1. Theory

Definition: Photoreflectance is a modulation spectroscopy in which a chopped pump laser periodically changes the surface electric field of a semiconductor, and a weak probe beam measures the resulting small change in reflectance against wavelength (Yastrubchak et al.; background knowledge).
Derivative spectra: Because the change is a derivative of the dielectric function, broad reflectance features become sharp lines at transition energies (background knowledge, Pollak).
Line shapes: Weak fields give Aspnes third-derivative lines; stronger fields give Franz-Keldysh oscillations whose period measures the field (Yastrubchak et al.; Nukeaw).
Contactless: No contacts or sample preparation are needed, and measurements are at room temperature (Yastrubchak et al.).

2. Working Principle

Pump: A laser above the gap creates carriers that screen the built-in surface field, chopped at tens of hertz (Yastrubchak et al.).
Probe: Monochromatic light from a lamp and monochromator is reflected to a photodiode (Yastrubchak et al.).
Detection: A lock-in amplifier measures $Δ R$ at the chopping frequency, and $R$ is recorded separately (background knowledge).
Fitting: Spectra are fitted with Aspnes functions or analysed for FKO extrema, sometimes by FFT (Yastrubchak et al.; Nukeaw).

3. Applications

Band-gap and alloy composition: Transition energies of III-V layers and alloys (Yastrubchak et al.).
Surface and interface fields: Built-in fields at surfaces and heterointerfaces from FKO (Nukeaw; background knowledge).
Heterostructures: Quantum-well and HEMT transitions on wafers before processing (background knowledge, Pollak).
Strain: Strain-induced shifts and splitting of light- and heavy-hole transitions (Yastrubchak et al.).

Modulation and Line Shapes

Name	Formula	Related Formulae	Variables/Constants
Modulated reflectance	$\frac{Δ R}{R} = α (ε_{1}, ε_{2}) Δ ε_{1} + β (ε_{1}, ε_{2}) Δ ε_{2}$	$\frac{Δ R}{R} = \frac{R _{o n} - R _{o f f}}{R _{o f f}}$	$Δ R / R : r e l a t i v e r e f l ec t an ce c han g e c a u se d b y t h e p u m p$ $α, β : S er a p hin coe f f i c i e n t s$ $Δ ε_{1}, Δ ε_{2} : c han g es o f t h e r e a l an d ima g ina r y d i e l ec t r i c f u n c t i o n$ $R_{o n}, R_{o f f} : r e f l ec t an ce w i t h p u m p o n an d o f f; Y a s t r u b c hak e t a l . : Δ R / R = α Δ ε_{1} w i t h t h e β t er m n e g l ec t e d n e a r t h e G a A s g a p; g e n er a l f or m (ba c k g r o u n d k n o w l e d g e, P o l l ak)$ $(co m p u t e d : a t y p i c a l 1 \times 1 0^{- 4} s i g na l o n 30 p er ce n t r e f l ec t an ce i s a c han g e o f 3 \times 1 0^{- 5} in ab so l u t e r e f l ec t an ce)$
Aspnes third-derivative line shape	$\frac{Δ R}{R} = Re [C e^{i ϑ} (ℏ ω - E_{C P} + iγ)^{- n}]$	$n = 2.5 (3-D band-to-band), n = 3 (2-D), n = 2 (excitonic)$	$C : am pl i t u d e$ $ϑ : p ha se$ $ℏ ω : p h o t o n e n er g y$ $E_{C P} : cr i t i c a l - p o in t (t r an s i t i o n) e n er g y$ $γ : b r o a d e nin g$ $n : cr i t i c a l - p o in t t y p e; Y a s t r u b c hak e t a l . : t hi s f or m w i t h n = 2.5 f or a t h r ee - d im e n s i o na l ban d - t o - ban d t r an s i t i o n, u se d f or t h e b e l o w - g a p f e a t u r e; N u k e a w : P R s p ec t r a ana l y se d w i t h t h e A s p n es f u n c t i o n; o t h er n v a l u es (ba c k g r o u n d k n o w l e d g e, P o l l ak)$ $(co m p u t e d : f or n = 2.5 t h e s i g na l f a l l s a s t h e in v er se 2.5 p o w er o f t h e d e t u nin g, so f e a t u r es a r e s ha r p, a f e w γ w i d e)$
Pump and probe	$E_{p u m p} = \frac{1239.84}{λ _{p u m p}} > E_{g}$	$f_{c h o p} \approx 70 Hz, Δ R from lock-in at f_{c h o p}, R from DC$	$E_{p u m p} : p u m p p h o t o n e n er g y$ $λ_{p u m p} : p u m p w a v e l e n g t h$ $f_{c h o p} : c h o pp in g f r e q u e n cy; Y a s t r u b c hak e t a l . : 488 nm a r g o n - i o n p u m p o f n o mina l 50 mW, 250 W ha l o g e n l am p an d m o n oc h r o ma t or a s p r o b e, S i p h o t o d i o d e, 70 Hz c h o pp in g, 2 mm s p o t s, r oo m t e m p er a t u r e; l oc k - in d e t ec t i o n (ba c k g r o u n d k n o w l e d g e)$ $(co m p u t e d : 488 nm i s 2.54 eV, w e l l ab o v e t h e 1.42 eV G a A s g a p)$

