott density, the initial high-density electron-hole plasma decays very rapidly (1.5 ps) through Auger annihilation to reach this value. In contrast to exciton formation, annihilation is found to be independent of lattice temperature, occurring while the plasma is still hot.

There has been some debate whether the description of the results in both the quantum wells and bulk materials in terms of a simplified two-component picture as a sum of a Drude and exciton transition is appropriate. Kaindl et al. (2009) validated this approach by calculating the exciton fractions for both excitation scenarios at different time delays using the Saha equation. Derived from basic thermodynamic relations, this equation assumes that excitons and free carriers reach a quasiequilibrium and their according densities can be deduced solely from the carrier temperature. Seasoned with literature values for the electron-phonon coupling strength the cooling dynamics can be calculated—which turned out to be consistent with the measured THz spectra.

On the other hand, it was shown recently that Coulomb correlations between electrons or holes are always present in the system, which can therefore not be completely adequately described as being composed of simply a mixture of an exciton gas and a free electron-hole plasma. A microscopic theory has suggested that one in fact has to distinguish three, rather than two, components: a continuum of ionized excitons, intermediate exciton states, and a Coulomb-correlated electron-hole plasma. It was shown that if this is not done correctly, it is difficult to quantitatively interpret the results in terms of relative contributions from the excitons and the electron-hole plasma (Kira and Koch, 2006; Koch et al., 2006).

Below a certain critical temperature an electron-hole gas can undergo a phase transition into an electron-hole liquid in the form of small droplets (EHD) that coexist with the gas phase of free carriers and excitons. The formation of a liquidlike phase is basically restricted to indirect semiconductors since the carrier lifetime is sufficiently long to reach the required charge carrier temperature and to spatially separate liquid and gas phases. In analogy to a real liquid the electron-hole liquid density only depends on its temperature and not the pump power (”pressure”). Instead, higher pump powers merely increase the volume ratio between liquid and gas fractions. The majority of studies on the properties of EHDs in indirect semiconductors such as Si and Ge originate from the 1970s and early 1980s (Jeffries, 1975; Shah et al., 1977). Because of technical limitations, these measurements were limited to the steady state; only recently have ultrafast spectroscopy methods permitted the time-resolved observation of EHD formation, most notably through photoluminescence measurements.

For silicon, the temporal buildup of excitons and electron-hole droplets after photoexcitation has recently been measured using THz TDS (Suzuki and Shimano, 2009). A pair of GaP crystals were used for terahertz generation and detection which provided a probe spectrum ranging from 0.5 to 6 THz (2–25 meV). The $e\mathrm{\text{-}}h$ phase diagram for silicon is shown in Fig. 18(a). The two dashed lines represent different calculations for the Mott density, i.e., the critical carrier density for the metal-insulator transition. Measurements were performed at charge carrier densities of $1\times 10{}^{16}{\mathrm{cm}}^{-3}$ and $1\times {10}^{17}{\mathrm{cm}}^{-3}$, i.e., below and above the Mott density, respectively. A sample temperature of 5 K assured that for both excitation scenarios the $e\mathrm{\text{-}}h$ system would reside in the two-phase region, i.e., an initial $e\mathrm{\text{-}}h$ plasma is expected to condense into excitons and droplets.

FIG. 18

The probe spectra for the high-density case at different pump-probe delays are shown in Fig. 18(b). At early times the response can roughly be described by the Drude model but excitonic signatures, recognizable by the appearance of the $1s\mathrm{\text{-}}2p$ transition at about 11 meV, are already visible. Within 400 ps the free-carrier response vanishes, accompanied by an increase of the exciton resonance and a broad high-frequency absorption. The exciton resonance subsequently decays and has all but disappeared after 2 ns whereas the magnitude of the broad high-frequency absorption stays practically constant. They assigned this broad absorption peak to $e\mathrm{\text{-}}h$ droplets and suggested the following scenario: after photoexcitation, free carriers simultaneously condense within 500 ps into excitons and EHDs. Subsequently, excitons diffuse into EHDs where they are efficiently screened by the $e\mathrm{\text{-}}h$ liquid and are hence not visible to the THz probe. After 2 ns, practically only EHDs are left in the system.

The EHD resonance can be understood by considering the droplet as being a conducting sphere surrounded by a dielectric (excitonic) medium. It is hence analogous to the observation of a plasmonic response of free carriers in nanoparticles (see the next section). They thus modeled the resonance using the Bruggemann formula as an effective medium description (see Sec. II D 2). Measurements at low charge carrier densities of $1\times {10}^{16}{\mathrm{cm}}^{-3}$ (not shown), i.e., below the Mott transition, yielded similar exciton condensation time scales of a few hundred picoseconds, with the EHD resonance, however, being much less pronounced and only visible as a slight side shoulder next to the exciton resonance.

The conversion of free carriers into excitons has also been observed in polymeric semiconducting materials. Semiconducting conjugated polymers have received considerable interest owing to their potential in technological applications, particularly in electronics. Despite their widespread optical applications, the nature of the photoexcitation physics in these materials has been subject to intense debate. One of the key questions that remained controversial was whether, initially upon photoexcitation, excitons or charge carriers are primarily formed. THz TDS was ideally suited to resolve this controversy, with its ability to monitor the evolution of free and bound charges on subpicosecond time scales following photoexcitation, through the time-dependent real and imaginary components of frequency-dependent conductivity. This allowed for an unambiguous assignment of the excited species during the excitation process. For the semiconducting polymer poly(2-methoxy-5-(2′-ethyl-hexyloxy)-p-phenylene vinylene) (MEH-PPV), it was found that only a very small fraction ( $<{10}^{-2}$) of photons resulted in direct excitation of free-electron-hole pairs, whereas the majority of the excitation resulted in exciton formation (Hendry, Schins et al., 2004). Free charges are generated from the dissociation of nascent, hot excitons, and are extremely short lived ( $\sim 1\mathrm{ps}$).

Akin to works on intersubband transitions in quantum wells, which are covered in Sec. IV B 1, THz measurements on intraexcitonic transitions have recently provided a new playing field to observe quantum phenomena such as stimulated emission, quantum beats (Huber et al., 2006), and Rabi cycling (Leinss et al., 2008). The key advantage here lies in its electro-optic detection scheme which permits the phase-resolved probing of pump-induced polarizations with a time resolution in the subcycle regime. This feature has so far been hard or impossible to achieve in the infrared or visible. In the following, we focus on the observation of Rabi cycles between intraexcitonic transitions in ${\mathrm{CuO}}_2$ by Leinss et al. (2008) since it provides another example of how high-field terahertz pulses can drive optical transitions in the nonlinear regime.

