Probing electron interactions in a two-dimensional system by quantum magneto-oscillations

We have experimentally studied the renormalized effective mass $m^{*}$ and Dingle temperature $T_D$ in two spin subbands with essentially different electron populations. Firstly, we found that the product $m^{*}{T}_D$ that determines the damping of quantum oscillations, to the first approximation, is the same in the majority and minority subbands even at a spin polarization degree as high as 66%. This result confirms the theoretical predictions that the interaction takes place at high energies $\sim E_F$ rather than within a narrow strip of energies $E_F\pm k_{B}T$. Secondly, to the next approximation, we revealed a difference in the damping factor of the two spin subbands, which causes skewness of the oscillation line shape. In the absence of the in-plane magnetic field $B_{\parallel }$, the damping factor $m^{*}{T}_D$ is systematically smaller in the spin-majority subband. The difference, quantified with the skew factor $\gamma =(T_{D\downarrow }-T_{D\uparrow })/2T_{D0}$ can be as large as 20%. The skew factor tends to decrease as $B_{\parallel }$ or temperature grow, or $B_{\perp }$ decreases; for low electron densities and high in-plane fields, the skew factor even changes sign. Finally, we compared the temperature and magnetic field dependencies of the magneto-oscillation amplitude with predictions of the interaction correction theory, and found, besides some qualitative similarities, several quantitative and qualitative differences. To explain qualitatively our results, we suggested an empirical model that assumes the existence of easily magnetized triplet scatterers on the Si/${\mathrm{SiO}}_2$ interface.

Figure 1

Calculated interaction corrections to the oscillation amplitude vs (a) $B_{\perp }$ field at $\mathrm{T}=0.19$ K and (b) temperature for $B=1$ T. Three representative density values are 2, 4, and 10 in units of ${10}^{11} \mathrm{cm}{}^{-2}$. $F_0^a$ values equal to $-0.41,-0.334$, and $-0.22$, respectively [22].

Figure 2

(a) The SdH oscillations $\delta {\rho }_{xx}$ vs $B_{\perp }$ in the absence of the parallel field. (b) The same oscillations normalized to the calculated amplitude of the first harmonic, Eq. (1), plotted vs the filling factor $\nu \propto 1/B_{\perp }$. Dots with connecting lines are the data, the continuous curve calculations with an empirical field-dependent $T_D$ (see below) with two parameters $T_{D0}=0.737$ K and $d_1=0.1$ T. The dashed line shows how the amplitude should vary according to Eq. (3). The canonical values for $g=2.57$ and $m^{*}=0.225m_e$ have been used [41]. The carrier density is in units of $\mathrm{cm}{}^{-2},r_s=2.61$.

Figure 3

The SdH oscillations $\delta {\rho }_{xx}/{\rho }_1^{\mathrm{LK}}$ normalized to the amplitude of the first harmonic calculated with a field independent $T_D=0.61\mathrm{K}$ vs filling factor $\nu \propto 1/B_{\perp }$ in the absence of the parallel field. Dots with connecting lines are the data, the dashed curves show the calculated oscillation envelope with empirical field-dependent $T_D=0.6(1-0.04/B_{\perp })$ K. The canonical values for $g=2.57$ and $m^{*}=0.225$ have been used [41]. The carrier density is $10.83\times {10}^{11} \mathrm{cm}{}^{-2}$ and $T=0.5$ K.

Figure 4

Normalized SdH oscillations $\delta {\rho }_{xx}/\delta {\rho }_1^{\mathrm{LK}}$ in the absence of the parallel field vs filling factor $\nu \propto 1/B_{\perp } [T=0.4$ K, carrier density $4.23\times {10}^{11} \mathrm{cm}{}^{-2}$, and $r_s\approx 4.0$] (a) over a wide range of fields $B_{\perp }=0.3\text{–}1.5$ T, and (b) oscillation fits corresponding to the sets of parameters shown in the figure.