Franz-Keldysh Oscillations and Fields

Name	Formula	Related Formulae	Variables/Constants
Electro-optic energy	$ℏ θ = (\frac{e ^{2} ℏ ^{2} F ^{2}}{2 μ})^{1/3}$	$F = \frac{2 μ ( ℏ θ ) ^{3}}{e ℏ}$	$ℏ θ : e l ec t r o - o pt i c e n er g y$ $e : e l e m e n t a r y c ha r g e$ $F : e l ec t r i c f i e l d$ $μ : in t er ban d r e d u ce d e f f ec t i v e ma ss; Y a s t r u b c hak e t a l . : t hi s d e f ini t i o n, ℏ θ_{hh} = 37.25 an d ℏ θ_{l h} = 48.29 meV f or L T - G a A s; in v er s i o n (co m p u t e d r e a r r an g e m e n t)$ $(a ss u m e d : m_{e} = 0.067, m_{hh} = 0.5, m_{l h} = 0.082 m_{0}; co m p u t e d : μ_{hh} = 0.0591 m_{0} g i v es F = 89.5 kV/cm, μ_{l h} = 0.0369 m_{0} g i v es 104.4 kV/cm)$
Franz-Keldysh oscillation extrema	$E_{j} = E_{g} + ℏ θ F_{j}$	$F_{j} = [\frac{3 π}{2} (j - \frac{1}{2})]^{2/3}$	$E_{j} : e n er g y o f t h e j - t h e x t r e m u m ab o v e t h e g a p$ $F_{j} : e f f ec t i v e in d e x; Y a s t r u b c hak e t a l . : t hi s ana l y s i s, t h e g a p f r o m t h e in t er ce pt o f E_{j} a g ain s t F_{j} an d t h e f i e l d f r o m t h e s l o p e, A i r y - f u n c t i o n m o d e l w i t h z an d w a r g u m e n t s; N u k e a w : F K O p er i o d in cr e a s in g w i t h f i e l d, f i e l d s o b t ain e d b y F F T o f t h e osc i l l a t i o n s$ $(co m p u t e d : F_{1} t o F_{4} = 1.77, 3.68, 5.18, 6.48; w i t h 1.426 eV an d 37.25 meV t h e seco n d e x t r e m u m l i es a t 1.563 eV)$
Surface field from Fermi-level pinning	$F_{s} = \frac{2 q N V _{s}}{ε _{s}}$	$V_{s} = V_{bi} - V_{p v} (reduced by pump photovoltage)$	$F_{s} : s u r f a ce f i e l d$ $N : d o p in g$ $V_{s} : s u r f a ce ban d b e n d in g$ $ε_{s} : p er mi tt i v i t y$ $V_{bi} : d a r k ban d b e n d in g$ $V_{p v} : p h o t o v o l t a g e in d u ce d b y t h e p u m p; d e pl e t i o n a pp r o x ima t i o n (ba c k g r o u n d k n o w l e d g e, P o l l ak)$ $(a ss u m e d : G a A s, N = 1 \times 1 0^{17} c m^{- 3}, V_{s} = 0.7 V, ε_{r} = 12.9; co m p u t e d : F_{s} = 140 kV/cm)$