Excitons in ${\mathrm{CuO}}_2$ possess a particularly large binding energy of 150 meV. Strong exchange interaction splits their $1s$ ground state by 12 meV into an ortho and a lower-lying para variety [see Fig. 19(a)] (O’Hara and Wolfe, 2000; Kubouchi et al., 2005). After pumping with 1.55 eV charge carriers are generated via two-photon excitation that show a free-carrier response at early times [see Fig. 19(b)]. Within 100 ps, these have largely condensed into excitons, as evidenced by the appearance of the $1s\mathrm{\text{-}}2p$ transitions of ortho (116 meV) and para (129 meV) excitons. In order to resonantly excite the $1s\mathrm{\text{-}}2p$ para transition, pulses with a photon energy of 129 meV and a width of 4 meV were generated by optical rectification of 0.2 mJ in a GaSe crystal. In this way, peak fields of up to $0.5\mathrm{MV}/\mathrm{cm}$ could be generated in the focal spot [see Fig. 19(c) upper panel).

FIG. 19

The response for six different peak fields are shown in Fig. 19(c), (i)–(vi). For low excitation fields [traces (i) and (ii)], the reemitted field reaches its maximum towards the end of the driving pulse and dephases within 0.7 ps. At higher excitation intensities [traces (iii)–(vi)]. however, the response does not simply scale in magnitude. Instead, the reemitted field increasingly rises more rapidly and its peak shifts to earlier times. For the highest intensities, it reaches its maximum before the peak of the pump pulse. At the same time, the peak amplitude saturates and a second less pronounced peak appears about 0.5 ps after the first peak. Apparently, the driving field saturates the $1s\mathrm{\text{-}}2p$ transition and induces a coherent nonlinearity that leads to an oscillatory behavior.

The authors analyzed the results employing a microscopic description of the intraexcitonic light-matter coupling [see Kira and Koch (2004, 2006)]. This involved the correction of the response for the ponderomotive current originating from existing free carriers and ionized excitons. The calculated dynamics of the $1s\mathrm{\text{-}}2p$ transition are plotted in Fig. 19(d). The population inversion (solid line) and the polarizability (shaded area) feature approximately two Rabi cycle periods and the inversion reaches peak values of 80%.

Pure crystalline semiconductors with high charge carrier mobilities constitute the main ingredient for most microelectronic devices. Such high degrees of crystallinity and purity are, however, often not desired or feasible due to cost considerations: low-cost thin films of amorphous silicon find, for instance, widespread use in thin-film transistors in the backplane of liquid-crystal displays. In solar cells, amorphous silicon can be processed as cost-saving thin films because due to its direct band gap it exhibits a larger absorption as compared to their crystalline counterparts with indirect band gap. Porous oxide nanoparticles form the main constituent in Gratzel-type solar cells (Oregan and Gratzel, 1991). Also, short lifetimes of optically excited charge carriers due to high defect concentrations can be beneficial in certain optoelectronic devices.

It is therefore apparent that “real-life” electronic materials can be found in varying compositions and structural appearances. Ultimately, their electronic properties on the nanometer scale, to which THz spectroscopy is sensitive to, depend on the degree to which delocalized “ideal” charge transport is disrupted and charge carriers become localized. In the Terahertz probe window the combined response of delocalized and localized charge carriers can lead to intricate spectra. The interpretation of these in terms of a meaningful microscopic charge transport picture remains challenging. In this section we introduce how these effects emerge and how they affect the THz response.

In order to introduce the role of localization effects and their physical interpretation we compare two studies on silicon, namely, on amorphous and polycrystalline silicon (Cooke et al., 2006; Fekete et al., 2009).

Cooke et al. (2006) compared the THz conductivities of silicon nanocrystallites embedded in an insulating silicon dioxide matrix (Si-NCs, grain sizes 3, 4, and 7 nm), nanocrystalline silicon (nc-Si, grain sizes around 20 nm), and an epitaxially grown silicon-on-sapphire thin film. The results are shown in Figs. 20(a) and 20(b). The complex conductivity of the silicon-on-sapphire sample shows the typical signatures of a Drude response: a real part that is positive and decreasing with frequency and a positive imaginary part. The crystallite samples however show signs of carrier localization, reminiscent to the response of polymer semiconductors introduced in Sec. III. The real part of the conductivity is positive and decreasing with frequency whereas the imaginary part is negative. The conductivity values at dc are lower, however not vanishing. Clearly, the conduction process over long distances is suppressed by the existence of grain boundaries, most notably by the ${\mathrm{SiO}}_2$ barriers in the Si-NC samples.

FIG. 20.

The study of the effects of disorder on conductivity has a rich scientific history. Generally, there are two regimes of interest: in the “weak” localization, electron motion is still considered diffusive, but electron wave interference caused by coherence after scattering from defects, introduces small deviations from the Drude conductivity. Such effects, first considered by Abrahams et al. (1979), generally only occur at low temperatures in conducting materials. “Strong” localization (Anderson, 1958) occurs when the degree of randomness of the impurities or defects is sufficiently large to inhibit diffusion. A large number of conductivity models have been developed to describe effects due to strong localization. The most well known of these are the variable range hopping model introduced by Mott (1968, 1969), which describes thermally assisted hopping between localized electron sites, and phonon-assisted tunneling of electrons (Ambegaokar et al., 1971), both of which are valid when electrons hop or tunnel a distance greater than the average impurity separation. The Drude-Smith model (Smith, 2001), because of its very general applicability, is a common choice for describing localization effects in THz measurements. Cooke et al. (2006) applied the Drude-Smith model to describe their data which is given by

This model was, among others (Mayou, 2000), developed to describe material systems in which long-range transport is suppressed by disorder. It lifts the constraint of the Drude model that charge carriers scatter isotropically, i.e., that their momentum randomizes completely. Instead, charge carriers are allowed to scatter in preferential directions by introducing a persistence of velocity parameter $c$ which can have a value between 0 and $-1$. It is implemented for every scattering event $j$ in the series term in the bracket of Eq. (12). A key assumption of the model is that the persistence of velocity is retained for only one collision, i.e., only the first scattering event $j=1$ is considered. Despite the crude approximation, this simple empirical model works remarkably well. It can reproduce the signatures characteristic for charge carrier localization, as model plots of the conductivity reveal in Fig. 20(c) (real part) and Fig. 20(d) (imaginary part). $c=0$ yields the common Drude response. A more negative $c$ value increasingly depresses dc conductivity and shifts the oscillator strength to higher frequencies. For $c=-1$, the dc conductivity vanishes completely and the conductivity maximum occurs at $\omega \tau =1$. Translated to the measurements of Cooke et al. (2006), the $c$ parameter can be seen as a measure for the fraction of electrons that bounce back into the grain when they scatter at the boundary and is thus an indicator for the degree of charge carrier localization. The authors obtained values of $-0.83$ for nc-Si and around $-0.97$ for Si-NCs.

