Nonlinear electron transport in GaAs/InGaAs asymmetric double-quantum-well pseudomorphic high-electron-mobility transistor structure

The pseudomorphic high-electron-mobility transistors (p-HEMTs) based on the InGaAs heterostructure are highly important because of their application in the microwave and millimetre frequency ranges, which are suitable for defence and commercial applications.¹⁾ The p-HEMT structures exhibit high power gain, excellent cut-off frequency, low noise, and extremely good dc characteristics suitable for power microwave and high-speed device applications.^2–9) The performance of a device depends on the improvement of sheet electron density and mobility. Much effort has been made to study the enhancement of the electron mobility in the InGaAs-based pseudomorphic channels by employing different structure geometry and doping profiles.^10–15)

A double quantum well is an interesting system because of the occurrence of tunneling coupling in addition to the quantum confinement of the carriers.^16,17) When two separate wells are brought closer to each other, their subband wave functions overlap through the central barrier, leading to the splitting of the subband energy levels. If the well widths are equal and also the doping concentrations have symmetric profiles, a symmetric coupled potential structure can be obtained in which the subband wave functions extend into both the wells equally. The lowest (bonding) and first excited (antibonding) wave functions exhibit completely symmetric and antisymmetric behaviors, respectively. However, the slight variations of the well widths or doping concentrations disturb the symmetry of the structure so that the resonance of the subband states is destroyed. As a result, an asymmetric double quantum well is formed, in which one of the two lowest subband wave functions almost lies in one well, while the other wave function lies in another well.^16,17) By varying the structure parameters or by applying an external electric field, the subband wave functions can be displaced from one well to the other.^16–18) The asymmetry of the structure parameters affects the optical properties of pseudomorphic double quantum wells substantially.^19–23)

In this work, we analyze the electron mobility as a function of the asymmetry of the structure parameters of a strained GaAs/In_xGa_1−xAs double quantum well. We consider that the barriers, lying towards the substrate and surface sides, are delta-doped with different doping concentrations, n_di and n_ds, respectively. The corresponding well widths are also made unequal as w_i and w_s. We calculate the subband energy levels and wave functions by using self-consistent solutions of the Schrödinger and Poisson's equations. The variations of the structure parameters change the subband energy levels, wave functions, and also the occupation of subbands. As a result, the scattering potentials are affected, thereby changing μ. In the case of more than one subband occupancy, their influence on the scattering potentials is mediated through the inter-subband effects. We consider the scatterings due to the ionized impurities (imp-), alloy disorder (al-), and interface roughness (ir-) to calculate μ. We use the random phase approximation to screen the scattering potentials. We show that the asymmetric combinations of doping concentrations and well widths lead to the interesting variations of the occupation of subbands. For example, when n_ds and w_s remain unchanged, the occupations of subbands vary from single to double and then again single-subband occupancy as a function of w_i. The nonlinearity of μ due to the transition of the occupation of subbands can be regulated by varying n_di. For n_di = 0, only the single subband is occupied with a cusplike variation of μ about a certain value of w_i having a faster rise towards higher values of w_i. By considering an increased value of n_ds, there is single-to double-subband occupancy only. For a low w_i, under the single-subband occupancy, the mobility increases with the increase in n_di. However, for a larger w_i, the variations of μ for different n_di values are different. We show that the nonlinear variation of mobility for different asymmetric combinations of well widths and doping concentrations is mostly due to the variations of the subband mobilities through changes in the scattering rate matrix elements due to the shifting of subband wave functions between the wells. In Sect. 2, we present the theory for the calculations. In Sect. 3, we discuss the results, while in Sect. 4, we provide a summary.

We consider the GaAs/In_xGa_1−xAs double-quantum-well HEMT structure shown in Fig. 1. The In_xGa_1−xAs well layers of widths w_i and w_s are separated by a thin GaAs barrier of width b, thereby allowing the coupling of eigenstates of the two wells. The GaAs side barriers are delta-doped with Si of widths d and doping concentrations n_di and n_ds at the spacer distance s from the interfaces along the substrate and surface sides of the structure, respectively. The transferred electrons in the wells interact with the ionized donors in the side barriers through Coulomb interaction, leading to band bending. The electrons are thus confined in the wells near the outer interfaces, leading to a coupled two-dimensional electron gas (2DEG).^24,25) The doping concentrations perpendicular to the interface plane can be written as

$$\begin{equation} {n_{\text{d}}(z) = \begin{cases} n_{\text{di}} & \text{$- (d + s + w_{\text{i}} + b/2 ) < z < - (s + w_{\text{i}} + b/2)$} \\ n_{\text{ds}} & \text{$(b/2 + w_{\text{s}} + s) < z < (b/2 + w_{\text{s}} + s + d)$}\\ 0 & \text{otherwise} \end{cases} .} \end{equation} \tag{ 1 }$$

