Model for the electron distribution in modulation-doped heterostructures with high density of surface states

We study heterostructures, consisting of parallel layers of GaAs and ${\text{Al}}_x{\text{Ga}}_{1-x}\text{As}$ containing $n$-doped sheets and, on both sides, GaAs cap layers with a high density of surface states. Assuming thermal equilibrium between the electrons in the surface states, the donor-induced states, and the eigenstates near the $\text{GaAs}/{\text{Al}}_x{\text{Ga}}_{1-x}\text{As}$ interface (IF), we obtain three different doping regimes. At low doping only the surfaces are charged with electrons, not the IF. At intermediate doping also a two-dimensional electron system (2DES) near the IF occurs. At high doping the electron densities of the surface layers and of the 2DES saturate, and a parallel channel occurs at the highly doped sheet, which accommodates the electrons resulting from further doping. Besides the results of a self-consistent Hartree-type calculation we present a quasiclassical approach, which gives very good results for the electron distribution over the sample in all three doping regimes. In the heavy-doping saturation regime also a simple electrostatic estimate is applicable.

Figure 1

Sketch (not to scale) of the conduction-band edge (without charge-induced band bending) of the ${\text{Al}}_x{\text{Ga}}_{1-x}\text{As}/\text{GaAs}$ heterostructure with pure GaAs $\left(x=0\right)$ in $w_l<z<w_r$ (the well) and in $s_l<z<c_l$ and $c_r<z<s_r$ (the caps). For the quantum-well structure we take $x_l=x_r\sim 0.3$, for the single-interface heterostructure $x_l\sim 0.3$ and $x_r=0$. Si doping is confined to the sheets at $z=d_l$ and $z=d_r$.

Figure 2

(Color online) Effective conduction-band edge $V\left(z\right)$ for a QW structure between $s_l=-390\text{nm}$ and $s_r=990\text{nm}$, with GaAs cap layers at $z<-375\text{nm}$ and $z>970\text{nm}$, and a GaAs well at $\left|z\right|<w_r=-w_l=10\text{nm}$. The Al-concentration in the other layers is $x_l=x_r=0.3$. Positive charges of area density $e{\nu }_l\times {10}^{11}{\text{cm}}^{-2}$ at $d_l=-75\text{nm}$ and $e{\nu }_r\times {10}^{11}{\text{cm}}^{-2}$ at $d_r=800\text{nm}$ describe the effect of $\delta$-doping and are indicated by the labels $\left({\nu }_l,{\nu }_r\right)$ at the graphs.

Figure 3

(Color online) $V\left(z\right)$ for the QW structure of Fig. 2 with doping densities $n_{Dl}=3.0\times {10}^{11}{\text{cm}}^{-2}$ and $n_{Dr}=2.4\times {10}^{11}{\text{cm}}^{-2}$. The inset shows $V\left(z\right)$, $E_F$, the energy eigenvalues, and $N_{\text{el}}\left(z\right)$ in the well region. Resulting electron densities: $n_{\text{el}}=1.24$, $n_{Sl}=1.75$, and $n_{Sr}=2.41$, all in ${10}^{11}{\text{cm}}^{-2}$.

Figure 4

(Color online) $V\left(z\right)$ for the structure of Fig. 2 in the region of PC and QW, for doping ${\nu }_{Dl}=5.5$ and ${\nu }_{Dr}=2.4$ (${\nu }_{\mu }$ means $n_{\mu }={\nu }_{\mu }\times {10}^{11}{\text{cm}}^{-2}$). Also shown are the Fermi energy $E_F$ and, in the QW and the PC, the electron-density and the low-energy eigenvalues $E_n^{\text{qw}}$ and $E_n^{\text{pc}}$, respectively. Calculated electron densities: ${\nu }_{\text{el}}^{\text{qw}}=2.62$, ${\nu }_{Sl}=2.07$, ${\nu }_{Sr}=2.41$, and ${\nu }_{\text{el}}^{\text{pc}}=0.80$.

Figure 5

(Color online) (a) Circles indicate the self-consistently calculated electron densities on surfaces $\left(n_{Sl},n_{Sr}\right)$, in the QW $\left(n_{\text{qw}}\equiv n_{\text{el}}^{\text{qw}}\right)$, and in the PC $\left(n_{\text{pc}}\equiv n_{\text{el}}^{\text{pc}}\right)$ as function of $n_{Dl}$ at fixed $n_{Dr}$ (dashed horizontal line). (b) Fermi energy $E_F$ and lowest energy eigenvalues in QW and PC obtained in the self-consistent calculation. The solid lines in (a) represent the quasiclassical approximation explained in Sec. 3 for $E_{\text{bqw}}=10\text{meV}$, $E_{\text{bpc}}=30\text{meV}$, the dashed lines for $E_{\text{bqw}}=E_{\text{bpc}}=0$.