Results and Regimes

Name	Formula	Related Formulae	Variables/Constants
Interface and film transitions	$E_{0}^{f i l m} - E_{0}^{in t er f a ce} = Δ E_{0}$	$Δ E_{0} = + 4 meV (1 per cent Mn), - 40 meV (6 per cent Mn)$	$E_{0} : f u n d am e n t a l g a p t r an s i t i o n; Y a s t r u b c hak e t a l . : in t er f a ce an d f i l m v a l u es o f 1.417 an d 1.426 eV (L T - G a A s), 1.421 an d 1.430 eV (1 p er ce n t M n), 1.420 an d 1.386 eV (6 p er ce n t M n), a 4 meV b l u e s hi f t an d 40 meV r e d s hi f t r e l a t i v e t o t h e r e f er e n ce f i l m, v er t i c a l s t r ain 3.99, 10.51 an d 42.02 \times 1 0^{- 4}$ $(co m p u t e d : 1.430 - 1.426 = + 4 meV an d 1.386 - 1.426 = - 40 meV, ma t c hin g t h e s t a t e d s hi f t s)$
Low- and intermediate-field regimes	$ℏ θ ≪ γ \Rightarrow low field (Aspnes form)$	$ℏ θ ≳ γ \Rightarrow Franz-Keldysh oscillations$	$r e g im e cr i t er i o n co m p a r in g e l ec t r o - o pt i c e n er g y w i t h b r o a d e nin g (ba c k g r o u n d k n o w l e d g e, P o l l ak); Y a s t r u b c hak e t a l . : A s p n es f or m f or t h e b e l o w - g a p f e a t u r e an d F K O ana l y s i s ab o v e t h e g a p$ $(co m p u t e d : w i t h ℏ θ = 37 meV, b r o a d e nin g b e l o w ab o u t 37 meV a l l o w s F K O)$

Important Questions

Ques: How does photoreflectance work?

By modulating the surface field and measuring how reflectance responds. Most semiconductor surfaces have a built-in field from Fermi-level pinning; a chopped pump above the gap creates carriers that partly screen it, so the dielectric function near transition energies changes at the chopping frequency (background knowledge). Yastrubchak et al. used a 488 $nm$ argon-ion pump of nominal 50 $mW$ , 2.54 $eV$ (computed), chopped at 70 $Hz$ , a halogen lamp and monochromator as probe, a silicon photodiode and 2 $mm$ spots. The relative change $Δ R / R$ is typically $1 \times 1 0^{- 5}$ to $1 \times 1 0^{- 3}$ , so a $1 \times 1 0^{- 4}$ signal on 30 per cent reflectance is only $3 \times 1 0^{- 5}$ in absolute terms (computed), which lock-in detection extracts. The result is a derivative-like spectrum with sharp features at each transition. Reference: O. Yastrubchak, J. Zuk, H. Krzyzanowska, J. Z. Domagala, T. Andrearczyk, J. Sadowski and T. Wosinski, ‘Photoreflectance Study of the Fundamental Optical Properties of (Ga,Mn)As Epitaxial Films’, arXiv:1012.4760 (fetch-tool summary); F. H. Pollak, ‘Modulation spectroscopy of semiconductors and semiconductor microstructures’, in Handbook on Semiconductors, vol. 2, Elsevier 1994 (background knowledge; not opened).

Ques: Why are photoreflectance spectra so much sharper than reflectance?

