Effect of density on microwave-induced resistance oscillations in back-gated GaAs quantum wells

TL;DR

This study investigates how carrier density influences microwave-induced resistance oscillations in GaAs quantum wells, revealing discrepancies with existing theories and highlighting the need for further research.

Contribution

It provides new experimental insights into the density dependence of MIROs in GaAs quantum wells, challenging current theoretical models.

Findings

MIRO amplitude increases with density

Oscillation extrema shift towards cyclotron resonance with density

Existing theories do not fully explain the observed behavior

Abstract

We report on microwave-induced resistance oscillations (MIROs) in a tunable-density 30-nm-wide GaAs/AlGaAs quantum well. We find that the MIRO amplitude increases dramatically with carrier density. Our analysis shows that the anticipated increase in the effective microwave power and quantum lifetime with density is not sufficient to explain the observed growth of the amplitude. We further observe that the fundamental oscillation extrema move towards cyclotron resonance with increasing density, which also contradicts theoretical predictions. These findings reveal that the density dependence is not properly captured by existing theories, calling for further studies.

Equations

$$\frac{\delta R(ϵ)}{R_0}=-2\pi ϵ{\lambda }^2\mathcal{P}\eta \sin 2\pi ϵ.$$

$$\mathcal{P}(ϵ)=\frac{{\mathcal{P}}^0}{2}\sum _{\pm }\frac{1}{(1\pm {ϵ}^{-1}{)}^2+{\beta }_{\omega }^2},{\mathcal{P}}^0=\frac{e^2{\mathcal{E}}_{\mathrm{ac}}^2{v}_F^2}{{ϵ}_{\mathrm{eff}}{\hslash }^2{\omega }^4}.$$

$$\frac{1}{{\tau }_{\mathrm{q}}}=\frac{1}{{\tau }_{\text{mbox}q0}}+\frac{1}{{\tau }_{\mathrm{ee}}}.$$

$$\frac{\hslash }{{\tau }_{\mathrm{ee}}}=\frac{\pi k_B^2{T}^2}{4E_F}\ln \frac{2\hslash v_F/a_B}{\pi k_{B}T},$$

$$\eta =\frac{\tau }{2{\tau }_{\star }}+\frac{2{\tau }_{\mathrm{in}}}{\tau },$$

Full text

Microwave-induced resistance oscillations (MIROs) appear in a two-dimensional (2D) electron gas (2DEG) (Zudov et al., 2001; Ye et al., 2001; Kärcher et al., 2016) (or a 2D hole gas (Zudov et al., 2014; Shi et al., 2014)) subjected to low temperature $T$, weak magnetic field $B$, and microwave radiation of frequency $f=\omega/2\pi$. It is well established experimentally that MIROs originate from the bulk of the 2DEG (Zhang et al., 2007; Hatke et al., 2008a, b; Bykov et al., 2010; Khodas et al., 2010; Fedorych et al., 2010; Andreev et al., 2011; Levin et al., 2015; Dorozhkin et al., 2016; Herrmann et al., 2016a; Herrmann et al., 2017). Theoretically, microwave photoresistance $\delta R$ due to MIROs can be described by (Dmitriev et al., 2005, 2009, 2012)

[TABLE]

Here, $R_{0}$ is the resistance at $B=0$, $\epsilon=\omega/\omega_{\mbox{\scriptsize{c}}}$, $\omega_{\mbox{\scriptsize{c}}}=eB/m^{\star}$ is the cyclotron frequency of the charge carrier with the effective mass $m^{\star}$, $\lambda=\exp(-\epsilon/2f\tau_{\rm q})$ is the Dingle factor, $\tau_{\rm q}$ is the quantum lifetime, ${\mathcal{P}}$ is the effective microwave power, and $\eta$ is the dimensionless parameter (discussed later in detail) which depends on the disorder characteristics and the inelastic relaxation. The above expression was obtained assuming $2\pi k_{B}T\gg\hbar\omega$ and is accurate away from the cyclotron resonance ($2\pi\epsilon\gg 1$), when the microwave power is not too high (${\mathcal{P}}\ll 1$), and when Landau levels are overlapping ($\lambda\ll 1$).

