Terahertz Time-Domain Spectrometer

1. Theory

• Definition: A terahertz time-domain spectrometer generates sub-picosecond pulses of terahertz radiation, passes them through or reflects them from a sample, and records the electric field against time, giving amplitude and phase over about 0.1 to 10 $\mathrm{THz}$ (Kubascik et al.; Whelan et al.).
• Free carriers: Terahertz photons, of a few millielectronvolts, interact mainly with free carriers, so the complex conductivity follows from the change in the transmitted pulse (Kubascik et al.; background knowledge).
• Thin films: For films much thinner than the wavelength and skin depth, the Tinkham formula gives the sheet conductance directly (Kubascik et al.).
• Drude analysis: Fitting the frequency dependence gives scattering time, and with the effective mass, mobility and carrier density, without contacts (Whelan et al.; Murata).

2. Working Principle

• Generation: Femtosecond laser pulses drive photoconductive antennas, nonlinear crystals, spintronic emitters or air plasmas (Kubascik et al.; Whelan et al.).
• Detection: Electro-optic sampling or photoconductive antennas measure the field at each delay of a probe pulse (Kubascik et al.).
• Reference: A bare substrate scan, or an internal substrate echo, serves as the reference (Kubascik et al.; Whelan et al.).
• Geometries: Transmission at normal incidence, or ellipsometric reflection at oblique incidence for conducting substrates (Murata).

3. Applications

• Wafer-scale mapping: Contactless sheet-conductance maps of graphene and thin metal films (Whelan et al.; Kubascik et al.).
• Semiconductor epitaxy: Carrier density and mobility of GaAs layers and GaN substrates (Murata).
• Anisotropy: Direction-dependent conductivity of epitaxial films (Kubascik et al.).
• Ultrafast dynamics: Optical-pump terahertz-probe studies of carrier lifetimes (background knowledge, Jepsen et al.).

Terahertz Conductivity

Name	Formula	Related Formulae	Variables/Constants
Terahertz units	$\lambda =\frac{c}{f},E=hf,\tilde{\nu }=\frac{f}{c}$	$1\mathrm{THz}=300\mu \mathrm{m}=4.14\mathrm{meV}=33.3\mathrm{c}{\mathrm{m}}^{-1}$	$\lambda :wavelength$ $f:frequency$ $E:photonenergy$ $\begin{array}{r}\tilde{\nu }:wavenumber;conversions(computed);\\ Kubasciketal.:analysisover0.2to2.5\\ \mathrm{THz};Whelanetal.:afibre-coupled\\ spectrometerat0.2to1.5\mathrm{THz}andan\\ air-plasmasystemat2to10\mathrm{THz}\end{array}$ $\begin{array}{r}(computed:1\mathrm{THz}photonscarry4.14\mathrm{meV},\\ farbelowsemiconductorgaps,sothey\\ probefreecarriers,notinterband\\ absorption)\end{array}$
Thin-film (Tinkham) transmission	$\sigma \left(\omega \right)=\frac{n_S\left(\omega \right)+n_A}{Z_{0}d}\left(\frac{1}{t^{\prime }\left(\omega \right)}-1\right)$	${\sigma }_s=\sigma d=\frac{n_S+1}{Z_0}\left(\frac{1}{t^{\prime }}-1\right),R_s=\frac{1}{{\sigma }_s}$	$\sigma :filmconductivity$ $n_S:substrateindex$ $n_A:ambientindex(1)$ $Z_0:free-spaceimpedance(about377\Omega )$ $\begin{array}{r}d:filmthickness,t^{\prime }:complex\\ transmissionoffilmonsubstraterelative\\ tobaresubstrate\end{array}$ ${\sigma }_s:sheetconductance$ $\begin{array}{r}{R}_s:sheetresistance;Kubasciketal.:this\\ equationasathin-filmapproximationfor\\ highlyconductivefilmswitherrorbelow\\ 0.02percent,substrateindices3.34\\ (GaP),3.6(GaAs)and3.42(Si);sheet\\ form(computedrearrangement)\end{array}$ $\begin{array}{r}(computed:t^{\prime }=0.5onsilicongives{\sigma }_s=\\ 11.7mS,R_s=85\Omega /\mathrm{sq})\end{array}$
Drude conductivity	$\sigma \left(\omega \right)=\frac{{\sigma }_0}{1-i\omega \tau },{\sigma }_0=\frac{n{e}^2\tau }{m^{\ast }}$	$\mu =\frac{e\tau }{m^{\ast }},{\omega }_p^2=\frac{n{e}^2}{{\epsilon }_0{\epsilon }_{\infty }{m}^{\ast }}$	${\sigma }_0:DCconductivity$ $\tau :scatteringtime$ $n:carrierdensity,m^{\ast }:effectivemass$ $\mu :mobility$ $\begin{array}{r}{\omega }_p:plasmafrequency,{\epsilon }_{\infty }:high-frequency\\ permittivity;Kubasciketal.:Drudeform,\\ \tau mostly0.32\mathrm{fs}forCuMnAs(upto25\mathrm{fs}\\ insomefilms),{\sigma }_{THz,0}of3.0to8.1kS/cm\\ nearlyconstantwithfrequency;Whelanet\\ al.:DCsheetconductivity,scattering\\ time,carrierdensity,mobilityandFermi\\ velocityfromDrudefits;mobilityand\\ plasmarelations(backgroundknowledge,\\ Jepsenetal.)\end{array}$ $\begin{array}{r}(computed:\mu =2000\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})with{m}^{\ast }=\\ 0.067m_{0}is\tau =76\mathrm{fs}and\omega \tau =0.48at1\\ \mathrm{THz};n=4\times 10^{18}\mathrm{c}{\mathrm{m}}^{-3}GaAshas{f}_p=21\mathrm{THz})\end{array}$