One must note one caveat when describing the THz conductivity in nanocrystalline materials which are described by a spatially varying conductivity. Such inhomogeneous materials must be treated differently from those that contain large numbers of microscopic local defects, such as impurities or grain boundaries, which are evaluated using scattering theories (Stroud, 1975). This is because a far-field THz measurement reflects an averaged conductivity of a spatially varying material and one has to link the local conductivity in nanocrystalline materials to the average conductivity obtained by far-field THz measurements. In other words, it is not only the change in local dielectric response that determines the overall dielectric response, but also the contrast in the dielectric response between the material of interest and its surroundings for the unexcited system. To make the connection between the local, spatially varying conductivities and averaged conductivities measured in the far field, requires an effective medium theory, such as those proposed by Garnett (1904) and Bruggemann (1935).

The problem of relating the far-field response to the local conductivity is abated when the dielectric response of the nanomaterial and the surroundings are identical. Such is the case for the experiments of Fekete et al. (2009), who conducted an extensive study on microcrystalline silicon prepared by plasma enhanced chemical vapor deposition. A schematic of the sample structure is depicted in Fig. 21(a). It consists of an amorphous silicon layer in which conically shaped crystallites of around 500 nm size are embedded. These crystallites in turn are comprised of small grains of several tens of nanometers in size. Conductivity measurements were performed in a temperature range between 20 and 300 K and modeled with the help of Monte Carlo simulations.

FIG. 21

Immediately following photoexcitation, the conductivity resembles a free-carrier response which rapidly vanishes within 0.6 ps. This decay is accompanied by the appearance of a conductivity shape characteristic of localized carriers, as shown before. They suggested that upon photoexcitation hot carriers are generated into delocalized bands which hence exhibit free-carrier properties. The carriers subsequently thermalize with the lattice and get trapped into shallow localized states at the small-grain boundaries within hundreds of femtoseconds.

The mobility of the free carriers at early times was estimated to be only $70{\mathrm{cm}}^2/\mathrm{Vs}$—an order of magnitude lower than bulk values. Furthermore, this mobility value and the free-carrier lifetime were independent of the sample temperature. These facts corroborate the speculation that the signal indeed originates from hot carriers whose temperature was estimated to be 1000 K. Charge carriers at such elevated temperatures experience increased scattering. That is primarily caused by their high kinetic energy rather than the lattice temperature—which explains the insensitivity to the steady-state sample temperature.

In order to understand the localization mechanism, they applied Monte Carlo simulations which included specific microscopic parameters that influence charge carrier transport. At first, they incorporated a charge carrier behavior similar to the Drude-Smith model (i.e., free charge carriers spatially confined within a grain that have a certain probability to be backscattered into the grain) The conductivity at room temperature could be reproduced [see Fig. 21(b), thick line; note that the graphs are plotted in terms of the mobility $g$ rather than the conductivity] assuming grain sizes of 20–30 nm, however, only with a physically unreasonable high backscattering ratio of 99%.³ Even more, the theoretically predicted temperature dependence was very different from the experimental results (thick lines).

In a second scenario it was assumed that charge carriers get trapped into shallow localized states which would result in carrier transport dominated by hopping, without a significant Drude-like contribution. In that sense, the peak position of the resonance in the frequency-dependent conductivity can be related to a characteristic frequency on which the conductivity saturates, i.e., on which the hopping process is most efficient. The model results are shown as thin lines in Fig. 21(b). The saturation frequency shifts to higher values at lower temperatures—from 10 THz at room temperature to 22 THz at 20 K. This suggests that a scattering mechanism which is activated at higher temperatures limits the upper hopping frequency and the maximum hopping mobility. It was suggested that scattering with optical phonons (located at 16 THz) could be the origin. At low temperatures, optical phonons are less populated; the electron-phonon scattering mechanism hence deactivated. This would push the conductivity saturation frequency to values past the optical phonon frequency.

Bulk plasmons are quantized, longitudinal oscillations of charge which occur in a plasma near the plasma frequency ${\omega }_p$. In a homogenous, three-dimensional material, one cannot couple optically to plasmon modes with a transverse electric field of a plane wave light source, and plasmon modes manifest themselves as nodes, rather than resonances, in the bulk dielectric function of a conductor (Kittel, 1995). Surface plasmons, first predicted in the 1950s by Ritchie (1957), are plasmons which are confined to interfaces between conductors and dielectrics. As is the case for bulk plasmons, surface plasmons at a flat metallic interface cannot be excited by plane wave light, because the wave vector of the plasmons is always larger than the wave vector of the incident light. This momentum mismatch can be overcome by scattering effects on rough surfaces (Kretschmann, 1972), for which coupling to surface plasmons occurs. Similarly, in inhomogeneous conducting materials one can couple to local plasmon oscillations. Semiconductors, on the other hand, exhibit significantly lower plasma frequencies than metals (Isaac, Barnes, and Hendry, 2008; Isaac, Rivas et al., 2008), so that one can expect coupling to THz plasmon oscillations in structured semiconductors. Such plasmon oscillations produce resonances in the effective dielectric function, similar to those expected from carrier localization effects (see preceding section). However, the physical origin of a plasmon resonance is very different from the single particle, localization effects described in the previous section: plasmon resonance is a many-body effect, and results in a coherent oscillation of all the conducting electrons.