One can also write the electron concentration distribution n(z) of the coupled system as²⁴⁾

$$\begin{equation} n(z) = \sum_{n}n_{n}|\psi_{n}(z)|^{2}, \quad - \infty < z < + \infty. \end{equation} \tag{ 2 }$$

The summation over n stands for different subbands and n_n is the number of electrons per unit area in the nth subband. The confinement potential V(z) = V_H(z) + V_S(z), where V_H(z) is the Hartree potential responsible for band bending and V_S(z) is the structure potential, which takes care of band offset near interfaces. V_H(z) can be obtained by solving Poisson's equation,²⁴⁾

$$\begin{equation} \frac{d^{2}V_{\text{H}}}{dz^{2}} = \frac{4\pi e^{2}}{\varepsilon_{0}}[n_{\text{d}}(z) - n(z)], \end{equation} \tag{ 3 }$$

where ε₀ is the dielectric constant. Within the effective mass approximation, the electron subband wave functions ψ_n(z) corresponding to the energy eigenvalues E_n satisfy the one-dimensional Schrödinger equation along the z-direction.²⁴⁾

$$\begin{equation} \left[\frac{-\hbar^{2}}{2m^{*}}\frac{d^{2}}{dz^{2}} + V(z) \right]\psi_{n}(z) = E_{n}\psi_{n}(z) \end{equation} \tag{ 4 }$$

We adopt the self-consistent solution of the one-dimensional Schrödinger and Poisson's equations to obtain ψ_n(z), E_n, and n_n.

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The subband transport lifetime τ_n can be written as τ_n = τ_n (E_Fn) since the electrons on the Fermi surface contribute to transport at the temperature T = 0 K. E_Fn = E_F − E_n is the Fermi energy of the nth subband, where E_F is the Fermi energy. E_Fn is related to the 2D electron density of the nth subband,²⁴⁾

$$\begin{equation} n_{n} = [m/(\pi\hbar^{2})]E_{\text{F}n}\theta(E_{\text{F}n}), \end{equation} \tag{ 5 }$$

where θ is the Heaviside step function. The surface electron density N_s can be expressed in terms of n_n as²⁴⁾

$$\begin{equation} N_{\text{s}} = \sum_{n=1}^{N}n_{n}, \end{equation} \tag{ 6 }$$

where N is the number of filled levels.

We obtain the multi-subband transport lifetime τ_n, which satisfies a coupled linear equation, by using the Boltzmann transport equation.^24,25) We consider occupancy up to two lowest subbands. Under double-subband occupancy, the inter-subband interaction plays a major role in the calculation of the subband lifetime.^26–30) In the case of the single lowest subband occupancy (n = 0), τ₀ can be written as²⁴⁾

$$\begin{equation} \frac{1}{\tau _{0}} = M_{00}, \end{equation} \tag{ 7 }$$

where

$$\begin{equation} M_{00} = \frac{m}{\pi\hbar^{3}}\int_{0}^{\pi} (1 - \cos\theta)|V_{00}^{\text{eff}}(q_{00})|^{2}\,d\theta. \end{equation} \tag{ 8 }$$

However, when the two lowest subbands are occupied (n = 0, 1), the subband life times τ₀ and τ₁ can be written as $M_{10} = [m/(\pi \hbar ^{3})]( - k_{\text{F}0}/k_{\text{F}1})\int_{0}^{\pi }d\theta \cos \theta |V_{10}^{\text{eff}}(q_{10})|^{2}$ s.²⁴⁾

$$\begin{align} \frac{1}{\tau_{0}} &= \frac{M_{00}M_{11} - M_{01}M_{10}}{M_{11} - M_{01}}, \end{align} \tag{ 9 }$$

$$\begin{align} \frac{1}{\tau_{1}} &= \frac{M_{00}M_{11} - M_{01}M_{10}}{M_{00} - M_{10}}, \end{align} \tag{ 10 }$$

where

$$\begin{align} M_{00} &= \frac{m}{\pi\hbar^{3}}\bigg[\int_{0}^{\pi}(1 - \cos\theta)|V_{00}^{\text{eff}}(q_{00})|^{2}d\theta \\&\quad+ \int_{0}^{\pi}|V_{01}^{\text{eff}}(q_{01})|^{2}d\theta\bigg], \end{align} \tag{ 11 }$$