Reference, Depth and Validation

Name	Formula	Related Formulae	Variables/Constants
Reference-free echo analysis	$\Delta t_{echo}=\frac{2n_S{d}_S}{c}$	$\delta f\approx \frac{1}{T_{window}}$	$\begin{array}{r}\Delta t_{echo}:delayofthefirstinternal\\ substratereflection\end{array}$ $d_S:substratethickness$ $\delta f:frequencyresolution$ $\begin{array}{r}{T}_{window}:timewindowanalysed;Whelanetal.:\\ ACconductivityfromthedirectly\\ transmittedpulseanditsinternal\\ reflectioninthesubstrate,avoidinga\\ referencescan,withpropagation\\ corrections;relations(background\\ knowledge)\end{array}$ $\begin{array}{r}(computed:a500\mu \mathrm{m}siliconsubstrategives\\ echoes11.4\mathrm{ps}apart,limitingresolution\\ toabout88\mathrm{GHz}ifthewindowstopsbefore\\ theecho)\end{array}$
Penetration and skin depth	${\delta }_{skin}=\sqrt{\frac{2}{\omega {\mu }_0\sigma }}$	${\delta }_{skin}\gg d\Rightarrow \text{thin-film formula valid}$	${\delta }_{skin}:skindepthfor\omega \tau \ll 1$ $\begin{array}{r}{\mu }_0:vacuumpermeability;form(background\\ knowledge);Murata:entrancelengthof5.8\\ \mu \mathrm{m}at1\mathrm{THz}andabouttentimesthe0.5\mu \mathrm{m}\\ filmthicknessforp-typeGaAs,giving\\ satisfactorysensitivity\end{array}$ $\begin{array}{r}(assumed:p-GaAsat1\times 10^{18}\mathrm{c}{\mathrm{m}}^{-3}with\mu =150\\ \mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s}),\sigma =2400S/m;computed:{\delta }_{skin}=\\ 10.3\mu \mathrm{m}at1\mathrm{THz})\end{array}$
Comparison with DC	${\sigma }_{DC}=\frac{l}{wd{R}_{DC}}$	$\frac{2\Delta \sigma }{\bar{\sigma }}=29.3\text{per cent (THz and DC, CuMnAs 20 nm on GaP)}$	${\sigma }_{DC}:DCconductivityofaHallbar$ $l,w:lengthandwidth$ $\begin{array}{r}{R}_{DC}:resistance,\Delta \sigma /\bar{\sigma }:relative\\ anisotropy;Kubasciketal.:Hallbars250\\ by50\mu \mathrm{m}atabout100\mu \mathrm{A},equalanisotropy\\ fromTHzandDCforCuMnAsonGaP,DCon\\ Sidivergingafter7to8monthsfrom\\ oxidation;Murata:Drudecarrierdensity\\ andmobilitywithinplusorminus10per\\ centofHallvaluesforGaAsfilms(4\times 10^{18}\\ n-type,1.0\mu \mathrm{m};1\times 10^{18}p-type,0.5\mu \mathrm{m})and\\ closeagreementonGaNsubstrates\end{array}$ $\begin{array}{r}(computed:5\times 10^{5}S/mis5kS/cm,withinthe\\ fitted3.0to8.1kS/cm)\end{array}$