To illustrate the origins of plasmon resonance, we consider the work of Nienhuys and Sundstrom (2005a), who measured THz photoconductivities of isolated silicon microspheres embedded in a polyvinylpyrrolidone matrix. These particles were fabricated by simply grinding a silicon wafer to a powder with particle sizes between 1 and $30\mu \mathrm{m}$. The conductivities for three different excitation fluences are shown in Fig. 22(a). Since the particles were completely isolated, the system lacked any pathway for long-range charge transport. It is therefore not surprising that the conductivity does not follow the Drude law and vanishes for probe frequencies approaching zero. Notably, the spectra feature a resonance whose center frequency shifts to higher values, proportionally to the square root of the charge carrier density. One should note that the Drude-Smith model cannot account for this behavior; its resonance frequency is solely defined by the scattering time (in addition, this model would not make sense physically: given that the particle dimensions are in the micron range and thus much larger than the charge carrier diffusion length on picosecond time scales, the relative amount of carriers that backscatter from the surface is negligible). A hopping model is also not appropriate: intraparticle transport is still expected to be bulklike, and interparticle transport—or transfer into the polyvinylpyrrolidone matrix—is practically impossible. It is therefore clear that these transport models, which describe localization of single particles in an inhomogeneous material, cannot account for the behavior shown in Fig. 22.

FIG. 22.

To understand the response, it is instructive to recall that the terahertz probe acts as an oscillating electric field that displaces electrons and holes in opposite directions. Because of the restricted size of the particles, charges will accumulate at the particle boundaries and form a space charge layer at the surface. The resulting dipole moment acts as an additional force on the motion of charge carriers (depolarization field) and has thus to be included as a restoring force in the differential equation of the damped harmonic oscillator introduced in Sec. II E 1 to derive the Drude model. The solution gives a conductivity ${\sigma }_{\mathrm{pl}}$ which is essentially represented by the Drude model whose resonance frequency has shifted from zero to higher frequencies (see Fig. 22):

${\tau }_D$ is the Drude scattering time, $s$ denotes a scaling factor that shifts the resonance frequency depending on the particle shape⁴ (in the case of spherical particles as considered here, $s=1/3$) and the dielectric constant of the particle and the surrounding medium (which were neglected in this study). One should note that Eq. (13) is a Lorentzian resonance [introduced in Sec. II E 2, Eq. (10)], defined in terms of conductivity, with a resonant frequency ${\omega }_0={\omega }_p\sqrt{s}$. Hence, the resonance frequency scales with the plasma frequency ${\omega }_p$ and with the square root of the charge carrier density, in accordance with the measured conductivities. The plasmon model for spherical particles as considered above gives the same resonance condition (i.e. ${\omega }_0={\omega }_p\sqrt{s}$) as is obtained by inserting the Drude conductivity [Eq. (9)] into Maxwell-Garnet effective medium theory [Eq. (4)] in the limits $\omega \tau \gg 1/{\tau }_D$, ${ϵ}_m=ϵ=1$ and the filling fraction $f\to 0$ (Hendry, Koeberg, O’Regan, and Bonn, 2006). This is because Maxwell-Garnet effective medium theory is derived from the Clausius-Mosotti relation (Feynman, 1970), and therefore takes into account the local depolarization field due to surrounding charge carriers. Indeed, it is well known that, in contrast to the Bruggemann’s approximation, Maxwell-Garnet effective medium theory reproduces plasmon resonance at optical frequencies in islandized metals (Riikonen et al., 2005). Strictly speaking, Maxwell-Garnet effective medium theory is correct only to first order expansion in $f$ (Spanier and Herman, 2000), and therefore gives exact results in the limit $f\to 0$. Despite this limitation, there is convincing evidence (Gittleman and Abeles, 1977) suggesting that Maxwell-Garnet theory may also satisfactorily reproduce plasmon resonances observed at optical frequencies in porous or granular metals, and has been shown to produce meaningful results (Granqvist and Hunderi, 1978) even for materials described by $f\sim 0.5$.

In addition to the observation of plasmon resonance in isolated silicon particles described above, there have been several studies where plasmon resonance has been observed in semiconductor nanowires. The diameter of most nanowire semiconductors fabricated today is larger than the exciton Bohr radius and electronic quantum confinement effects are thus not observed. For a general review of charge carrier dynamics in semiconductor nanowires, see Prasankumar et al. (2009). Most nanowires fabricated so far have not yet reached lateral sizes small enough to exhibit quantum confinement effects. Their response in the terahertz region is thus expected to be of plasmonic nature as well. Parkinson et al. (2007) investigated charge carrier dynamics in GaAs nanowires. After photoexcitation, the conductivity vanishes rapidly within 1–2 ps, an indication of severe trapping of charge carriers in surface trap states. Interestingly, the accompanied decrease in charge carrier density is reflected as a redshift of the plasmon frequency within the probe spectrum (see Fig. 23, plasma frequency indicated with arrow). Around 300 fs after photoexcitation the localized surface plasmon mode has build up at a frequency of 1 THz. Its peak shifts within the next picosecond to lower frequencies until it is outside the probe window and all charge carriers have been trapped.

FIG. 23.

In a follow-up paper Parkinson et al. (2009) correlated the influence of growth conditions of GaAs nanowires and their overcoating with a larger-band gap material on the charge-carrier mobility and lifetime.

The influence of depolarization fields on the THz response of nanowires depends on the sample dimensions with respect to the probe polarization: this was explicitly demonstrated by Strait et al. (2009). Germanium nanowires ( $l=10\mu \mathrm{m}$, $d=80\mathrm{nm}$) had been transferred in an aligned fashion from the growth substrate onto quartz. The high degree of ordering is observable in the dark-field optical micrograph image in Fig. 24(a). The sample was oriented in such a way that the THz polarization had an angle of 45° with respect to the nanowire axes. A polarizer was used to select certain polarization components, indexed by the angle $\Theta$. An angle of 90° corresponds to the polarization being perpendicular to the axis whereas for $\Theta =0$ it is parallel [see sketch in Fig. 24(b)]. The dynamics of the photoinduced transmission change are shown in Fig. 24(b). The decay is due to carrier recombination, most likely via surface defect states. The transmission change, i.e., the real conductivity, is largest for polarizations along the nanowire axis and decreases with increasing $\Theta$ until for 90° it has vanished completely. They applied Eq. (13) to explain the results. For $\Theta =0$, the scaling factor $s$ was set to $(d/L)({ϵ}_s/{ϵ}_0{)}^{1/2}$. Since $d\ll L$, the scaling factor pushed the plasmon frequencies below 300 GHz and hence below the THz window for all charge carrier densities. This implies that the conductivity approximately resembles the bulk plasmon mode, i.e., the traditional Drude response with a peak at dc. For polarizations perpendicular to the nanowire axis, $s$ was set to $[{ϵ}_s/({ϵ}_s+{ϵ}_0){]}^{1/2}$. This yielded plasmon frequencies of tens of THz. Since the width of the plasmon peak is relatively narrow due to the high carrier scattering times in germanium, no absorption can be found in the THz window.

