Measurements of the Electric Field of Zero-Point Optical Phonons in GaAs Quantum Wells Support the Urbach Rule for Zero-Temperature Lifetime Broadening

We study a specific type of lifetime broadening resulting in the well-known exponential “Urbach tail” density of states within the energy gap of an insulator. After establishing the frequency and temperature dependence of the Urbach edge in GaAs quantum wells, we show that the broadening due to the zero-point optical phonons is the fundamental limit to the Urbach slope in high-quality samples. In rough analogy with Welton’s heuristic interpretation of the Lamb shift, the zero-temperature contribution to the Urbach slope can be thought of as arising from the electric field of the zero-point longitudinal-optical phonons. The value of this electric field is experimentally measured to be $3\mathrm{kV}{\mathrm{cm}}^{-1}$, in excellent agreement with the theoretical estimate.

Figure 1

(a) Photoluminescence spectrum of a 15 nm wide high-quality $\mathrm{GaAs}/\mathrm{AlGaAs}$ QW. (b) The absorption spectra inferred from such PL spectra obtained at three different excitation powers. (c) Low-energy tails of the absorption spectra have a range where they are exponential. Furthermore, the curves measured at different temperatures extrapolate to a focus. (d) Temperature dependence of $\sigma$. Solid line is the fit to Eq. (2) with $\hslash {\mathrm{\Omega }}_p=6.8\mathrm{meV}$ and the prefactor ${\sigma }_0[2k_B/\hslash {\mathrm{\Omega }}_p]=0.0443$.

Figure 2

Absorption spectra of different QW samples (inferred from PL in samples S1 and S2 and photoconductivity in sample S3) measured at 4 K. Sample S1 has MOVPE-grown high-quality three uncoupled QWs of widths 15, 5, and 2 nm. Sample S2 has metal-organic vapor phase epitaxy (MOVPE)-grown poor-quality four uncoupled QWs of widths 10, 5, 3, and 2 nm. Sample S3 is a high-quality molecular beam epitaxy (MBE)-grown sample with 50 uncoupled QWs each of thickness 12 nm.

Figure 3

(a) Electroabsorption spectra at different $F_{\text{ex}}$ electric fields (inferred from photoconductivity) at 4 K. The values of $F_{\text{ex}}$ are the same as in (b). Each of the curves has been arbitrarily scaled such that the absorption coefficients have the same value at 1.560 eV. (b) Natural logarithm of the low-energy part of the heavy-hole excitonic resonance plotted versus the photon energy for different values of $F_{\text{ex}}$. The linear fits to the curves yield the slope to be $614\pm 6$, $540\pm 10$, $473\pm 8$, $422\pm 6$, and $393\pm 3{\mathrm{eV}}^{-1}$ for $F_{\text{ex}}=0.5,1,1.5,2,2.5\mathrm{kV}{\mathrm{cm}}^{-1}$, respectively. The location of the Urbach focus $E_0$ [see Eq. (4)] is depicted by a square. (c) The slope of the logarithm of the absorption tail times the total electric field ($|F_{\text{in}}|+|F_{\text{ex}}|$) for different values of the external field $F_{\text{ex}}$ is approximately constant if $F_{\text{in}}=3\mathrm{kV}{\mathrm{cm}}^{-1}$. The error bars are the errors in the fits in (b).