Because modulation measures a derivative of the dielectric function. A field change shifts and broadens each critical point slightly, and $Δ R / R$ , through the Seraphin coefficients, records how $ε_{1}$ and $ε_{2}$ change, which is largest right at the transition and falls quickly away from it (background knowledge, Pollak). In the low-field limit, the line shape is Aspnes’s third-derivative form, $Re [C e^{i ϑ} (ℏ ω - E_{C P} + iγ)^{- n}]$ , with n = 2.5 for a three-dimensional band-to-band transition (Yastrubchak et al.), n = 3 for two-dimensional and n = 2 for excitonic transitions (background knowledge). Fitting gives the transition energy and broadening to a few millielectronvolts, even at room temperature where ordinary reflectance shows only a weak shoulder. Reference: O. Yastrubchak, J. Zuk, H. Krzyzanowska, J. Z. Domagala, T. Andrearczyk, J. Sadowski and T. Wosinski, ‘Photoreflectance Study of the Fundamental Optical Properties of (Ga,Mn)As Epitaxial Films’, arXiv:1012.4760 (fetch-tool summary); F. H. Pollak, ‘Modulation spectroscopy of semiconductors and semiconductor microstructures’, in Handbook on Semiconductors, vol. 2, Elsevier 1994 (background knowledge; not opened); J. Nukeaw, ‘Fast Fourier Transform Photoreflectance’, NECTEC Technical Journal no. 8 (abstract only) (fetch-tool summary).

Ques: How are Franz-Keldysh oscillations used to measure electric fields?

From the spacing of their extrema. In stronger fields, the absorption above the gap oscillates, with the j-th extremum at $E_{j} = E_{g} + ℏ θ F_{j}$ and $F_{j} = [3 π (j - 1/2) /2]^{2/3}$ (Yastrubchak et al.); the first four indices are 1.77, 3.68, 5.18 and 6.48 (computed). Plotting the extremum energies against $F_{j}$ gives the gap as the intercept and the electro-optic energy $ℏ θ = (e^{2} ℏ^{2} F^{2} /2 μ)^{1/3}$ as the slope, from which the field follows. For LT-GaAs, Yastrubchak et al. report 37.25 $meV$ for the heavy-hole branch, which corresponds to about 89.5 $kV/cm$ with assumed GaAs masses (computed). Nukeaw shows that FFT of the oscillations gives the field directly, the period growing with field. Reference: O. Yastrubchak, J. Zuk, H. Krzyzanowska, J. Z. Domagala, T. Andrearczyk, J. Sadowski and T. Wosinski, ‘Photoreflectance Study of the Fundamental Optical Properties of (Ga,Mn)As Epitaxial Films’, arXiv:1012.4760 (fetch-tool summary); J. Nukeaw, ‘Fast Fourier Transform Photoreflectance’, NECTEC Technical Journal no. 8 (abstract only) (fetch-tool summary).

Ques: Why do heavy- and light-hole oscillations give different fields?

They should give the same field, but each depends on an assumed reduced mass. Yastrubchak et al. report electro-optic energies of 37.25 $meV$ for the heavy-hole and 48.29 $meV$ for the light-hole transition in LT-GaAs. With assumed masses of 0.067 for electrons, 0.5 for heavy holes and 0.082 for light holes, the reduced masses are 0.0591 and 0.0369, giving 89.5 and 104.4 $kV/cm$ (computed), 17 per cent apart. Since the field scales as $μ^{1/2}$ , a 36 per cent error in one reduced mass would explain it (computed), and hole masses in GaAs are direction-dependent and uncertain (background knowledge). The heavy-hole value is more likely reliable, as it is less sensitive to the light-hole mass, but both should be quoted with the masses used (my reading). Reference: O. Yastrubchak, J. Zuk, H. Krzyzanowska, J. Z. Domagala, T. Andrearczyk, J. Sadowski and T. Wosinski, ‘Photoreflectance Study of the Fundamental Optical Properties of (Ga,Mn)As Epitaxial Films’, arXiv:1012.4760 (fetch-tool summary); F. H. Pollak, ‘Modulation spectroscopy of semiconductors and semiconductor microstructures’, in Handbook on Semiconductors, vol. 2, Elsevier 1994 (background knowledge; not opened).

Ques: How large is the surface field, and does the pump change it?