Instrument

Name	Formula	Related Formulae	Variables/Constants
Spot size and diffraction	$D_{spot}\approx \frac{1.22\lambda }{NA}$	$D_{FWHM}\approx 800\text{to}865\mu \mathrm{m}(\text{Kubascik et al.})$	$D_{spot}:focalspot$ $\begin{array}{r}NA:numericalaperture;Kubasciketal.:\\ about800\mu \mathrm{m}inMethods,about860\mu \mathrm{m}inthe\\ maintext,865plusorminus63\mu \mathrm{m}and828\\ \mu \mathrm{m}insupplementaryfigures(seetheQ/A),\\ beamwanderabout40\mu \mathrm{m};diffraction\\ relation(backgroundknowledge)\end{array}$ $\begin{array}{r}(computed:at1\mathrm{THz}withNA=1the\\ diffractionlimitis366\mu \mathrm{m})\end{array}$
Emitter and detection	$E_{THz}\left(t\right)\text{sampled by delaying the probe pulse}$	$\tilde{E}\left(\omega \right)=\int E\left(t\right)e^{i\omega t}dt,t^{\prime }=\frac{{\tilde{E}}_{sample}}{{\tilde{E}}_{ref}}$	$E_{THz}(t):terahertzfieldtransient$ $\begin{array}{r}\tilde{E}:itsFouriertransform;Kubasciket\\ al.:spintronicemitterpumpedat1030\mathrm{nm}\\ with170\mathrm{fs}pulses,0.15mJ/c{\mathrm{m}}^2,10\mathrm{kHz},\\ electro-opticsamplingin2\mathrm{mm}110-cut\\ GaP,0.2degreerotationstage;Whelanet\\ al.:fibre-coupledandair-plasmasystems;\\ time-domainsamplingandtransform\\ (backgroundknowledge,Jepsenetal.)\end{array}$ $\begin{array}{r}(computed:a0.2to2.5\mathrm{THz}bandspans\\ 0.83to10.3\mathrm{meV})\end{array}$

Important Questions

Ques: What does a terahertz time-domain spectrometer measure?

The electric field of a picosecond terahertz pulse after it interacts with a sample, giving amplitude and phase at each frequency. A femtosecond laser drives an emitter, such as the spintronic emitter pumped at 1030 $\mathrm{nm}$ used by Kubascik et al., and a delayed probe pulse samples the field in an electro-optic crystal, 2 $\mathrm{mm}$ of GaP in their case. Fourier transforming sample and reference transients gives the complex transmission (background knowledge, Jepsen et al.). At 1 $\mathrm{THz}$ the wavelength is 300 $\mu \mathrm{m}$ and the photon energy 4.14 $\mathrm{meV}$ (computed), so the radiation probes free carriers rather than interband transitions. Whelan et al. used a fibre-coupled system from 0.2 to 1.5 $\mathrm{THz}$ and an air-plasma system from 2 to 10 $\mathrm{THz}$. Reference: P. Kubascik et al., ‘Terahertz probing of anisotropic conductivity and morphology of CuMnAs epitaxial thin films’, arXiv:2303.15268 (fetch-tool summary); P. R. Whelan et al., ‘Reference-free THz-TDS conductivity analysis of thin conducting films’, Opt. Express 28, 28819 (2020) (abstract) (fetch-tool summary); P. U. Jepsen, D. G. Cooke and M. Koch, ‘Terahertz spectroscopy and imaging - Modern techniques and applications’, Laser Photonics Rev. 5, 124 (2011) (background knowledge; not opened).

Ques: How is sheet conductance obtained from the transmission?

With the thin-film, or Tinkham, formula. For a conducting film much thinner than the wavelength on a substrate of index $n_S$, $\sigma d=[(n_S+1)/Z_0](1/t^{\prime }-1)$, where $t^{\prime }$ is the transmission relative to the bare substrate and $Z_0$ about 377 $\Omega$ (Kubascik et al.). On silicon, with index 3.42, a relative transmission of 0.5 means 11.7 mS, a sheet resistance of 85 $\Omega /\mathrm{sq}$ (computed). Kubascik et al. describe it as an approximation for highly conductive films, with error below 0.02 per cent for CuMnAs, whose conductivity is about $5\times 10^5$ $S/m$. The film must also be thinner than the skin depth, about 10 $\mu \mathrm{m}$ for moderately doped GaAs at 1 $\mathrm{THz}$ (assumed conductivity; computed). Reference: P. Kubascik et al., ‘Terahertz probing of anisotropic conductivity and morphology of CuMnAs epitaxial thin films’, arXiv:2303.15268 (fetch-tool summary); P. U. Jepsen, D. G. Cooke and M. Koch, ‘Terahertz spectroscopy and imaging - Modern techniques and applications’, Laser Photonics Rev. 5, 124 (2011) (background knowledge; not opened).

