Localization landscape theory of disorder in semiconductors. II. Urbach tails of disordered quantum well layers

Urbach tails in semiconductors are often associated to effects of compositional disorder. The Urbach tail observed in InGaN alloy quantum wells of solar cells and LEDs by biased photocurrent spectroscopy is shown to be characteristic of the ternary alloy disorder. The broadening of the absorption edge observed for quantum wells emitting from violet to green (indium content ranging from 0% to 28%) corresponds to a typical Urbach energy of 20 meV. A three-dimensional absorption model is developed based on a recent theory of disorder-induced localization which provides the effective potential seen by the localized carriers without having to resort to the solution of the Schrödinger equation in a disordered potential. This model incorporating compositional disorder accounts well for the experimental broadening of the Urbach tail of the absorption edge. For energies below the Urbach tail of the InGaN quantum wells, type-II well-to-barrier transitions are observed and modeled. This contribution to the below-band-gap absorption is particularly efficient in near-ultraviolet emitting quantum wells. When reverse biasing the device, the well-to-barrier below-band-gap absorption exhibits a red-shift, while the Urbach tail corresponding to the absorption within the quantum wells is blue-shifted, due to the partial compensation of the internal piezoelectric fields by the external bias. The good agreement between the measured Urbach tail and its modeling by the localization theory demonstrates the applicability of the latter to compositional disorder effects in nitride semiconductors.

Figure 1

(a) Schematic of the InGaN/GaN solar cell structures grown by MOCVD. (b) Contact geometry of the processed devices.

Figure 2

Responsivity measurements by BPCS of (a) $m$-plane UV GaN/AlGaN LED, and (b)–(f) $c$-plane ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}/\mathrm{GaN}$ solar cells with indium composition ranging between 6% and 28%. The curves correspond to different applied biases to the devices (the negative sign indicates a reverse bias). Three regions having a different bias dependence and named A, B, C are enclosed by boundaries. The PL peak energy of the solar cells measured at 0 V is marked as stars. The schematic in (f) shows the conduction band diagram of a $c$-plane InGaN/GaN QW structure in which the QW electric field is partly compensated under application of a reverse bias.

Figure 3

(a)–(c) Change in responsivity measured by BPCS as a function of applied bias at exciting photon energies belonging to the A, B, C regions of the spectra for ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ QW structures with different indium composition. The photon energies corresponding to the different curves are marked as A, B, C in Fig. 2.

Figure 4

(a) Dependence on applied bias of the Urbach energy $E_U$ determined in the B region of BPCS spectra for ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ QW layers with different mean indium content. (b) Mean values of $E_U$ averaged over the applied biases and plotted as a function of the mean indium content of the corresponding InGaN/GaN solar cell.

Figure 5

(a) Responsivity measurements by BPCS of an ${\mathrm{In}}_{0.11}{\mathrm{Ga}}_{0.89}\mathrm{N}/\mathrm{GaN}$ solar cell at 300 K (continuous lines) and 5 K (dashed lines) as a function of applied bias. The low-temperature curves are blue-shifted due to the increase in band-gap energy. The A, B, C regions are enclosed by boundaries. (b)–(d) Change in responsivity as a function of applied bias at 300 K (continuous lines) and 5 K (dashed lines) for exciting photon energies belonging to the different regions and marked in (a) as A, B, C and ${\mathrm{A}}^{\prime },{\mathrm{B}}^{\prime },{\mathrm{C}}^{\prime }$, at 300 and 5 K, respectively. (e) Different mechanisms of carrier extraction from the QW. At low temperature, tunneling outside the QW (T) may occur via type-II transitions. At room temperature, in addition to tunneling, absorption in the QW followed by thermionic emission (TE) may also occur.

Figure 6

(a), (b) EL spectra measured at 5-mA injection and BPCS measurements of an ${\mathrm{In}}_{0.11}{\mathrm{Ga}}_{0.89}\mathrm{N}/\mathrm{GaN}$ solar cell (continuous lines) and a commercial UV LED (dashed lines). (c), (d) EL spectra measured at 5-mA injection and BPCS measurements of an ${\mathrm{In}}_{0.28}{\mathrm{Ga}}_{0.72}\mathrm{N}/\mathrm{GaN}$ solar cell (continuous lines) and a commercial bluish-green LED (dashed lines).