Figure 5

Normalized SdH oscillations $\delta {\rho }_{xx}/\delta {\rho }_1^{\mathrm{LK}}$ in the absence of the parallel field vs filling factor $\nu \propto 1/B_{\perp }$ [carrier density is $4.15\times {10}^{11} \mathrm{cm}{}^{-2}$. $r_s\approx 4.0$, and $T=0.19$ K] (a) in the wide range of fields $B_{\perp }=0.3\text{–}1.5$ T, and (b) oscillation fits corresponding to the sets of parameters shown in the figure.

Figure 6

Normalized SdH oscillations $\delta {\rho }_{xx}/\delta {\rho }_1^{\mathrm{LK}}$ in the absence of the parallel field vs filling factor $\nu \propto 1/B_{\perp }$ [electron density is $6.18\times {10}^{11} \mathrm{cm}{}^{-2} \left(r_s=3.3\right),g^{*}=2.9$]. Dots are the data, the continuous curve is a fit with $\gamma =+0.07$, and the dashed curve is a fit for $\gamma =0$.

Figure 7

Normalized SdH oscillations $\delta {\rho }_{xx}/\delta {\rho }_1^{\mathrm{LK}}$ in the absence of the parallel field vs filling factor $\nu \propto 1/B_{\perp }$ [electron density is $2.63\times {10}^{11} \mathrm{cm}{}^{-2} \left(r_s=5.07\right),g^{*}=3.72$ (canonical value 3.24)]. Dots are the data, the continuous curve is a fit with $\gamma =+0.09$.

Figure 8

SdH oscillations in the presence of the parallel field: (a) $\rho \left(B_{\perp }\right)$ data for $B_{\parallel }=1.5$ T and for almost the same density as that in Fig. 2. The temperature is $T=0.15$ K. (b) Normalized oscillations $\delta \rho /\delta {\rho }_1^{\mathrm{LK}}$ (dots) and their fitting (line) with Eq. (1) using $T_D=0.87$ K, $d_1=0,\gamma =0$, and $g^{*}=2.65$.

Figure 9

Examples of SdH oscillations in the presence of the parallel field: (a) for $B_{\parallel }=2.15$ and (b) for $3.4$ T. The temperature $T=0.1$ K. Carrier density is in units of ${10}^{11} \mathrm{cm}{}^{-2}$.

Figure 10

Examples of fitting with Eq. (1): (a) $n=3.76,B_{\parallel }=2.15$ T, and (b) $n=1.815,B_{\parallel }=2.5$ T. The temperature is $T=0.1$ K and the density is in units of ${10}^{11} \mathrm{cm}{}^{-2}$. Dots are the normalized data $\delta {\rho }_{xx}/{\rho }_1^{\mathrm{LK}}$, curves are the fits with adjustable $T_D,d_1,\gamma$; $m^{*}\left(n\right)$ and $g^{*}\left(n\right)$ are the canonical values from Refs. [5, 22]. $T_{D0}=0.77$ K, $\gamma =0.8\%$, and $d_1=0.07$ for panel (a). $T_{D0}=0.87$ K, $\gamma =4$%, and $d_1=0.17$ for panel (b). Note that all ${\rho }_{xx}$ minima on panel (b) correspond to the “spin gaps,” $\nu =4i-2$ [43].

Figure 11

Examples of fitting SdH oscillations for sample Si3-10 at $n=6.178\times {10}^{11} \mathrm{cm}{}^{-2}$: (a) $B_{\parallel }=0.56$ T, $T_{D0}=0.43$ K, and $T=0.15$ K. (b) blow-up of the low field range $\nu =16$–48 of the same data. (c) $B_{\left|\right|}=2.6$ T and $T_{D0}=0.45$ K. Dots show the normalized data, the continuous and dash-dotted curves show fittings with various $\gamma$ values. Vertical arrows point at two nodes.

Figure 12

Examples of fitting SdH oscillations for sample Si3-10 at $B_{\left|\right|}=3.0$ T: for $n=1.67\times {10}^{11} \mathrm{cm}{}^{-2},\zeta =45\%$ (top) [54], and $n=1.3\times {10}^{11} \mathrm{cm}{}^{-2},\zeta =66\%$), $T_{D0}=0.67$ K (bottom). Dots show the normalized data, continuous and dashed curves show fittings with various $\gamma$ values.