Figure 6

(Color online) Effective conduction-band edges of a heterostructure with a ${\text{Al}}_{0.3}{\text{Ga}}_{0.7}\text{As}/\text{GaAs}$ interface at $z=-10\text{nm}$ and undoped GaAs in $-10\text{nm}<z<990\text{nm}$ for three doping strengths $n_{Dl}={\nu }_{Dl}\times {10}^{11}{\text{cm}}^{-2}$ in the plane $z=-75\text{nm}$. The ${\text{Al}}_{0.3}{\text{Ga}}_{0.7}\text{As}$ layer is located in $-375\text{nm}<z<-10\text{nm}$, with a GaAs cap layer at $-390\text{nm}<z<-375\text{nm}$. The inset shows, for ${\nu }_{Dl}=5.5$, the band-edge minima and the electron density $N_{\text{el}}\left(z\right)$ at the interface and in the parallel channel.

Figure 7

(Color online) (a) Electron densities in surface states ($n_{Sl}$ and $n_{Sr}$), in the 2DES near the interface $\left(n_{\text{if}}\right)$ and in the parallel channel $\left(n_{\text{pc}}\right)$ as functions of the doping density $n_{Dl}$, for $n_{Dr}=0$. Filled circles: self-consistent calculations; lines: quasiclassical approximation (see text). (b) Self-consistently calculated lowest energy eigenvalues $E_n^{\text{if}}$ and $E_n^{\text{pc}}$ in the potential minima near the IF and at the PC, respectively, and the Fermi energy.

Figure 8

(Color online) Effective conduction-band edges of a heterostructure as in Fig. 6, but with a $\delta$ doping of areal density $n_{Dr}=2.5\times {10}^{11}{\text{cm}}^{-2}$ at $d_r=800\text{nm}$, for three doping strengths $n_{Dl}={\nu }_{Dl}\times {10}^{11}{\text{cm}}^{-2}$ in the plane $z=-75\text{nm}$. The inset shows, for ${\nu }_{Dl}=6$, the band-edge minima and the electron density $N_{\text{el}}\left(z\right)$ at the interface and in the parallel channel. About 4% of the electrons in the 2DES at the interface occupy the second subband.

Figure 9

(Color online) (a) Electron densities in surface states ($n_{Sl}$ and $n_{Sr}$), in the 2DES near the IF (total $n_{\text{if}}$ and in the second subband $n_{\text{el}-\text{sb}2}$), and in the PC $\left(n_{\text{pc}}\right)$ as functions of the doping density $n_{Dl}$, for fixed $n_{Dr}$. Filled circles represent results of self-consistent calculations, lines are just guides to the eye. (b) Lowest energy eigenvalues $E_n^{\text{if}}$ and $E_n^{\text{pc}}$ in the potential minima near the IF and at the PC, respectively, and Fermi energy; $n_{\text{el}-\text{sb}2}>0$ if $E_2^{\text{if}}<E_F$.

Figure 10

(Color online) Electron densities in surface states ($n_{Sl}$ and $n_{Sr}$), in the 2DES near the IF $\left(n_{\text{if}}\right)$, and in the PC $\left(n_{\text{pc}}\right)$ as functions of the doping density $n_{Dl}$ of the left doping sheet in a single-interface heterostructure with undoped GaAs buffer of $1\mu \text{m}$ thickness. ${\text{Al}}_x{\text{Ga}}_{1-x}\text{As}$ layers (with $x=0.24$) of three different thicknesses dAl are considered, all with a 40 nm thick spacer.

Figure 11

(Color online) Electron densities on surfaces ($n_{Sl}$ and $n_{Sr}$), of the 2DES near the IF $\left(n_{\text{if}}\right)$, and in the PC $\left(n_{\text{pc}}\right)$ as functions of the doping density $n_{Dl}$ of the left doping sheet in a single-interface heterostructure. The GaAs buffer is $1\mu \text{m}$, the ${\text{Al}}_{0.24}{\text{Ga}}_{0.76}\text{As}$ layer is 365 nm and the following GaAs cap is 15 nm thick. Upper panel: fixed spacer thickness of 65 nm in the ${\text{Al}}_{0.24}{\text{Ga}}_{0.76}\text{As}$ and three doping densities $n_{Dr}={\nu }_{Dr}\times {10}^{11}{\text{cm}}^{-2}$ of the right doping sheet at 500 nm from the interface. Lower panel: undoped substrate, $n_{Dr}=0$, and three values of the spacer thickness in the ${\text{Al}}_{0.24}{\text{Ga}}_{0.76}\text{As}$.