To date, MIROs have been observed in three kinds of material systems, namely, GaAs/AlGaAs (Zudov et al., 2001; Ye et al., 2001), Ge/SiGe (Zudov et al., 2014; Shi et al., 2014), and MgZnO/ZnO (Kärcher et al., 2016) heterostructures, and the dependence on $\epsilon$, $\delta\rho_{\omega}\propto-\epsilon\lambda^{2}\sin 2\pi\epsilon$, has been verified in many experiments. In addition, it was established that while MIROs at high microwave power significantly deviate from Eq. (1), they can still be well described within the same theoretical framework after generalization to an arbitrary radiation intensity (Khodas et al., 2010; Hatke et al., 2011a; Shi et al., 2017a). At the same time, experiments also revealed situations when existing theory is inadequate, e.g., in describing the measured dependencies on radiation polarization (Smet et al., 2005; Herrmann et al., 2016a) and on temperature (Shi et al., 2016). Limitations of the theory were also identified in the regime of separated Landau levels (Hatke et al., 2011b) and in the radiation-induced modification of Shubnikov$-$de Haas oscillations (Shi et al., 2015).

One important parameter, whose role has remained largely unexplored, is the carrier density $n_{\rm e}$. While it has been recently demonstrated that $n_{\rm e}$ affects $\epsilon$, presumably through interaction-induced renormalization of the effective mass $m^{\star}$ (Shchepetilnikov et al., 2017; Fu et al., 2017), it should also modify other quantities, e.g., ${\mathcal{P}}$ and $\eta$, entering Eq. (1). Since the density dependencies of ${\mathcal{P}}$ and $\eta$ are both known theoretically, MIRO measurements as a function of $n_{\rm e}$ should provide an important test to existing microscopic description of microwave photoresistance.

In this Rapid Communication we investigate the effect of the carrier density on the MIRO amplitude employing a tunable-density 2DEG (Watson et al., 2015; Shi et al., 2017b; Fu et al., 2017; Qian et al., 2017). We find that the quantum lifetime depends on the carrier density only weakly, in agreement with the recent study investigating Shubnikov$-$de Haas oscillations in a similar device (Qian et al., 2017). Our main finding, however, is a significant growth of the MIRO amplitude with the carrier density. The analysis shows that this growth cannot be accounted for by the anticipated density dependence of ${\mathcal{P}}\eta$ entering the prefactor of Eq. (1) (Khodas and Vavilov, 2008; Zhang et al., 2014; Dmitriev et al., 2005, 2009, 2012). Furthermore, we find that the fundamental extrema move towards $\epsilon=1$ which also contradicts theoretical expectations. Both of these findings indicate that our understanding of the role of density in microwave photoresistance remains limited calling for further investigations.

Our 2DEG resides in a 30-nm GaAs/AlGaAs quantum well located about 200 nm below the sample surface. The structure is doped in a 2-nm GaAs quantum well at a setback of 63 nm on a top side. The in situ back gate consists of an $n^{+}$ GaAs layer situated 850 nm below the bottom of the quantum well. Ohmic contacts were fabricated at the corners and midsides of the lithographically defined $1\times 1$ mm2 van der Pauw mesa. The density of the 2DEG was varied from $n_{\rm e}\approx 1.41$ to $2.87\times 10^{11}$ cm*-2*. Over this density range, the low-temperature electron mobility increased from $\mu\approx 0.37$ to $\mu\approx 1.1\times 10^{7}$ cm2V*-1s-1* (not, a), roughly following $\mu\propto n_{\rm e}^{\alpha}$, with $\alpha\approx 1.5$ (not, b; Sammon et al., 2018). Microwave radiation, generated by a synthesized sweeper, was delivered to the sample immersed in liquid 3He via a rectangular (WR-28) stainless steel waveguide. The resistance $R$ was measured using a standard low-frequency (a few Hz) lock-in technique.