$$\begin{align} M_{11} &= \frac{m}{\pi\hbar^{3}}\bigg[\int_{0}^{\pi}(1 - \cos\theta)|V_{11}^{\text{eff}}(q_{11})|^{2}d\theta \\&\quad+ \int_{0}^{\pi}|V_{10}^{\text{eff}}(q_{10})|^{2}d\theta \bigg], \end{align} \tag{ 12 }$$

$$\begin{align} M_{01} &= \frac{m}{\pi\hbar^{3}}\left(-\frac{k_{\text{F}1}}{k_{\text{F}0}}\right) \int_{0}^{\pi}d\theta\cos\theta |V_{01}^{\text{eff}}(q_{01})|^{2}, \end{align} \tag{ 13 }$$

$$\begin{align} M_{10} &= \frac{m}{\pi\hbar^{3}}\left(-\frac{k_{\text{F}0}}{k_{\text{F}1}}\right) \int_{0}^{\pi}d\theta\cos\theta|V_{10}^{\text{eff}}(q_{10})|^{2}, \end{align} \tag{ 14 }$$

$q_{n1} = [k_{\text{F}n}^{2} + k_{\text{F}1}^{2} - 2k_{\text{F}n}\ k_{\text{F}1}\cos \theta ]^{1/2}$, and k_Fn = (2mE_Fn/ħ²)^1/2. We consider nonphonon scattering mechanisms due to ionized impurities, interface roughness, and alloy disorder. We adopt the random phase approximation to screen the scattering potentials. The screened ionized impurity scattering potential can be written as

$$\begin{align} &{|V_{nm}^{\text{imp}}(q)|^{2}} \\&\quad= {\frac{4\pi^{2}e^{4}n_{\text{di}}}{\varepsilon_{0}^{2}q^{2}}\int_{-(b/2 + w_{\text{i}} + s + d )}^{-(b/2 + w_{\text{i}} + s)}dz_{i}\left|\sum_{n'm'}\varepsilon_{nm,n'm'}^{-1}(q)p_{n'm'}(q,z_{\text{i}})\right|^{2}} \\ &\qquad + {\frac{4\pi^{2}e^{4}n_{\text{ds}}}{\varepsilon_{0}^{2}q^{2}}\int_{(b/2 + w_{\text{s}} + s)}^{(b/2 + w_{\text{s}} + s + d)}dz_{i}\left| \sum_{n'm'}\varepsilon_{nm,n'm'}^{-1}(q)p_{n'm'}(q,z_{i})\right|^{2},} \end{align} \tag{ 15 }$$

where

$$\begin{equation} P_{n'm'}(q,z_{i}) = \int_{-\infty}^{\infty}dz\psi_{n'}(z)\psi_{m'}(z)e^{-q|z - z_{i}|}. \end{equation} \tag{ 16 }$$

The screened interface roughness scattering potential can be written as

$$\begin{align} &|V_{nm}^{\text{ir}}(q)|^{2} \\&\quad= V_{0}^{2}\pi\Lambda^{2}\Delta^{2}e^{-q^{2}\Lambda^{2}/4} \left|\sum_{n'm'}\varepsilon_{nm,n'm'}^{-1}(q)\psi_{n'}(z)\psi_{m'}(z)|_{z = z_{I}}\right|^{2}. \end{align} \tag{ 17 }$$

Similarly, the screened alloy disorder scattering potential can be written as

$$\begin{align} |V_{nm}^{\text{al}}(q)|^{2} &= [a^{3}\delta V^{2}x(1 - x)/4] \\&\quad\times \int dz \left| \sum_{n'm'}\varepsilon_{nm,n'm'}^{-1}(q)\psi_{n'}(z)\psi_{m'}(z)\right|^{2}. \end{align} \tag{ 18 }$$

$\varepsilon _{nm,n'm'}^{ - 1}$ is the inverse of the dielectric screening matrix.²⁴⁾ In practical numerical calculations, the sums over n', and m' are normally considered to be over the occupied subbands.³⁰⁾ Impurity scattering is responsible for this owing to the ionized impurities lying in the side barriers. Alloy scattering arises from the In_xGa_1−xAs wells, unlike the GaAs/Al_xGa_1−xAs structures, where it occurs from the barrier regions.²⁵⁾ Interface roughness scattering is assumed to occur from the interfaces lying towards the substrate side.^26,29,30) Z_I is the position corresponding to the interfaces I₁ and I₃, which are considered as rough (Fig. 1). Note that, when the double subband is occupied, the expression for M₀₀ [Eq. (11)] contains an additional (inter-subband) term compared with that in Eq. (8). We also note that the contribution of the intra-subband term [first term in Eq. (11)] is also different from that in Eq. (8) through the dielectric screening matrix because of the change in the occupation of subbands. The subband transport lifetime τ_n^imp/ir/al can be related to the subband transport mobility μ_n^imp/ir/al, where μ_n^imp/ir/al = (e/m*)τ_n^imp/ir/al. The mobility due to a particular scattering mechanism (imp/ir/al) can be calculated as

$$\begin{equation} \mu^{\text{imp/ir/al}} = \sum_{n}n_{n}\mu_{n}{}^{\text{imp/ir/al}}\Big/\sum_{n}n_{n} \end{equation} \tag{ 19 }$$

