Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy

Time-resolved, pulsed terahertz spectroscopy has developed into a powerful tool to study charge carrier dynamics in semiconductors and semiconductor structures over the past decades. Covering the energy range from a few to about 100 meV, terahertz radiation is sensitive to the response of charge quasiparticles, e.g., free carriers, polarons, and excitons. The distinct spectral signatures of these different quasiparticles in the THz range allow their discrimination and characterization using pulsed THz radiation. This frequency region is also well suited for the study of phonon resonances and intraband transitions in low-dimensional systems. Moreover, using a pump-probe scheme, it is possible to monitor the nonequilibrium time evolution of carriers and low-energy excitations with sub-ps time resolution. Being an all-optical technique, terahertz time-domain spectroscopy is contact-free and noninvasive and hence suited to probe the conductivity of, particularly, nanostructured materials that are difficult or impossible to access with other methods. The latest developments in the application of terahertz time-domain spectroscopy to bulk and nanostructured semiconductors are reviewed.

Figure 1

Schematic representation of an optical pump-THz probe experimental setup. A train of ultrashort laser pulses enters the setup from the upper left corner and is split into three: the excitation beam, and two beams for the generation and detection of THz pulses. A mechanical chopper is placed in either the excitation or generation beam, depending on the type of experiment. In the balanced detection scheme, the THz field-induced rotation of the polarization of the detection pulse is measured as indicated. Two optical delay lines are used to control the delay times between the three pulses.

Figure 2

Schematic of typical sample excitations achieved in an optical pump-THz probe experiment. Experiments with (a) homogeneous excitation and (b) interfacial excitation can be treated analytically.

Figure 3

Drude response for two regimes of scattering rates relative to the probe frequency window. Upper row: complex dielectric function and conductivity for a relatively low scattering rate $\gamma$; lower row: the same for a very high scattering rate. All solid (dotted) lines indicate real (imaginary) part of the response. The dotted lines indicate zero point. ${\gamma }_0$ is a frequency unit. The frequency axes of the conductivity plots are normalized for ${\gamma }_0$, while for the dielectric functions they are normalized for the screened plasma frequency ${\omega }_{sp}$ and ${ϵ}_{\infty }$ is set to 1.

Figure 4

Resonant response with different line widths and resonance frequencies relative to the probe frequency window. Upper row: complex dielectric response and conductivity for fully resonant response; middle row: the low-frequency side of the resonance; lower row: completely off-resonant response. All solid (dotted) lines indicate real (imaginary) part of the response. The dotted lines indicate zero point. ${\gamma }_0$ is a frequency unit. All frequency axes are normalized for ${\gamma }_0$. The dielectric response has been displaced by $-{ϵ}_{\infty }$ for clarity.

Figure 5

Time dependence of signals in a (a) transient photoconductivity measurement and (b) THz-TDS measurement, both for a Drude electron gas undergoing a step increase in density at time zero.

Figure 6

Number of generated electron-hole pairs per photon in bulk PbS ($E_{\mathrm{gap}}=0.42\mathrm{eV}$) as a function of photon energy (circles). Also shown is the results of a device measurement (sloping solid line) from 266. The inset shows the conductivity and the fit to the Drude model and corresponding plasma frequency and scattering time. From 238.

Figure 7

Real and imaginary parts of the conductivity of an $n$-doped Si wafer with fits to the Cole-Davidson model (solid line) and the Drude model (dashed line). From 127.

Figure 8

Conductivity of an organic crystal pentacene at room temperature; the dashed lines are guides to the eye for a flat, frequency-independent response. From 226.

Figure 9

Real conductivity (filled squares) and imaginary conductivity (open squares) of an MEH-PPV/PCBM blended sample, measured 10 ps after photoexcitation. Lines are the results of model calculations. From 107.

Figure 10

Plot of the scattering rate $\gamma$ as a function of time after photogeneration of carriers in GaAs (corresponding to the frequency of the peak of the imaginary part of the conductivity). Variations as a function of time are attributed to phonon-assisted intravalley relaxation processes. Inset (a) shows the THz absorption assisted by LO-phonon absorption and emission. Inset (b) shows that after the electron distribution has relaxed below the longitudinal optical phonon frequency, ${\omega }_{\mathrm{LO}}$, THz absorption is only assisted by LO-phonon absorption. From 18.