In Fig. 1(a) we present $R/R_{0}$ vs $1/\epsilon$ for three different densities, $n_{\rm e}\approx 1.41$ (bottom trace), $2.15$ (middle trace), and $2.87\times 10^{11}$ cm*-2* (top trace), measured at $T=1.5$ K under irradiation by microwaves of $f=34$ GHz. The data show that MIROs become significantly stronger with increasing density. We next extract the oscillatory correction $\delta R/R_{0}$ and present the result in Fig. 1(b) as a function of $\epsilon$. All three data sets reveal an expected periodicity with $\epsilon$, in agreement with Eq. (1).

According to Eq. (1), the MIRO amplitude is proportional to the effective microwave power, which was shown to be (Chiu et al., 1976; Khodas and Vavilov, 2008)

[TABLE]

Here, $\beta_{\omega}\equiv(\omega\tau_{\rm em})^{-1}+(\omega\tau)^{-1}$, $\tau=(m^{\star}/e)\mu$ is the momentum relaxation time, $\tau_{\rm em}^{-1}=n_{\rm e}e^{2}/2\sqrt{\epsilon_{\mathrm{eff}}}\epsilon_{0}m^{\star}c$ (Chiu et al., 1976) is the radiative decay rate, $2\sqrt{\epsilon_{\mathrm{eff}}}=\sqrt{\varepsilon}+1$ defines the effective dielectric constant $\epsilon_{\mathrm{eff}}$, $\varepsilon=12.8$ is the dielectric constant of GaAs, $v_{F}$ is the Fermi velocity, and $\mathcal{E}_{\mathrm{ac}}$ is the microwave electric field. The density dependence of $\beta_{\omega}$ has been recently verified in time-resolved measurements of the cyclotron resonance (Zhang et al., 2014). Within the density range studied in our experiment, $\tau\gg\tau_{\rm em}$ and $\beta_{\omega}\approx(\omega\tau_{\rm em})^{-1}\propto n_{\rm e}$. However, $\beta_{\omega}$ remains much smaller than unity and, as a result, ${\mathcal{P}}$ increases with $n_{\rm e}$ for all $\epsilon$ except in close proximity to $\epsilon=1$. We will see, however, that the anticipated increase in ${\mathcal{P}}$ is rather small and contributes little to the growth of MIRO shown in Fig. 1.

The growth of MIRO with $n_{\rm e}$ observed in Fig. 1 can, in principle, stem from $\tau_{\rm q}$ (entering $\lambda$) or $\eta$. Both of these parameters are readily available from the Dingle analysis. Following Eq. (1), we introduce a reduced MIRO amplitude $\mathcal{A}=|\delta R|_{\max}/2\pi\epsilon{\mathcal{P}}R_{0}$ (not, d), where $|\delta R|_{\max}$ is the MIRO amplitude, and present it in Fig. 2 as a function of $\epsilon$ for $n_{\rm e}=1.41$ (circles), $2.15$ (squares), and $2.87\times 10^{11}$ cm*-2* (triangles). Fits to the data with $\mathcal{A}_{0}\exp(-\epsilon/f\tau_{\rm q})$ (solid lines) yield $\tau_{\rm q}\approx 19.5$, $20.0$, and $21.2$ ps, respectively, indicating a slight increase of $\tau_{\rm q}$ with $n_{\rm e}$. In contrast, the intercept of the Dingle plots, $\mathcal{A}_{0}$, grows substantially with $n_{\rm e}$. As we show below, theory predicts that under our experimental conditions $\mathcal{A}_{0}$ can only decrease with $n_{\rm e}$.