Ques: How do Drude fits give mobility and carrier density?

From the frequency dependence of the complex conductivity. In the Drude model ${\sigma }_0/(1-i\omega \tau )$, the real part falls and the imaginary part rises with frequency on a scale set by $\tau$, so a broadband fit gives both ${\sigma }_0$ and $\tau$; with the effective mass, $\mu =e\tau /m^{\ast }$ and $n={\sigma }_0/(e\mu )$ (background knowledge). The fit works best when $\omega \tau$ is near one in the band: a GaAs layer with 2000 $\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$ has $\tau =76$ $\mathrm{fs}$ and $\omega \tau =0.48$ at 1 $\mathrm{THz}$ (computed). For CuMnAs, Kubascik et al. report $\tau$ mostly 0.32 $\mathrm{fs}$, giving $\omega \tau$ of only 0.002 at 1 $\mathrm{THz}$ (computed), so the conductivity is flat and $\tau$ is poorly determined. Whelan et al. extract mobility, density and Fermi velocity for graphene this way. Reference: P. Kubascik et al., ‘Terahertz probing of anisotropic conductivity and morphology of CuMnAs epitaxial thin films’, arXiv:2303.15268 (fetch-tool summary); P. R. Whelan et al., ‘Reference-free THz-TDS conductivity analysis of thin conducting films’, Opt. Express 28, 28819 (2020) (abstract) (fetch-tool summary); P. U. Jepsen, D. G. Cooke and M. Koch, ‘Terahertz spectroscopy and imaging - Modern techniques and applications’, Laser Photonics Rev. 5, 124 (2011) (background knowledge; not opened).

Ques: How can the measurement avoid a reference scan?

By using the substrate’s own internal reflection as the reference. Whelan et al. compare the directly transmitted pulse with the transient that reflects once back and forth inside the substrate, which passes through the film again, so the ratio carries the film’s conductivity without a separate bare-substrate scan. This removes errors from substrate thickness variation and from drift between scans, which matter in wafer mapping. The echo arrives $2n_S{d}_S/c$ after the main pulse, 11.4 $\mathrm{ps}$ for 500 $\mu \mathrm{m}$ of silicon (computed), and if the analysis window must stop before it, the frequency resolution is limited to about 88 $\mathrm{GHz}$ (computed). Whelan et al. note that propagation correction terms are needed for accurate parameters. Reference: P. R. Whelan et al., ‘Reference-free THz-TDS conductivity analysis of thin conducting films’, Opt. Express 28, 28819 (2020) (abstract) (fetch-tool summary).

Ques: How well does terahertz analysis agree with Hall and DC measurements?

Within about 10 per cent for semiconductors, and closely for anisotropy in metals. Murata used terahertz time-domain spectroscopic ellipsometry, at shallow incidence for large polarization changes, on GaAs films on semi-insulating GaAs: n-type at $4\times 10^{18}$ $\mathrm{c}{\mathrm{m}}^{-3}$ and 1.0 $\mu \mathrm{m}$, p-type at $1\times 10^{18}$ $\mathrm{c}{\mathrm{m}}^{-3}$ and 0.5 $\mu \mathrm{m}$. Drude fits gave carrier density and mobility within plus or minus 10 per cent of Hall values, and GaN substrates also agreed closely. Kubascik et al. found the same 29.3 per cent anisotropy in CuMnAs on GaP from terahertz and from DC Hall bars 250 by 50 $\mu \mathrm{m}$; on silicon the DC values diverged after 7 to 8 months, attributed to oxidation. Contactless methods thus avoid contact artefacts but rely on known effective masses. Reference: Murata Manufacturing, ‘Noncontact Measurement of Physical Properties of Semiconductors by Terahertz Time-Domain Spectroscopic Ellipsometry’, metamorphosis no. 15, Paper Review 1 (fetch-tool summary); P. Kubascik et al., ‘Terahertz probing of anisotropic conductivity and morphology of CuMnAs epitaxial thin films’, arXiv:2303.15268 (fetch-tool summary).

Ques: Is the terahertz spot about 800 or 865 microns?