Figure 13

$B_{\parallel }$ field dependencies of the skew factor $\gamma$: (a) for $n=9.53\times {10}^{11} \mathrm{cm}{}^{-2}$ extracted for a single node, (b) for $n=6.178\times {10}^{11} \mathrm{cm}{}^{-2}$ for two nodes (upper curve for larger $B_{\perp }$ and lower curve for smaller $B_{\perp }$). (c) Schematic diagram of the Landau levels for two spin subbands, in the vicinity of two sequential nodes.

Figure 14

(a) SdH oscillations fitting at low densities and in the presence of strong $B_{\parallel }$ field. $T=0.4$ K and $\zeta \left(B_{\mathrm{total}}\right)\approx$ 35%. (b) Typical temperature dependence of $\gamma$. Density is given in units of ${10}^{11} \mathrm{cm}{}^{-2}$. Sample Si6-14.

Figure 15

Typical temperature dependencies of $\gamma$ and $d_1$ measured at fixed $B_{\parallel }=1.984$ T. Density is given in units of ${10}^{11} \mathrm{cm}{}^{-2}$. Sample Si6-14.

Figure 16

The dependence $T_D\left(T\right)$ measured at $B_{\parallel }=0$ and $n=10.83\times {10}^{11} \mathrm{cm}{}^{-2}$. Full symbols show $\delta T_D=T_D\left(T\right)-T_D\left(0\right)$ for three $B_{\perp }$ fields, the connecting solid lines are guide to the eye. $T_D\left(0\right)=0.586$ K was subtracted to simplify comparison with the theory. Three dashed curves show the theoretical $\delta T_D^{*}\left(T\right)$ dependencies (divided by a factor of 10), calculated from Eq. (6) for $B_{\perp }=1$, 0.85 and 0.55 T (from bottom to top). For comparison, the dependence $\rho (T,B=0)$ is shown with empty symbols. Note a 0.1% reproducibility of $\delta \rho /\rho$ measured for the same sample Si6-14 several years apart.

Figure 17

Comparison with theory of the $T_D\left(T\right)$ variations measured at $B_{\parallel }=0$ for (a) $n=4.37\times {10}^{11} \mathrm{cm}{}^{-2}$ and (b) for $n=2.63\times {10}^{11} \mathrm{cm}{}^{-2}$. Symbols with connecting lines designate in (a) $\delta T_D=T_D-0.225$ K values measured at three $B_{\perp }$ fields and in (b) $\delta T_D=T_D-0.3$ K values measured at four $B_{\perp }$ fields. The curves are calculated from Eq. (6) for the same fields (from bottom to top) and reduced by $10\times$ in (a) and by $3\times$ in (b). For comparison, empty circles show also $\delta \rho /\rho$ variations with temperature. Sample Si6-14.

Figure 18

$T_D$ variations with temperature. Density $n=2.71\times {10}^{11} \mathrm{cm}{}^{-2}$ and $B_{\parallel }=1.984$ T. Full symbols are for $\delta T_D=T_D-0.275$ K, measured at five different $B_{\perp }$ fields. Sample Si6-14.

Figure 19

Full symbols show $T_D$ variations with temperature for three values of the $B_{\perp }$ field. Density $n=1.77\times {10}^{11} \mathrm{cm}{}^{-2}$ and $B_{\parallel }=3.375$ T. Sample Si6-14.

Figure 20

Full symbols show $\delta T_D=T_D-0.44$ K variations with temperature for two values of the $B_{\perp }$ field. Density $n=2.23\times {10}^{11} \mathrm{cm}{}^{-2}$ and $B_{\parallel }=3.36$ T. Empty circles, for comparison, depict $\delta \rho /\rho$. Sample Si6-14.

Figure 21

Variations of $T_D$ with in-plane field. The temperature is $0.05$ K and the density is $n=2.38\times {10}^{11}\mathrm{cm}{}^{-2}$. Empty circles show $\delta \rho /\rho$, full symbols show $T_D$ measured at different $B_{\perp }$ fields, corresponding to the two nodes and one antinode. Sample Si6-14.