After repeating the Dingle analysis for other $n_{\rm e}$, we present the density dependence of $\tau_{\rm q}$ (circles) in Fig. 3. A slight increase of $\tau_{\rm q}$ with $n_{\rm e}$ appears to contradict a recent study (Qian et al., 2017), which has found a saturation of $\tau_{\rm q}$ at $n_{\rm e}\approx 2\times 10^{11}$ cm*-2* and a monotonic decrease at higher $n_{\rm e}$. This discrepancy can be alleviated by recalling that Shubnikov$-$de Haas oscillations employed in Ref. Qian et al., 2017 yield only impurity contributions to the quantum lifetime $\tau_{\mbox{\scriptsize{q0}}}$ (Martin et al., 2003; Adamov et al., 2006). The quantum lifetime obtained from MIROs, on the other hand, is reduced by electron-electron scattering. More specifically (Chaplik, 1971; Giuliani and Quinn, 1982; Ryzhii and Suris, 2003; Ryzhii et al., 2004; Hatke et al., 2009; Dmitriev et al., 2009),

[TABLE]

Under the conditions of our experiment the electron-electron scattering rate is given by (Chaplik, 1971; Giuliani and Quinn, 1982; Dmitriev et al., 2005)

[TABLE]

where $E_{F}$ is the Fermi energy and $a_{B}\approx 11$ nm is the Bohr radius in GaAs. Using Eqs. (3) and (4) we compute $\tau_{\mbox{\scriptsize{q0}}}$ and present the results (squares) in Fig. 3. The results show that the impurity-limited quantum lifetime decreases slightly with $n_{\rm e}$, in general agreement with Ref. Qian et al., 2017. We note, however, that in our experiment most of this decrease takes place at densities below $\approx 2\times 10^{11}$ cm*-2*.

As already mentioned, Fig. 2 also reveals a significant increase of the intercept of the Dingle fits, given by $\mathcal{A}_{0}$, with increasing $n_{\rm e}$. Since $\mathcal{A}_{0}\propto\eta$ (not, e), this increase reflects the increase in $\eta$, provided that the density dependence of ${\mathcal{P}}$ is accurately described by Eq. (2). To quantify this increase we introduce a parameter $\kappa=\mathcal{A}_{0}(n_{\rm e})/\mathcal{A}_{0}(n_{l})$, where $n_{l}=1.41\times 10^{11}$ cm*-2* is the lowest density studied. As shown in Fig. 4, $\kappa$ (circles) increases by a factor of about 3 over the investigated density range. This finding is unexpected since, as we show next, one should anticipate a decrease of $\eta$ with increasing $n_{\rm e}$.

The dimensionless scattering rate $\eta$ is given by (Dmitriev et al., 2009)

[TABLE]

where the first (second) term represents displacement (Durst et al., 2003; Lei and Liu, 2003; Vavilov and Aleiner, 2004) (inelastic (Dmitriev et al., 2005, 2009)) contribution. Here, $\tau_{\rm in}\approx 0.82\tau_{\rm ee}$ (Dmitriev et al., 2005) and $\tau/2\tau_{\star}$ (not, f) can vary between $\tau/2\tau_{\star}=6(\tau/\tau_{\mbox{\scriptsize{q0}}}+3)^{-1}$ (smooth disorder limit) and $\tau/2\tau_{\star}=3/2$ (sharp disorder limit) according to the mixed-disorder model (Vavilov et al., 2007; Dmitriev et al., 2009). As a result, the relative change in $\eta$ (or $\mathcal{A}_{0}$) with $n_{\rm e}$ is expected to fall between $\kappa_{\rm sm}$ and $\kappa_{\rm sh}$, given by $\eta(n_{\rm e})/\eta(n_{l})$ evaluated in the smooth and the sharp disorder limit, respectively. On a qualitative level, the decrease of $\tau/2\tau_{\star}$ with $n_{\rm e}$ can be expected whenever $\tau_{\rm q}/\tau\ll 1$, i.e., when small angle scattering dominates, which is the case for all modern high-mobility GaAs quantum wells. This decrease should occur because $\tau_{\star}^{-1}$ (not, f) is less sensitive to small-angle scattering than $\tau^{-1}$ and because the characteristic scattering angle decreases with density.

