Valley splitting theory of $\mathrm{Si}\mathrm{Ge}∕\mathrm{Si}∕\mathrm{Si}\mathrm{Ge}$ quantum wells

We present an effective mass theory for $\mathrm{Si}\mathrm{Ge}∕\mathrm{Si}∕\mathrm{Si}\mathrm{Ge}$ quantum wells, with an emphasis on calculating the valley splitting. The theory introduces a valley coupling parameter $v_v$ which encapsulates the physics of the quantum well interface. The new effective mass parameter is computed by means of a tight binding theory. The resulting formalism provides rather simple analytical results for several geometries of interest, including a finite square well, a quantum well in an electric field, and a modulation doped two-dimensional electron gas. Of particular importance is the problem of a quantum well in a magnetic field, grown on a miscut substrate. The latter may pose a numerical challenge for atomistic techniques such as tight binding, because of its two-dimensional nature. In the effective mass theory, however, the results are straightforward and analytical. We compare our effective mass results with those of the tight binding theory, obtaining excellent agreement.

Figure 1

(Color online) Comparison of effective mass and tight binding results for the two lowest eigenstates in a ${\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}∕\mathrm{Si}∕{\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}$ quantum well of width $9.5\mathrm{nm}$. (Only half the eigenfunctions are shown.) The solutions correspond to the same orbital subband, but different valley states. Bottom: ground state. Top: excited state. Solid lines: effective mass theory. Circles: tight-binding theory. Dashed line: quantum well boundary.

Figure 2

The valley coupling parameter $v_v$ and the quantum well “set-back” parameter $\delta$, defined in Eq. (27), as a function of the conduction band offset $\Delta E_c$. The results are obtained for a symmetric, finite square well by comparing the EM and TB theories.

Figure 3

(Color online) Valley splitting in a finite square well. We compare effective mass results (solid curve) with tight binding results (circles), as a function of the quantum well widths $L$ and $L^{\prime }$, respectively (see text). The TB data points occur at integer multiples of the TB unit cell. We assume a ${\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}∕\mathrm{Si}∕{\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}$ quantum well, corresponding to $\Delta E_c=160\mathrm{meV}$.

Figure 4

(Color online) Effective mass results for the valley splitting in an electric field, as a function of the quantum well width. Five $E$ fields are shown, from bottom to top: $E=0$, 1.04, 2.08, 3.12, and $4.16\mathrm{MV}∕\mathrm{m}$. The well width $L$ is scaled by the TB unit cell size $a∕2=2.716\mathrm{\AA }$. Results are shown for a ${\mathrm{Si}}_{0.8}{\mathrm{Ge}}_{0.2}∕\mathrm{Si}∕{\mathrm{Si}}_{0.8}{\mathrm{Ge}}_{0.2}$ quantum well, analogous to Fig. 1(a) of Ref. 25. Inset: $E$-field geometry, with confinement potential $V\left(z\right)=V_{\mathrm{QW}}\left(z\right)+V_{\varphi }\left(z\right)$ and variational wave function $F\left(z\right)$ given in Eq. (29). Here, $z_b=z_t-\pi ∕k$.

Figure 5

(Color online) Typical 2DEG structure for a $\mathrm{Si}\mathrm{Ge}∕\mathrm{Si}∕\mathrm{Si}\mathrm{Ge}$ quantum well. (a) Heterostructure, left to right: strain-relaxed SiGe substrate and barrier, strained silicon quantum well, strain-relaxed SiGe barrier, $n^{+}$ SiGe doping layer, and SiGe spacer layer. (b) Conduction band profile, showing the envelope function $F\left(z\right)$ and the Fermi energy $E_F$.

Figure 6

(Color online) Tilted quantum well geometry, with crystallographic axes $(x,y,z)$ and rotated axes $(x^{\prime },y^{\prime },z^{\prime })$, assuming $y=y^{\prime }$. The effective tilt angle is $\vartheta$, and the atomic step size is $s$.

Figure 7

(Color online) Valley splitting on a tilted quantum well, in a magnetic field. The lower (blue) curve shows the exponential suppression of the valley splitting, Eq. (55), arising from perfectly uniform steps. The upper (green) curve shows the linear behavior arising from the “plateau" model of disordered steps. Inset shows a localized step fluctuation (or “wiggle") of area $A$, as analyzed in the text.