FIG. 24

Two recent investigations report the dark and photoconductivity of nanoporous InP structures (Lloyd-Hughes et al., 2008; Merchant et al., 2008). The samples were fabricated by electrochemical etching of $n$-doped InP wafers, generating pores perpendicular to the surface. A scanning electron microscope image and a schematic are shown in Fig. 25(a). The pores had diameters of around 60 nm and were separated by a distance of around 100 nm. The dark conductivities, i.e., of the extrinsic electrons introduced by doping, of three samples with different degrees of porosity featured the common signs of localized charge transport: a real part that is positive and increasing with frequency and a negative, increasing imaginary part. The real part was further suppressed at higher porosities, accompanied by an increased imaginary part. Clearly, a higher porosity impairs charge transport. Interestingly however, the conductivity of photoexcited charge carriers displays a free-carrier response with mobility values that are independent of the excitation density and are of roughly half the bulk value.

FIG. 25

In order to explain the observations they calculated the two-dimensional carrier density, assuming an electron doping density of ${10}^{18}{\mathrm{cm}}^{-3}$. Additionally, it was assumed that electronic states at the pore surface pin the Fermi level above its bulk value. The bands thus bend upwards near the pores, creating a surface layer depleted of electrons. The calculated spatial charge carrier density distribution is shown in Fig. 25(b): the carrier densities in regions close to the pore approach zero and are hence of low conductivity. The increasing prevalence of depleted areas in the higher-porosity samples thus explains the decreasing conductivity trend. The localized conductivity response can now be understood by visualizing the areas with a significant amount of charge carriers as completely isolated from each other by depleted zones, hence lacking any percolation pathway between them. These carrier islands exhibit localized surface plasmon modes with plasmon frequencies much higher than the THz probe bandwidth.

The scenario after photoexcitation is shown in Fig. 25(c). Here the color legend indicates the change in charge carrier density as compared to the dark conductivity. As can be seen, the charge density in formerly depleted regions close to the pores increases whereas it decreases in formerly populated regions. The band bending has thus been flattened out and charge carriers have been distributed more homogeneously. In fact, no entirely depleted areas can be found anymore. Charge transport is thus well percolated, in agreement with the measured free-carrier response.

Since plasmon resonance and carrier localization in inhomogeneous semiconductors can both result in conductivity resonances in the THz frequency region, it is often difficult to determine the physical origin of any resonance behavior in these materials. To illustrate this point, we summarize below two works published in recent years which investigated the THz response of nanoporous ${\mathrm{TiO}}_2$ films (Turner et al., 2002; Hendry, Koeberg, O’Regan, and Bonn, 2006). These studies illustrated how one can interpret THz measurements on the same system in different ways.

First published was the work by Turner et al. (2002), who studied the THz photoconductivity of a porous material composed from sintered ${\mathrm{TiO}}_2$ nanoparticles (25 nm radius). The ${\mathrm{TiO}}_2$ matrix was sensitized with photoactivated dye molecules, from which carriers were injected into the ${\mathrm{TiO}}_2$ matrix. The photoconductivity, measured at 77 K, exhibited a clear resonance in the THz spectrum. They interpreted this response as a localization effect, fitting their frequency-dependent conductivity using the Drude-Smith model (Smith, 2001). This fit yielded a mobility for the porous material of $1.5{\mathrm{cm}}^2/\mathrm{Vs}$, compared to the bulk value for ${\mathrm{TiO}}_2$ of $56{\mathrm{cm}}^2/\mathrm{Vs}$.

In later work on a similar system, Hendry, Koeberg, O’Regan, and Bonn (2006) also measured the THz photoconductivity of sintered, 25 nm ${\mathrm{TiO}}_2$ nanoparticles. There were, however, a number of subtle differences from the original experiments by Turner et al. (2002). First, in order to remove effects due to injection, carriers were introduced by directly photoexciting an unsensitized ${\mathrm{TiO}}_2$ with both 266 and 400 nm light. This means that both electrons and holes were excited in the material. By comparing directly the photoconductivity of the nanoporous material with that measured for a homogeneous ${\mathrm{TiO}}_2$ single crystal (Hendry, Wang, Shan et al., 2004), they were able to determine that the THz mobility of electrons in the porous sample was $\sim {10}^{-2}{\mathrm{cm}}^2/\mathrm{Vs}$ at 77 K. The contribution from the holes was negligible due to the very large effective mass of holes in ${\mathrm{TiO}}_2$. On increasing the temperature of the sample to 300 K, a reduction in mobility of only a factor of 2 (compared to a change of almost 2 orders of magnitude in the bulk ${\mathrm{TiO}}_2$ material) was observed. More notably, they also observed a conductivity resonance in the THz conductivity of the nanoporous material. By varying the excitation intensity, they observed that the resonance frequency depended on carrier density, which is a strong indication of a plasmon resonance. They therefore employed Maxwell-Garnet effective medium to describe the observed plasmon resonance. This model also explained the relative insensitivity of the THz conductivity to temperature, since the plasmon resonance frequency is determined only by the carrier density, and not the scattering rate.

While both Turner et al. (2002) and Hendry, Koeberg, Wang et al. (2006) observed similar resonances in their THz conductivity spectra of nanoporous ${\mathrm{TiO}}_2$, two different explanations for the physical origin of the response were given: as an intraparticle, single-carrier response governed by boundary scattering, on the one hand, and as a Drude response buried in an effective medium, on the other hand, resulting in a plasmon resonance. This illustrates how THz conductivities ought to be evaluated with care, as one essentially has to distinguish how local, single-carrier properties (Drude, hopping, backscattering) and local field or many-body contributions (dielectric screening, depolarization fields) affect the overall, far-field response. A proper understanding of both the local intraparticle behavior and local field effects due to inhomogeneities in the dielectric response are thus imperative.

To address the issue of local conductivity, Nemec, Kuzel, and Sundstrom (2009) performed Monte Carlo simulations to calculate the conductivities of charge carriers in spherical particles of varying sizes. Only a few parameters entered the simulation: the particle dimension $d$, the thermal velocity $v_{th}$, and the carrier momentum scattering time $\tau$ (and hence the mean free path $l_f=v_{\mathrm{th}}\tau$). The parameter $p_r$ denotes the probability that upon scattering with the particle boundary a charge carrier is reflected back into the particle ( $p_r$ is thus akin to the $c$-parameter in the Drude-Smith model). The degree of localization was expressed as the parameter $\alpha$, defined as the ratio between the mean free path and the particle dimension. They concluded that the shape of the conductivities only depend on $\alpha$ and $p_r$.