Figure 11

Normalized THz electric fields transmitted through a GaAs sample before (thin line) and 10 ps after photoexcitation (open circles) under low (a) and high (b) electric-field strengths. The solid lines represent the calculated fraction of photoexcited electrons in the central $\Gamma$ valley as a function of time. From 274.

Figure 12

THz pump and THz probe experiment on doped InSb: (a) Spectrally averaged THz absorption as a function of pump-probe delay at a sample temperature of 80 K; (b) frequency-dependent absorption coefficient at a sample temperature of 200 K, featuring an increasing Drude part and a rising phonon population after cooling of hot electrons. From 112.

Figure 13

(a) Imaginary part of the inverse dielectric function of InP vs frequency for various pump-probe delays. The grey region indicates the Reststrahlenband; (b) visualization of the transition of bare charges to a correlated screened charge carrier plasma; (c) plot of the buildup time $\tau$ for plasmon-phonon coupling vs excitation density; (d) spectral positions of the $L_{+}$ and $L_{-}$ modes for various excitation densities. From 119.

Figure 14

Self-induced polaron potential of a polaron at (a) weak and (b) strong electric fields. Left side: contour plot of the polaron potential. Right side: polaron potential in coordinates relative to the quasiequilibrium position of the electron (dot); the circles illustrate the electron oscillation within the polaron potential at high electric fields. (c) Measured transients for a pump-probe delay of 77 fs; the bottom graph shows the time-integrated transmission change $\Delta T/T$; (d) transmission change $\Delta T/T$ as a function of pump-probe delay $\tau$. From 78.

Figure 15

(a) Linear absorption spectrum and refractive index of a rubrene crystal; the inset shows its molecular structure; (b) frequency-resolved real part of the transmsission change $\Delta E/E$ as a function of pump-probe delay; (c) close-up of the transmission change of the resonance around 15.5 THz; (d) experimental (squares) and modeled (solid line) shift of the resonance frequency vs pump-probe delay. From 165.

Figure 16

Transient terahertz spectra of charge carriers in GaAs quantum wells at different pump-probe delays: (a) Nonresonant photoexcitation into the continuum at a sample temperature of 6 K shows the formation of excitons from an unbound charge carrier plasma; (b) resonant generation of exciton states at a sample temperature of 40 K and subsequent ionization of excitons. The respective left sides of the plots show the real part of the conductivity, the right side the real part of the dielectric constant of the charge carrier gas. From 140.

Figure 17

(a) Evolution of the real (solid line) and imaginary (dashed line) conductivity after photoexcitation of low charge carrier densities in ZnO at 30 K; (b) complex conductivity 15 ps after photoexcitation showing the Drude response of free carriers (solid and dashed lines are real and imaginary conductivity following the Drude model); (c) complex conductivity 200 ps after photoexcitation showing the off-resonant response of intraexcitonic transitions (solid and dashed lines are real and imaginary conductivity of an excitonic response). From 101.

Figure 18

(a) Phase diagram of an electron-hole system in silicon; (b) real part of the dielectric function (left) and conductivity (right) of the photoexcited plasma at high excitation densities for various pump-probe delays. From 276.

Figure 19

(a) Sketch of the first intraexcitonic transitions of ${\mathrm{CuO}}_2$ orthoexcitons and paraexcitons in momentum space; (b) pump-induced absorption change $\Delta \alpha$ for various pump-probe delays featuring the 1s–2p transitions of ortho and para excitons; (c) upper panel: profile of the exciting THz pulse; lower panel: Reemitted THz fields for driving field strengths reaching from $0.07\mathrm{MV}/\mathrm{cm}$ (i) to $0.5\mathrm{MV}/\mathrm{cm}$ (vi); (d) calculated inversion (solid line) and polarizability (shaded area) of the paraexciton $1s\mathrm{\text{−}}2p$ two-level system. From 182.

Figure 20

(a) Real part and (b) imaginary part of the conductivities of photoexcited charge carriers in bulk and nanocrystalline silicon; the lower panels show exemplary plots of the Drude-Smith model for various values of $c$: (c) real conductivity; (d) imaginary conductivity—both vs frequencies normalized to $\omega \tau$. From 50.

Figure 21

(a) Scheme of the composition of the microcrystalline samples under investigation by 73, (b) Mobility spectra of charge carriers in microcrystalline silicon for temperatures of 300, 100, and 20 K at a pump-probe delay of 50 ps. Upper panel: real part, lower panel: imaginary part. Symbols: experimental data, thick lines: Monte Carlo simulations incorporating backscattering, thin lines: fits to hopping model. From 73.