Kubascik et al. give about 800 $\mu \mathrm{m}$ in the methods, about 860 $\mu \mathrm{m}$ in the main text, and 865 plus or minus 63 and 828 $\mu \mathrm{m}$ in supplementary figures. These are consistent within the stated 63 $\mu \mathrm{m}$ uncertainty, and differences probably reflect separate measurements of the beam on different days or at different positions (my reading); 865 $\mu \mathrm{m}$, with its uncertainty, is the best-documented value. The spot is set by diffraction: even at 1 $\mathrm{THz}$ with a numerical aperture of 1, the limit is about 366 $\mu \mathrm{m}$ (computed), and lower frequencies focus to larger spots, so terahertz maps resolve hundreds of micrometres, not micrometres. The paper also reports one-fold anisotropy from about 40 $\mu \mathrm{m}$ beam wander and two anisotropy values, 9.1 and 10.7 per cent, for GaAs-based films. Reference: P. Kubascik et al., ‘Terahertz probing of anisotropic conductivity and morphology of CuMnAs epitaxial thin films’, arXiv:2303.15268 (fetch-tool summary); P. U. Jepsen, D. G. Cooke and M. Koch, ‘Terahertz spectroscopy and imaging - Modern techniques and applications’, Laser Photonics Rev. 5, 124 (2011) (background knowledge; not opened).

Ques: What limits terahertz conductivity measurements on semiconductor wafers?

Substrate conductivity, film thickness relative to skin depth, and assumed effective masses. The substrate must transmit terahertz radiation, so semi-insulating or lightly doped substrates are used; a heavily doped substrate absorbs the pulse, which is why Murata uses reflection ellipsometry at oblique incidence. Murata quotes an entrance length of 5.8 $\mu \mathrm{m}$ at 1 $\mathrm{THz}$ and found the p-type film’s entrance length about ten times its 0.5 $\mu \mathrm{m}$ thickness, giving good sensitivity. For heavily doped layers the plasma frequency can enter the band; $4\times 10^{18}$ $\mathrm{c}{\mathrm{m}}^{-3}$ GaAs gives about 21 $\mathrm{THz}$ (computed), above typical bandwidths. Converting conductivity to mobility needs the effective mass, so errors in mass scale directly into mobility (background knowledge). Reference: Murata Manufacturing, ‘Noncontact Measurement of Physical Properties of Semiconductors by Terahertz Time-Domain Spectroscopic Ellipsometry’, metamorphosis no. 15, Paper Review 1 (fetch-tool summary); P. U. Jepsen, D. G. Cooke and M. Koch, ‘Terahertz spectroscopy and imaging - Modern techniques and applications’, Laser Photonics Rev. 5, 124 (2011) (background knowledge; not opened).

References

1. P. Kubascik, A. Farkas, K. Olejnik, T. Troha, M. Hyvl, F. Krizek, D. C. Joshi, T. Ostatnicky, J. Jechumtal, E. Schmoranzerova, R. P. Campion, J. Zazvorka, V. Novak, P. Kuzel, T. Jungwirth, P. Nemec and L. Nadvornik, “Terahertz probing of anisotropic conductivity and morphology of CuMnAs epitaxial thin films,” arXiv:2303.15268. Available: https://arxiv.org/pdf/2303.15268
2. P. R. Whelan, Q. Shen, D. Luo, M. Wang, R. S. Ruoff, P. U. Jepsen, P. Boggild and B. Zhou, “Reference-free THz-TDS conductivity analysis of thin conducting films,” Optics Express, vol. 28, no. 20, 28819, 2020, doi: 10.1364/OE.402447.
3. P. U. Jepsen, D. G. Cooke and M. Koch, “Terahertz spectroscopy and imaging - Modern techniques and applications,” Laser & Photonics Reviews, vol. 5, pp. 124-166, 2011 (cited from background knowledge).
4. Murata Manufacturing, “Noncontact Measurement of Physical Properties of Semiconductors by Terahertz Time-Domain Spectroscopic Ellipsometry,” metamorphosis no. 15, Paper Review 1. Available: https://corporate.murata.com/technology/techmag/metamorphosis15/paper/01-02

Related pages in this vault: Spectroscopic Ellipsometer, Hall Effect System, Four-Point Probe, Spectral Reflectometer, Photoreflectance Spectrometer, FTIR Spectrometer, Raman Spectrometer, Scanning Near-Field Optical Microscope (SNOM) and TERS, Picosecond Ultrasonic Film Metrology.