Quantum confinement induced by strain relaxation in an elliptical double-barrier $\mathrm{Si}∕{\mathrm{Si}}_x{\mathrm{Ge}}_{1-x}$ resonant tunneling quantum dot

Starting with a double-barrier $p\text{−}\mathrm{Si}∕{\mathrm{Si}}_{0.75}{\mathrm{Ge}}_{0.25}$ resonant tunneling heterostructure, we fabricated sub-$100\text{−}\mathrm{nm}$ elliptical quantum dots. Sidewall strain relaxation in the ${\mathrm{Si}}_x{\mathrm{Ge}}_{1-x}$ layer induces a lateral confining potential that quantizes heavy hole (HH) and light hole (LH) states in the SiGe quantum well, leading to fine structure in the HH-LH $I\left(V\right)$ resonant tunneling curves at low temperature. In this paper, we present the magnetotunneling $I(V,B)$ characteristics of heavy holes and light holes in magnetic fields $B$ parallel to the tunneling current. From the evolution of the fine structure, we observe the competition between the strain-induced lateral confinement potential and the magnetic confinement, from which we infer lateral potentials of HH and LH different from those of previously studied cylindrically symmetric dots. The experimental data are in qualitative agreement with inhomogeneous strain-induced HH and LH potential obtained via a full three-dimensional finite-element strain simulation.

Figure 1

(Color online) (a) Smooth $I\left(V\right)$ characteristics of a large device with diameter $D=1.5\mu \mathrm{m}$ for both bias polarities. Arrows show the calculated threshold and peak bias values for the HH and LH peaks in both polarities. (b) Calculated HH and LH subbands $E\left(k_{\perp }\right)$ dispersion at full strain $(\epsilon =-1)$ and partially relaxed strain $(\epsilon =-0.4)$, corresponding to 60% strain relaxation, in the ${\mathrm{Si}}_x{\mathrm{Ge}}_{1-x}$ quantum well. The energy is referred to the Fermi level $E_{\mathrm{F}}$ in the emitter.

Figure 2

(Color online) (a) SEM top contact micrograph of the elliptical quantum dot under study; long axis is $\sim 80\mathrm{nm}$, short axis is $\sim 60\mathrm{nm}$. (b) HH and LH $I\left(V\right)$ tunneling peaks of the elliptical quantum dot at $B=0$ and $T=1.7\mathrm{K}$, together with expanded $(25\times )$ views of the HH peak and the $dI∕dV$ characteristics of the LH peak. Arrows are used to point out the position of the LH peak in the $I\left(V\right)$ and $dI∕dV$ characteristics. (c) Expanded view of the HH $I\left(V\right)$ peak, and the corresponding $dI∕dV$ characteristics; arrows used to point out the fine structure features. For comparison, the smooth HH $I\left(V\right)$ peak of the large $D=1.5\mu \mathrm{m}$ device of Fig. 1a is also shown (dotted line).

Figure 3

(Color online) (a) Model used to simulate the strain relaxation in our quantum dot; only half of the structure is simulated due to Ge content symmetry about the midplane of the SiGe quantum well. (b) Strain profile by finite element simulation. In our notation, full biaxial strain corresponds to $\epsilon \equiv -1$, fully relaxed strain is $\epsilon \equiv 0$. Regions $A$ are the most strained regions; $B$ is the bottom of the ringlike region; $C$ is the least strained region. (c) Cross sections of the partially relaxed strain in the quantum well in $X$ and $Y$ directions; points $A,B,C$ are the same as in (b).

Figure 4

(Color online) By retaining only the radial ${\epsilon }_{xx}$ and ${\epsilon }_{yy}$ components that are most affected by strain relaxation, calculated HH and LH potential energies in the quantum well in $X$ and $Y$ directions using the six-band Luttinger-Kohn Hamiltonian are shown here. Points $A,B,C$ are the same as in Figs. 3b, 3c.

Figure 5

(Color online) HH $I(V,B)$ characteristics of elliptical quantum dot in forward bias (a) and reverse bias (b) under constant $B$ from $0\text{to}10\mathrm{T}$ with step $1\mathrm{T}$ parallel to the current. The $I\left(V\right)$ curves with different magnetic fields are offset vertically for clarity; arrows are used to point out the fine structure positions. HH fine structure positions in positive bias polarity are plotted against $B$ in (c).

Figure 6

(Color online) (a) LH $I(V,B)$ characteristics of elliptical quantum dot in forward bias under constant $B$ parallel to the current. The $I\left(V\right)$ curves with different magnetic fields are offset vertically for clarity; arrows are used to point out the superimposed fine structure positions. (b) The corresponding $dI∕dV$ characteristics are plotted.

Figure 7

(Color online) Numerically calculated HH densities of states in the elliptical quantum dot assuming no energy broadening at $B=\left(\mathrm{a}\right)$ 0 and (b) $10\mathrm{T}$. States originating in the minimum potential regions $A$ and the higher-lying ringlike regions $B$ are indicated separately. (c) The same densities of states calculated with $\Gamma =1.5\mathrm{meV}$ Gaussian energy broadening for $B=0$ and $10\mathrm{T}$ (lower curves), as well as the $B=10\mathrm{T}$ density of states with a larger $\Gamma =2.5\mathrm{meV}$ energy broadening (top curve). The energy scale is converted to applied bias at the top of (c) to facilitate direct comparison with the experimental $I\left(V\right)$ data of Fig. 5a.