The results for completely isolated particles (corresponding to $p_r=1$ and $c=-1$ in the Drude-Smith model) are shown in Fig. 26(a). For $\alpha$ going to infinity, i.e., $d\gg l_f$, the conductivity is represented by the Drude model. For $\alpha <10$, the dc conductivity significantly drops and a resonance appears. The center frequency of this peak is related to the ballistic round-trip time of carriers in the sphere ${\tau }_{\mathrm{rt}}=2d/v_{\mathrm{th}}$. For smaller particles, this round-trip time decreases which is reflected in a blueshift of the resonance frequency for decreasing $\alpha$. Additionally, the conductivity amplitude decreases with $\alpha$. The conductivity evolution is in principle similar to the case of $p_r=0.5$ [Fig. 26(b)] with the exception that the real part does not vanish at dc for all values of $\alpha$, proving that long-range transport is enabled.

FIG. 26.

One should note, however, that the spectra calculated by Nemec et al. (2009) still represent local, single-carrier properties of the material. In order to obtain the effective response of the whole system one would still have to either incorporate carrier-carrier interactions into the calculation or alternatively apply an appropriate effective medium theory. Nevertheless, we still can consider, in a general manner, the effects that carrier-carrier interactions would have: for a nanoporous material in which the interparticle transport is inefficient (i.e., one which exhibits a localized, single particle conductivity resonance), carrier-carrier interactions would introduce an additional restoring force for carriers near the interface, pulling them towards the center of the particle. Such a restorative effect would shift the spectral weight of the low-frequency, localized resonance, to higher frequencies (see Sec. II E 2). This means that on increasing the strength of carrier-carrier interactions (by increasing the carrier density) one can expect to observe the evolution from a localized resonance to plasmon resonance in the same material. This problem of resolving plasmon resonance from localization effects is obviously not restricted to ${\mathrm{TiO}}_2$ but can also occur in other nanostructured materials that possess domain dimensions comparable to the carrier mean free path.

QWs are often formed in semiconductors by sandwiching a material between two layers of another wider band gap material such as GaAs/AlAs. These structures typically are grown by either molecular beam epitaxy or chemical vapor deposition with a control of the layer thickness down to monolayers. The potential well in the growth direction confines electrons and holes in the plane and creates subbands [see Fig. 27(a)]. Within the subbands, charge carriers behave as a free quasi-two-dimensional electron gas (Shik, 1999).

FIG. 27.

Initial THz-TDS experiments on QWs measured basic linear properties of intersubband (IS) transitions, e.g., their dispersion (Heyman et al., 1998), induced polarization (Kersting et al., 2000), and subband population dynamics (Bratschitsch et al., 2002). Note that these early experiments did not probe the electric field via electro-optic sampling but by a heterodyne detection scheme in which the interferometric signal between the probe, and a reference pulse was detected with a bolometer or MCT detector. A typical sample consists of a stack of a few dozens of quantum wells grown on a single substrate. A prism is attached to it that efficiently couples the probe light into the sample at a tilted angle [see Fig. 27(b)] In that way a $p$-polarized THz probe is sensitive to both the in-plane IS transitions and the two-dimensional free charge carrier plasma. The interference between the two contributions, manifested as a Fano resonance, was observed only recently in QWs (Golde et al., 2009).

The sample studied consisted of undoped GaAs QWs of 8.2 nm diameter that featured the $1\to 2$ IS transition at 27 THz. A 100 fs optical pump pulse at 1.56 eV was used to populate the first conduction subband and a MIR probe pulse generated in GaSe to measure the response. The pump-induced differential transmission of the probe field is shown in Fig. 28(a). It features the induced IS polarization with the common exponentially decreasing envelope due to the free induction decay. However, a clear beating can be observed as well. The origin of the beating is revealed in the frequency domain as a broad absorption peak centered at 20 THz next to the IS transition at 27 THz [Fig. 28(b)]. The spectrum features typical signatures of a Fano resonance: an asymmetric line shape with a dip on the low-frequency side. It is caused by the coherent superposition of the polarizations of a two-level system (the IS transition) and a continuum (the 2D electron gas). The interference is destructive on the red side of the IS resonance, hence causing the undershoot; whereas, it is constructive on the blue side, resulting in a broadening. Interestingly, the plasma contribution vanishes in the absorption spectrum, leaving only the bare IS transition [Fig. 28(c)]. This indicates that the plasma contribution is nonabsorptive in the probed spectral range and hence only its phase behavior responsible for the modulation of the overall transmission. This again corroborates the coherent nature of the mechanism.

FIG. 28

Transition dipole moments between subbands are comparably strong due to the large extension of the envelope wave function along the $z$ direction. This allows one to drive IS transitions into the nonlinear regime at relatively low electric field strengths (Luo, Reimann, Woerner, and Elsaesser, 2004). As a result, Rabi oscillations can be observed more clearly and at lower electric fields as compared to the case of intraexcitonic transitions (Luo, Reimann, Woerner, Elsaesser et al., 2004). This was observed in $n$-doped GaAs quantum wells that had a charge carrier density such that the Fermi level lies in the gap of the lowest subbands, i.e., only the first subband is populated. The $1\to 2$ transition at 24 THz was pumped with a narrow-band MIR pulse generated in GaSe. The dynamics of the reemitted field in the linear regime are similar to the exciton study in Sec. III B, hence we focus on the nonlinear regime. Figures 29(a) and 29(b) show the dynamics for driving fields of $25$ and $50\mathrm{kV}/\mathrm{cm}$ in peak amplitude, respectively. The Rabi effect is clearly visible as an envelope modulation of the reemitted field. The time period of that beating decreases for higher driving fields. In fact, the modulation frequency (Rabi frequency) scales linearly with the electric field of the pump pulse [see Fig. 29(c)]. A closer look on the phase relation between the driving field and the reemitted field in each of the Rabi half-cycles reveals the alternating nature of the process between absorption and stimulated emission: a phase difference of 180°, indicative of absorption and hence buildup of population, in one half-cycle, is followed by a phase difference of 0° in the next half-cycle, caused by stimulated emission and hence depopulation of the excited state. The relation between these parameters is sketched in a simulation in Fig. 29(d).