Figure 7

(a) Schematic of QCFK absorption in a QW. In presence of an electric field, ${\vec{E}}_{\text{QW}}$ below-gap absorption at energy $h\nu <E_g$ is allowed by the overlap of the electron and hole wave functions. (b) Simulation of QCFK absorption curves as a function of applied bias for an ${\mathrm{In}}_{0.17}{\mathrm{Ga}}_{0.83}\mathrm{N}/\mathrm{GaN}$ MQW solar cell structure including thickness fluctuations among the different QWs. Alloy disorder is not taken into account in these simulations.

Figure 8

(a) Schematic of the typical InGaN/GaN QW structure used in the 3D absorption model. (b) The simulated InGaN QW incorporates a disk that is one monolayer thick representing a well width fluctuation. (c) “Effective” electron and hole confining potentials obtained from the LL theory for an ${\mathrm{In}}_{0.11}{\mathrm{Ga}}_{0.89}\mathrm{N}/\mathrm{GaN}$ structure at 0 V and computed self-consistently with the Poisson-DD equations. The maps shown here correspond to 2D cuts of the 3D landscapes in the middle plane of the QW.

Figure 9

Urbach tails of ${\mathrm{In}}_{0.17}{\mathrm{Ga}}_{0.83}\mathrm{N}/\mathrm{GaN}$ structures: (a) computed from the 3D absorption model including compositional disorder and QW electric field, and (b) experimentally measured by BPCS. Curves correspond to different applied biases to the structure. The dashed line is shown as a reference and corresponds to an Urbach energy of 20 meV.

Figure 10

Absorption curves for InGaN/GaN structures of different indium composition at 0 V computed using the 3D theoretical model incorporating compositional disorder and QW electric field as a function of different smoothing parameters (dashed line). The responsivity curves measured by BPCS of InGaN/GaN solar cells of different indium composition at 0 V are also shown (continuous lines).

Figure 11

(a), (b) Simulated conduction band potential maps of InGaN QWs for two different indium compositions (11% and 28%). All 2D cuts in this figure are performed just above $z=141.5\mathrm{nm}$ (see Fig. 8), i.e., at the bottom of the QW. (c) 1D cuts along the dashed lines of the $E_c$ maps showing the fluctuating potential. (d), (e) Electron effective confining potentials calculated from the LL theory based on the $E_c$ maps shown in (a) and (b). (f) 1D cuts along the dashed lines of $1/u_e$. The amplitude of the fluctuations is much reduced with respect to that of the corresponding $E_c$ potentials. The simulated maps and 1D profiles corresponding to the case of holes are shown in (g)–(l). The circular-shaped fluctuation observed in the center of (a), (b), (d), (e), (g), (h), (j), (k) is due to the single-monolayer disk fluctuation (see Fig. 8).

Figure 12

Standard deviations characterizing the amplitude of the fluctuations in InGaN QWs of different indium composition for (a) the conduction and valence band potentials, and (b) the corresponding electron and hole effective confining potentials.

Figure 13

(a) Schematic of type-II well-to-barrier transitions in an InGaN/GaN QW. In presence of barrier and QW electric fields, the electron and hole final states, ${\psi }_e$ and ${\psi }_{h,0}$, are Airy functions with tails extending in the gap and allowing below-gap absorption at photon energies $h\nu <E_{g,\text{QW}}$. (b) Schematic explaining why type-II transitions have a higher probability in InGaN/GaN structures with low indium content. Band diagrams are calculated with a 1D Poisson-DD solver and show realistic QW electric fields. (c) Simulated type-II absorption curves for an ${\mathrm{In}}_{0.11}{\mathrm{Ga}}_{0.89}\mathrm{N}/\mathrm{GaN}$ QW as a function of applied bias. (d) Change in the simulated type-II absorption as a function of bias at a photon energy marked as C in (c).