Effect of Quantum Hall Edge Strips on Valley Splitting in Silicon Quantum Wells

We determine the energy splitting of the conduction-band valleys in two-dimensional electrons confined to low-disorder Si quantum wells. We probe the valley splitting dependence on both perpendicular magnetic field $B$ and Hall density by performing activation energy measurements in the quantum Hall regime over a large range of filling factors. The mobility gap of the valley-split levels increases linearly with $B$ and is strikingly independent of Hall density. The data are consistent with a transport model in which valley splitting depends on the incremental changes in density $eB/h$ across quantum Hall edge strips, rather than the bulk density. Based on these results, we estimate that the valley splitting increases with density at a rate of $116\mu \mathrm{eV}/{10}^{11}{\mathrm{cm}}^{-2}$, which is consistent with theoretical predictions for near-perfect quantum well top interfaces.

Figure 1

(a) Cross-section schematic of a Si/SiGe heterostructure field effect transistor. (b) HAADF-STEM image of the strained Si quantum well and nearby ${\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}$ with superimposed HAADF-STEM intensity profile (blue line). The heterostructure growth direction $z$ is indicated by a black arrow (c) Mobility $\mu$ and (d) conductivity ${\sigma }_{xx}$ as a function of density $n$ at a temperature of 110 mK, measured at the cold finger of the dilution refrigerator. The black line in (d) is a fit to percolation theory. (e) Resistivity ${\rho }_{xx}$ as a function of filling factor $\nu$ measured at $n=4.0\times {10}^{11}{\mathrm{cm}}^{-2}$. Different colors correspond to different temperatures from 110 (dark blue) to 450 mK (orange). The inset reports the Arrhenius plot and fit to extract ${\mathrm{\Delta }}_v$ for $\nu =5$. (f) Single particle Landau level energy diagram. Valley split levels correspond to odd integer filling factors $\nu$, Zeeman split levels to $\nu =(4k\text{−}2)$ ($k=1,2,3,\dots$), whereas spin and valley degenerate Landau levels correspond to $\nu =4k$. The shaded areas represent the single-particle level broadening $\mathrm{\Gamma }$ due to disorder.

Figure 2

Activation energy ${\mathrm{\Delta }}_v$ for odd-integer filling factors $\nu$ measured as a function of magnetic field $B$ and Hall density $n$ (filled circles). The blue plane defined by the equation ${\mathrm{\Delta }}_v=c_{B}B+c_{n}n-\mathrm{\Gamma }$ with $c_B=28.1\mu \mathrm{eV}/\mathrm{T}$, $c_n=0.1\mu \mathrm{eV}/{10}^{11}{\mathrm{cm}}^{-2}$, and $\mathrm{\Gamma }=37.5\mu \mathrm{eV}$ is a fit to the experimental data points with coefficient of determination $R^2=0.993$.

Figure 3

(a) Schematic representation of the charge density profile $n(x)$ on the left-hand side of a Hall bar shaped HFET for the case of $\nu =3$, in units of the density $n_B=eB/h$ corresponding to one completely filled Landau level. The edge of the Hall bar is at $x=0$. The 2DEG is divided into compressible (blue) and incompressible (pink) strips. (b) Energy-level diagram, including valley and Zeeman splittings. Landau-level splittings are not present for the case of ${\nu }_{\mathrm{bulk}}=3$ shown here, but they would occur for larger ${\nu }_{\mathrm{bulk}}$ values. Valley splittings are assumed to be proportional to the local value of $n$. Filled, partially filled, and empty Landau levels are indicated by filled, half-filled, and empty circles, respectively. Our model of activated transport incorporates activation and tunneling processes across the alternating compressible and incompressible strips. The thick black arrow indicates the location where the valley splitting takes its characteristic value, $E_{v0}$. The valley splitting increases by an amount $E_{v0}$ in each of the compressible strips. (c) Agreement between experimental (filled circles) and simulated (open circles) data points of valley splitting $E_v$ as a function of density $n_B=eB/h$. The dashed line is the expected valley splitting dependence on density for a disorder-free quantum well top-interface as calculated in Ref. [25].