FIG. 29

Apart from the more fundamental scientific insights that quantum wells can provide on quantum phenomena (Wang et al., 2007), recent interest on the application side has been motivated (among others) by their use in quantum-cascade lasers, which consist of a stack of multiple quantum well heterostructures (Williams, 2007). QCLs promise a cheap and compact source for coherent terahertz and MIR radiation. Compared to traditional semiconductor lasers, they rely on a similar design philosophy in the sense that a bias voltage is applied to the structure in order to align the energy bands and create inversion. However, inversion is achieved between intraband (i.e., subband) transitions rather than interband transitions. Furthermore, a recombined electron can be reused by tunneling from the lowest state (injector state) in one QW to the upper laser state (collector state) in the adjacent QW, again building up inversion (cascade; see sketch in Fig. 30(a)]. For more information about the device architecture, see Williams (2007).

FIG. 30

A time-domain technique like electro-optic sampling is perfectly suited to unambiguously study lasing dynamics in a phase-resolved manner since it allows one to detect the relative phase between probe and emitted fields. Kroll, Darmo et al. (2007) studied the polarization dynamics of AlGaAs/GaAs THz QCLs that lased at a frequency of 2.9 THz. The THz probe pulse was generated by a photoconductive antenna, coupled into the QCL waveguide and its transmission detected by EO sampling. Figure 30(b) shows the electric field after transmission through an unbiased sample. It features the input pulse followed by small oscillations due to weak absorption in the waveguide. The situation changes when a bias is applied, as depicted in Fig. 30(c) which shows the difference signal between unbiased and biased signal. It is now a result of changed absorption (the first single-cycle pulse) and gain (the subsequent oscillation lasting several picoseconds). The transition from initial absorption to stimulated emission can be recognized by a phase jump between the initial pulse and the oscillating part, indicated by the arrow. This behavior is visible in the frequency domain as well: the absorption spectrum in Fig. 30(d) shows a broad absorption peak between 1.0 and 2.2 THz, which is caused by the differences in the charge carrier distribution between unbiased and biased conditions. A narrow negative absorption peak, i.e., gain, is observed at the lasing frequency of 2.9 THz. It is accompanied by a flipped phase behavior, as compared to absorption, around the resonance which displays the negative part on the low-frequency side. This is an unambiguous evidence for stimulated emission.

Other THz studies on QCLs focused on gain dynamics below and above lasing threshold (Kuehn et al., 2008; Jukam et al., 2009), the charge carrier mobility and density (Lloyd-Hughes et al., 2009), observed spatial hole burning (Kroll, Darmo et al., 2007), spectral gain narrowing (Jukam et al., 2008), and identified the charge injection process from the injector layer into the upper laser state as being coherent, i.e., through resonant tunneling (Eickemeyer et al., 2002).

Quantum dots are semiconductors with charge carriers confined in all three dimensions (Efros and Rosen, 2000; Harrison, 2002). In QDs the electron and hole energy states become discrete due to the 3D spatial confinement. As a result, QDs have optical responses very different from the bulk material from which they are made. What is more, the QD optical response is tunable: by varying their size (and therefore degree of electronic confinement), the band gaps of QDs can be modified and tuned to cover the entire visible to near-infrared spectral range (Woggon, 1997). As such, they are promising materials for a variety of electro-optic applications, such as LEDs and photovoltaic devices. Knowledge of the processes immediately following photoexcitation is therefore essential, and there has been much interest recently in exciton dynamics in QDs [see Klimov (2007), and references therein], especially in exciton decay, exciton cooling, multiexciton dynamics, and the possible formation of multiexcitons by carrier multiplication.

Many studies of carrier dynamics in semiconductor quantum dots have been performed using ultrafast optical spectroscopy (Klimov, 2007). Most of these studies, typically based on transient absorption or time-resolved fluorescence, have focused on interband transitions in the optical range. The observed optical dynamics in the visible are dictated by both electron and hole dynamics. Infrared transient absorption measurements have been used to study specifically electron dynamics by probing electron intraband transitions. As we will show in the following, THz spectroscopy has proven to be a fruitful approach for the study of exciton and carrier dynamics in quantum dots and quantum dot assemblies.

In quantum dots, the spacing of the discrete electron and hole energy levels is strongly dependent on the size of the quantum dot. The two relevant length scales of the problem are the QD diameter $D$ and the exciton Bohr radius $a_B$ that is the natural physical separation in a bulk crystal between an electron in the conduction band and its corresponding hole in the valence band. Strong confinement in small QDs [ $D<a_B$, see Fig. 31(a)] results in high confinement energies and relatively large energy spacing between the levels, while larger QDs have more closely spaced energy levels. For even larger QDs [ $D\gg a_B$, see Fig. 31(b)], the confinement is weak, and charge carriers can move within the boundaries of the QD volume. For assemblies of quantum dots, the electron or hole wave functions between different QDs can couple, and energy and charge transfer can occur within the system [Fig. 31(c)].

FIG. 31

In the limit of strong confinement, the energy-level spacing depends on the degree of confinement, but also on the band masses of the electrons and holes: a rule of thumb is that the heavier the quasiparticle the more closely spaced the energy levels. In quantum dots made of II–VI materials such as CdSe, the hole band mass is several times that of the electron, so that the valence electronic levels are more closely spaced than the electron levels [see Fig. 32(a)].

FIG. 32

THz spectroscopy was first used to investigate the nature and dynamics of photoexcitations in semiconductor quantum dots by Beard et al. (2002a), for colloidal CdSe quantum dots of various sizes. They note that for QDs in the strong confinement regime ( $<5\mathrm{nm}$ diameter), the photoinduced change in the THz dielectric function is purely real (finite imaginary conductivity; zero real conductivity) and the magnitude increases with the fourth power of the particle radius.

The origin of such a response (purely real dielectric response of photoexcited QDs) was subsequently revealed by Wang et al. (2006) and Dakovski et al. (2007). They argued that for QDs with strong confinement both electrons and holes occupy discrete energy levels separated by at least tens of meV (Efros and Rosen, 2000), i.e., energies significantly larger than the photon energies used to probe the levels ( $1\mathrm{THz}=4\mathrm{meV}$); see Fig. 32(a). This means that THz photons with energy below the intraband transitions of electrons and holes interact with with them nonresonantly. The nonresonant nature of the interaction dictates that no absorption of THz light can occur, which explains the experimental observation of a photoinduced change in the THz dielectric function that is purely real (imaginary conductivity). The amount of photoinduced increase in the dielectric response can be directly related to the polarizability ( $\alpha$) of the exciton.