Typically tens to hundreds of kilovolts per centimetre, and yes. With the Fermi level pinned near midgap, band bending of about 0.7 $V$ in GaAs doped at $1 \times 1 0^{17}$ $c m^{- 3}$ gives $2 q N V_{s} / ε_{s}$ , about 140 $kV/cm$ (assumed values; computed). The pump itself creates a photovoltage that reduces the band bending, so the measured field is lower than in the dark and depends on pump intensity (background knowledge, Pollak). Low pump power and extrapolation to zero intensity are used for accurate fields. The 89.5 $kV/cm$ found from Yastrubchak et al.’s heavy-hole data on low-temperature-grown GaAs lies in this range (computed). In heterostructures, FKO from different interfaces can be separated, giving interface fields and hence charge densities. Reference: F. H. Pollak, ‘Modulation spectroscopy of semiconductors and semiconductor microstructures’, in Handbook on Semiconductors, vol. 2, Elsevier 1994 (background knowledge; not opened); O. Yastrubchak, J. Zuk, H. Krzyzanowska, J. Z. Domagala, T. Andrearczyk, J. Sadowski and T. Wosinski, ‘Photoreflectance Study of the Fundamental Optical Properties of (Ga,Mn)As Epitaxial Films’, arXiv:1012.4760 (fetch-tool summary).

Ques: What did photoreflectance show for (Ga,Mn)As?

Small shifts of the gap transition that track Mn content and strain. Yastrubchak et al. fitted interface and film transitions: 1.417 and 1.426 $eV$ for low-temperature GaAs, 1.421 and 1.430 $eV$ with 1 per cent Mn, and 1.420 and 1.386 $eV$ with 6 per cent Mn. Relative to the GaAs film these are a 4 $meV$ blue shift and a 40 $meV$ red shift (computed check), with vertical strain rising from 3.99 to $42.02 \times 1 0^{- 4}$ . The 6 per cent film’s electro-optic energies, 57.53 and 34.37 $meV$ , differ from GaAs, indicating changes in field or effective mass. The example shows the millielectronvolt resolution of room-temperature photoreflectance on thin epitaxial films. Reference: O. Yastrubchak, J. Zuk, H. Krzyzanowska, J. Z. Domagala, T. Andrearczyk, J. Sadowski and T. Wosinski, ‘Photoreflectance Study of the Fundamental Optical Properties of (Ga,Mn)As Epitaxial Films’, arXiv:1012.4760 (fetch-tool summary).

Ques: When is the Aspnes form valid, and when are FKO seen?

It depends on the field compared with the broadening. When the electro-optic energy is much smaller than the broadening $γ$ , the field only perturbs the transition and the low-field third-derivative form applies; when $ℏ θ$ approaches or exceeds $γ$ , oscillations appear above the gap (background knowledge, Pollak). With $ℏ θ = 37$ $meV$ , broadening below about 37 $meV$ allows FKO to be resolved (computed), which room-temperature GaAs films often satisfy. Yastrubchak et al. used both regimes in one spectrum: the Aspnes form for the feature below the gap and FKO analysis above it. Heavily doped or strongly disordered layers broaden lines and suppress FKO, while undoped layers on doped substrates give long, clear oscillation trains (background knowledge). Reference: O. Yastrubchak, J. Zuk, H. Krzyzanowska, J. Z. Domagala, T. Andrearczyk, J. Sadowski and T. Wosinski, ‘Photoreflectance Study of the Fundamental Optical Properties of (Ga,Mn)As Epitaxial Films’, arXiv:1012.4760 (fetch-tool summary); F. H. Pollak, ‘Modulation spectroscopy of semiconductors and semiconductor microstructures’, in Handbook on Semiconductors, vol. 2, Elsevier 1994 (background knowledge; not opened).

References

O. Yastrubchak, J. Zuk, H. Krzyzanowska, J. Z. Domagala, T. Andrearczyk, J. Sadowski and T. Wosinski, “Photoreflectance Study of the Fundamental Optical Properties of (Ga,Mn)As Epitaxial Films,” arXiv:1012.4760. Available: https://arxiv.org/pdf/1012.4760
F. H. Pollak, “Modulation spectroscopy of semiconductors and semiconductor microstructures,” in Handbook on Semiconductors, vol. 2, M. Balkanski, Ed. Amsterdam: Elsevier, 1994 (cited from background knowledge).
J. Nukeaw, “Fast Fourier Transform Photoreflectance,” NECTEC Technical Journal, no. 8 (abstract). Available: https://www.nectec.or.th/NTJ/No8/No8_short_1.html

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Photoreflectance Spectrometer