We adopt Matthiessen's rule to calculate the total mobility μ.²⁴⁾

We study the effects of the asymmetric variations of well widths and doping concentrations on the multi-subband electron mobility of the strained GaAs/In_0.2Ga_0.8As double-quantum-well structure. The potential barrier height V₀, which depends on the conduction band offset, is obtained from the strain-dependent band gap between In_xGa_1−xAs and GaAs. For the mole fraction x = 0.2, V₀ = 140 meV.³¹⁾ The strain-dependent electron effective mass m is 0.62 m₀.³²⁾ The interface roughness parameters are the mean height of the roughness, Δ = 2.83 Å, and the correlation length Λ = 100 Å. The alloy disorder scattering potential δV is 530 meV.²⁶⁾ We consider the spacer width s = 100 Å, the doping layer width d = 20 Å, and the central barrier width b = 60 Å.

We study the mobility as a function of asymmetry in well widths by considering w_s = 100 Å and varying w_i. The doping profiles are made symmetric, n_di = n_ds = 0.25 × 10¹⁸ cm⁻³. The mobility of the individual scattering mechanisms, μ^imp/ir/al, and the total mobility μ (cm² V⁻¹ s⁻¹) are presented in Fig. 2 as a function of w_i (Å). With the increase in w_i, the occupation of subbands changes from single to double, and then again, single-subband occupancy. Near the transition of occupation of subbands, a discontinuity in mobility occurs owing to inter-subband effects. There are abrupt drops (rises) in μ, μ^imp, and μ^ir near the transition from single- to double- (double- to single-) subband occupancy. The discontinuity in μ^imp is more than that in μ^ir. On the other hand, note that the change in μ^al has an opposite trend. With the increase in w_i, μ almost increases. Under the double-subband occupancy, μ is mostly governed by imp-scattering. For smaller well widths (w_i < 90 Å), when a single subband is occupied, the mobility is governed by all the scattering mechanisms, imp-, al-, and ir-scatterings, whereas towards larger w_i values (w_i > 90 Å), the effect of ir-scattering is minimized.

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In Fig. 3, we present the subband mobilities μ₀^imp/ir/al and μ₁^imp/ir/al as a function of w_i to explain the difference between the changes in μ^imp, μ^ir, and μ^al at the transition of occupation of subbands as described in Fig. 2. We note that the drop in μ₀^imp is more than that in μ₀^ir near the change in occupation from single- to double-subband occupancy. In the case of μ₀^al, the lowest drop is observed. Regarding μ₁^imp/ir/al, it manifests within the double-subband occupancy. There is a marked variation among different subband mobilities as a function of w_i mediated by inter-subband effects. For imp-scattering, μ₁^imp starts from zero, becomes maximum, and then decreases such that μ₀^imp ≫ μ₁^imp. On the other hand, in the case of ir- and al- scatterings, μ₁^ir/al is substantial and of the same order of μ₀^ir/al. This difference in the variation of subband mobilities, μ₀^imp/ir/al and μ₁^imp/ir/al, is based on the change in scattering potential through the dielectric screening matrix involving the intra- and inter-subband terms [Eqs. (15)–(18)]. As a result, there are net drops in μ^imp and μ^ir, but a small rise in μ^al near the transition from single- to double-subband occupancy. The discontinuity in mobility near the transition from double- to single-subband occupancy can similarly be analyzed.

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In Fig. 4, we present μ as a function of the well width by considering symmetric well widths, w_i = w_s, but considering asymmetric doping concentrations, i.e., n_di = 0.0, 0.25, 0.5, 1.0, and 2.0 × 10¹⁸ cm⁻³ for a fixed n_ds (n_ds = 1.0 × 10¹⁸ cm⁻³). We note that for n_di = 0, the variation of w_i facilitates the occurrence of single-subband then double-subband and again single-subband occupancy as a function of w_i as in Figs. 2 and 3. However, for all other doping concentrations, the double subband is occupied in the entire w_i range. In general, as the doping concentration increases, the mobility increases as long as the occupation of subbands remains unchanged. In the inset of Fig. 4, we present μ^imp/ir/al for n_di = 0 and n_ds =1.0 × 10¹⁸ cm⁻³. For the entire w_i range, al- and imp-scatterings dominate the mobility, except in the smaller well width range (w < 70 Å), where the contribution of ir-scattering is effective. Unlike the GaAs/Al_xGa_1−xAs systems, here in the GaAs/In_xGa_1−xAs structures, al-scattering contributes substantially since the channel layer is an alloy.