In a QD, the energy levels of electron and holes scale with the quantum dot radius $R$ as $\propto R^{-2}$, while the Coulomb interaction scales as $\propto R^{-1}$. In the limit of strong confinement, i.e., small $R$, the confinement energies far exceed the Coulomb interactions and the electron and hole comprising an exciton are therefore largely uncorrelated (Efros and Rosen, 2000). Consequently, one can treat the exciton polarizability as the sum of separate contributions from the electron and the hole. The polarizability is a measure of the extent to which the electron and hole wave functions are deformed in an applied electric field. This deformation requires mixing of higher lying states, and occurs more readily when these states are energetically close by. Owing to smaller energy-level spacing, the hole therefore dominates the polarizability at frequencies below the intraband transition energies, giving rise to a real and spectrally flat susceptibility [see Fig. 32(b)]. Typically, the magnitude of the hole polarizability is around ${10}^4{\AA }^3$, scaling as $\alpha \propto R^4$ [see Fig. 32(c)]. A more detailed analysis, taking into account the nonparabolicity of the electronic bands and the electron-hole interaction, yields $\alpha \propto R^{3.6}$. Subsequent studies on PbSe quantum dots (Dakovski et al., 2007) revealed that similar conclusions can be drawn for PbSe QD systems. However, the THz response is no longer dominated by the holes due to the comparable band masses of electron and hole and corresponding similar level spacing for conduction and valence states in lead salts.

In comparison, by investigating the response in the range of 2–7 THz of negatively charged CdSe QDs in solution, Mandal and Chikan (2007) were able to experimentally determine the electron polarizability in 3.2 and 6.3 nm diameter QDs to be $0.5\pm 0.1\times {10}^3$ and $14.6\pm 0.3\times {10}^3{\AA }^3$, respectively, around an order of magnitude smaller than the corresponding polarizabilities of the holes.

THz TDS is evidently capable of selectively probing the hole populations in II–VI (e.g., CdSe) quantum dots. By monitoring the hole polarizability as a function of time after excitation, one can evaluate population dynamics of the hole states. This principle has been exploited by Pijpers et al., (2007, 2008) to study carrier multiplication in highly excited, small gap, InAs QDs. Moreover, coupling femtosecond optical pump-THz probe measurements with ultrafast optical spectroscopies (such as time-resolved luminescence or transient absorption) can therefore help separate hole from electron dynamics, which is generally challenging. Hendry, Koeberg, Wang et al. (2006) employed a 400 nm optical pump-THz probe scheme, coupled with time-resolved, band gap luminescence measurements, to extract cooling rates of photoexcited hot holes and hot electrons in colloidal CdSe QDs. These cooling rates are inferred from the rise times of the experimental signals ${\tau }_{\mathrm{rise}}$, for various QD diameters; see Fig. 33. A clear step in the luminescence rise time is observed for diameters $>3.5\mathrm{nm}$. For particles larger than this diameter, the excitation photon energy (3.2 eV) is sufficient to excite electrons into the $1P$ excited state, which subsequently relax into the $1S$ state. The delayed appearance of ground state electrons is reflected in a slower rise of the time-resolved photoluminescence response. Interestingly, a simultaneous slowing down in the THz response is also observed. This observation of delayed hole cooling upon higher electronic excitation signifies energy transfer from electrons to holes (the holes are reexcited upon electron cooling, resulting in an apparent slowing down of hole cooling). These observations constituted the first direct evidence of electron-to-hole (or “Auger”) energy transfer as a key electron cooling mechanism in QDs. A quantitative analysis of the data using a simple rate equation model (lines in figure), allowed for the determination of hole cooling and electron-to-hole energy transfer times of $330\pm 50\mathrm{fs}$ and $1\pm 0.15\mathrm{ps}$, respectively.

FIG. 33.

While for CdSe QDs in the strong confinement regime ( $<5\mathrm{nm}$ diameter), the photoinduced change in the THz dielectric function was observed to be purely real (finite imaginary conductivity; zero real conductivity), in the weak confinement regime ( $>5\mathrm{nm}$ diameter), a complex dielectric response was observed (finite imaginary and real conductivity). Beard et al. (2002a) interpreted the results for the larger particles in terms of a modified Drude transport theory, adapted to include a term accounting for surface scattering.

Larger conductivities have been observed for self-assembled or arrayed (Muller et al., 2008) QD assemblies. The use of THz to study QD assemblies is motivated by the notion that THz TDS is very sensitive to the degree of electronic coupling between the quantum dots (Beard et al., 2003; Crooker et al., 2003; Cooke et al., 2004; Murphy et al., 2006). Strong coupling will allow electrons or holes to “hop” between QDs, giving rise to a real component of the complex conductivity (or equivalently an imaginary component of the permittivity or susceptibility) even at frequencies below the intraband transitions of isolated QDs. Initial studies on quantum dot molecules (Boucaud et al., 2000), consisting of two vertically coupled InAs self-assembled quantum dots separated by a GaAs barrier, revealed the electronic coupling between the dot states through an intraband absorption at THz frequencies. In laterally ordered chains of self-assembled InGaAs quantum dots (Cooke et al., 2004), a large anisotropy in the transient photoconductive response was observed depending on the polarization of the THz probe pulse with respect to the orientation of the dot chains, indicating that the increased THz response indeed arises from the interdot coupling.

In arrays (or QD solids), THz TDS allows both the degree of interdot electronic coupling and the carrier dynamics (affected by the delocalization of carriers between dots) to be extracted simultaneously. In disordered arrays of 3.2 nm diameter InP QDs, Beard et al. (2003) reported a sixfold increase in the transient photoconductivity when the average QDs are separated by 0.9 nm compared to arrays with 1.8 nm separation (the different separation is introduced by modifying the capping layer of the QD). This change in the transient response was interpreted using the Drude-Smith transport model (see Sec. IV A), a derivative of the Drude model which takes into account varying degrees of charge localization. By varying pump and probe delays, Beard et al. (2003) also found triexponential decay kinetics for the photospecies, with longer trapping times observed for closer spacing, attributed to the increased tunneling probability between closely spaced QDs, allowing carriers to escape from trap cites. Similarly, Murphy et al. (2006) observed similar effects in PbSe QD arrays.