Figure 22

(a) Complex conductivity of photoexcited charge carriers in silicon microparticles for three pump fluences (upper panel: real part, lower panel: imaginary part) and (b) exemplary conductivity plots of the Drude model and a plasmon resonance. From 218.

Figure 23

(a) Real part and (b) imaginary part of the conductivity of photoexcited charge carriers in GaAs nanowires for various pump-probe delays. A redshift of the localized surface plasmon resonance frequency is observed (indicated by the arrows): trapping at surface states decreases the charge carrier density over time and hence the plasmon frequency. From 230.

Figure 24

(a) Dark-field optical micrograph of oriented germanium nanowires. (b) Measured differential transmission for various polarizer angles $\Theta$. Inset: Sketch of the geometry of nanowire orientation, THz electric field, and polarizer angle $\Theta$. From 272.

Figure 25

(a) Scanning electron micrograph of the surface of nanoporous InP; the inset shows a three-dimensional cartoon of the porres. Simulations of the spatial distribution of the carrier density in nanoporous InP at 100 K: (b) Before photoexcitation regions close to the pores are depleted of charge carriers due to band bending. (c) After photoexcitation, the bands flatten out and charge carriers are distributed more homogenously. From 187.

Figure 26

Mobility spectra of localized particles calculated by Monte Carlo simulations for various values of $\alpha$ (upper panels: real part, lower panels: imaginary part). (a) $p_r=1$, (b) $p_r=0.5$. From 213.

Figure 27

(a) Sketch of the electron wave functions in subbands of quantum wells; (b) schematic of a optical pump-THz probe experiment on multiple-quantum-wells samples. From 85 and 192.

Figure 28

Fano effect in intersubband transitions of GaAs quantum wells: the differential transmission of the THz probe field in time shows a beating (a) and a broad peak in the normalized transmission spectrum, additionally to the intersubband resonance (b). The absorption spectrum only features the IS transition (c). From 85.

Figure 29

Rabi oscillations of intersubband transitions for two different electric-field strengths of the driving field [(a) and (b)]: the thin line indicates the driving field, the thick lines the reemitted field. (c) Rabi frequency as a function of electric-field amplitude. (d) Calculated time evolution of the applied field (thin line), reemitted field (thick line), and population inversion (shaded area). From 192.

Figure 30

Terahertz time-domain spectroscopy of a THz quantum-cascade laser: (a) Schematic of a quantum-cascade laser; (b) THz probe transmitted through an unbiased QCL; (c) change in the transmitted probe through a biased QCL; (d) absorption spectra, and (e) phase of the bias-modulated probe. From 172.

Figure 31

Different regimes of charge carriers in quantum dot (assemblies): (a) In the limit of strong confinement, the QD radius is smaller than the exciton Bohr radius and strong confinement occurs. (b) For weak confinement, charge carriers can move within the confined space of the nanocrystal. (c) For QD assemblies, electronic coupling between adjacent quantum dots (owing to the finite overlap of charge carrier wave functions) allows for finite electrical conductivity over macroscopic distances.

Figure 32

(a) Energy-level scheme for a QD in the limit of strong confinement. All transitions occur at energies exceeding that of the low-frequency THz spectrometer (small arrows). As a result, the excitonic transitions are interrogated off-resonant. (b) The resulting change in the real $\Delta {\chi }^{\prime }$ and imaginary $\Delta {\chi }^{\prime \prime }$ parts of the photoinduced sheet susceptibility of a CdSe QD dispersion. The light lines are experimental data and the dashed line represents a purely real, frequency-independent change in the susceptibility that corresponds to purely imaginary conductivity. (c) Polarizability of single excitons in photoexcited CdSe QDs as a function of the QD radius. The experimental data (circles) are well described with a simple $R^4$ scaling (solid line). The QD polarizability was calculated by perturbative multiband tight-binding calculations (dashed line). From 287.

Figure 33

Cooling times of CdSe nanoparticles as a function of mean particle diameter. From 106.

Figure 34

Schematic representation of the graphene band structure. Dotted arrows: interband transitions. Solid bent arrow: intraband transitions.

Figure 35

Complex dielectric response of unexcited films of high pressure CO grown NTs. From 144.