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In Fig. 5, we vary both well widths and doping concentrations and present μ (cm² V⁻¹ s⁻¹) as a function of w_i (Å) for w_s = 100 Å and n_ds = 0.25 × 10¹⁸ cm⁻³, considering different values of n_di (n_di = 0.0, 0.25, 0.5, 1.0, and 2.0 × 10¹⁸ cm⁻³). As n_di increases, the asymmetry between the doping concentrations increases. The well width range for the occurrence of double-subband occupancy increases. However, the magnitude of drop in mobility near the transition of occupation of subbands decreases. For n_di ≥ 1 × 10¹⁸ cm⁻³, the double subband is occupied even at w_i = 60 Å. In the case of n_di = 0, there is single-subband occupancy throughout the entire w_i range with a dip in mobility around w_i = 110 Å. Unlike the other mobility curves, where inter-subband effects play a role in determining μ, here, in the case of n_di = 0, the mobility is governed by the lowest intra-subband scattering, mostly imp-scattering. To explain the dip in mobility, we note that the variation of w_i changes the structure potential. Around w_i = 110 Å, the resonance of subband states and hence the minimum of anticrossing of subband energy levels occur. Accordingly, the Fermi energy level of the lowest subband E_F0 has a minimum around w_i = 110 Å. Since the imp-scattering potential [Eq. (15)] is inversely proportional to E_F0, the mobility is being affected through the intrasubband scattering [Eq. (7)]. We further note that, around the resonance, the distribution of subband wave functions varies considerably.^24–26) The lowest subband wave function ψ₀, which almost lies in the right well for w_i < 110 Å, is pushed towards the left well when w_i > 110 Å, leading to a decrease in the effect of the imp-scattering potential. Therefore, the net effect of the above two mechanisms leads to a dip in mobility around w_i = 110 Å with a small increase for w_i < 110 Å but a large increase towards w_i > 110 Å.

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In Fig. 6, we present μ (cm² V⁻¹ s⁻¹) as function of w_i (Å) for w_s = 100 Å and n_di = 0.0, 0.25, 0.5, 1.0, and 2.0 (× 10¹⁸ cm⁻³) by considering a higher value of n_ds (n_ds =1.0 × 10¹⁸ cm⁻³) compared with Fig. 5. As in the earlier case (Fig. 5), here, also a single subband is occupied from the beginning of w_i, i.e., at w_i = 50 Å for n_di = 0, 0.25, and 0.5 × 10¹⁸ cm⁻³. The mobility increases with the increase in n_di. The change from single- to double-subband occupancy occurs at a decreased value of w_i with the increase in n_di. For n_di =1.0 and 2.0 (× 10¹⁸ cm⁻³), two subbands are occupied at w_i = 50 Å. However, unlike in Fig. 5, here, no change from double-subband to single-subband occupancy is obtained. This is because the larger band bending due to the higher n_ds also leads to double-subband occupancy at larger w_i values.

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We note that the decrease in mobility at the onset of the occupation of the second subband also causes the decrease in the current since the surface electron density N_s, which is the sum of the electron densities of the occupied subbands, remains unchanged [Eq. (6)]. This trend in the change in the current is also maintained by varying the well widths since the increase in the well width causes only the overall increase in the mobility owing to the reduction in the effect of the scatterers.

The principal work involved in this paper is to relate the enhancement of electron mobility to the asymmetric variations of the structure parameters in a GaAs/In_xGa_1−xAs double-quantum-well pseudomorphic high-electron-mobility transistor (p-HEMT). We calculate the subband energy levels and wave functions through the self-consistent solution of the Schrödinger and Poisson's equations and use them to determine the electron mobility for the multi-subband occupied system. We show that suitable choices of well widths and doping concentrations lead to interesting changes in the occupation of subbands causing nonlinear trends in mobility through intersubband effects. The nonlinearity in the mobility depends on the changes in the scattering potentials due to the shifting of the subband wave functions from one well to the other because of the asymmetric variations of the structure parameters. Our results can be utilized for the analysis of the device performance of GaAs/In_xGa_1−xAs coupled double-quantum-well p-HEMT structures.