As shown in Fig. 4, both $\kappa_{\rm sh}$ (squares) and $\kappa_{\rm sm}$ (triangles) monotonically decrease with $n_{\rm e}$. The decrease in $\kappa_{\rm sh}$ with $n_{\rm e}$ occurs solely due to the weakening of the inelastic contribution, given by the second term in Eq. (5). This weakening, in turn, owes to a superlinear increase in the momentum relaxation time $\tau$, which wins over the slightly sublinear increase in $\tau_{\rm in}$ [see Eq. (4)]. In the smooth disorder limit, characterized by $\kappa_{\rm sm}$, the decrease becomes larger due to the growing ratio of $\tau/\tau_{\mbox{\scriptsize{q0}}}$ which enters the denominator of the displacement contribution [first term in Eq. (5)]. We thus conclude that regardless of the exact disorder characteristics, theoretical predictions are in contrast with the experimentally obtained $\kappa$ (circles) which shows a significant increase over the density range studied (not, g). Our findings were confirmed by measurements using $f=39.5$ GHz in another sample which are discussed in Supplemental Material (not, h).

We next examine the effect of density on the positions of the MIRO extrema near the cyclotron resonance. As noted from the data in Fig. 1(b), these extrema move closer towards $\epsilon=1$ with increasing density. To examine this behavior quantitatively, we introduce a parameter $\varphi=(\epsilon^{-}-\epsilon^{+})/2$, where $\epsilon^{-}$ ($\epsilon^{+}$) is the position of the fundamental minimum (maximum). We then present obtained $\varphi$ (circles) in Fig. 5(a) as a function of $n_{\rm e}$ and observe that it monotonically decreases with $n_{\rm e}$. Similar to $\tau_{\mbox{\scriptsize{q0}}}$, the decrease is more pronounced at lower densities. Theory, however, predicts just the opposite behavior; as illustrated in Fig. 5(a), the calculated values of $\varphi_{\rm sh}$ (squares) and $\varphi_{\rm sm}$ (triangles), representing sharp and smooth disorder limits, respectively, both increase with $n_{\rm e}$. The expected growth of $\varphi_{\rm sh}\approx\varphi_{\rm sm}$ with $n_{\rm e}$ occurs, for the most part, due to the increase in $\beta_{\omega}$, dominated by $\tau_{\rm em}^{-1}\propto n_{\rm e}$, which controls the sharpness of ${\mathcal{P}}$ near $\epsilon=1$. Indeed, as shown in Fig. 5(b), ${\mathcal{P}}(\epsilon)$ is considerably sharper at $n_{\rm e}=1.41\times 10^{11}$ cm*-2* (solid line) than at $n_{\rm e}=2.87\times 10^{11}$ cm*-2* (dotted line).

It is known that the phase reduction can occur with increasing ${\mathcal{P}}$ due to contributions from multiphoton processes (Hatke et al., 2011a; Shi et al., 2017a). This scenario, however, can be ruled out since ${\mathcal{P}}$, in fact, decreases with $n_{\rm e}$ near $\epsilon=1$ within the investigated density range. As shown in Fig. 5(c), ${\mathcal{P}}$ at the fundamental MIRO extrema (solid and open circles) exhibits a slight overall decrease within the studied density range, similar to ${\mathcal{P}}$ at $\epsilon=1$ (solid line). At higher MIRO orders, ${\mathcal{P}}$ monotonically increases, as illustrated by dashed and dotted lines computed for $\epsilon=2$ and $\epsilon=3$, respectively. This increase in ${\mathcal{P}}$ occurs because ${\mathcal{P}}_{0}\propto n_{\rm e}$ while $\beta_{\omega}$ remains relatively small within the studied density range.

One somewhat uncertain parameter is $\epsilon_{\mathrm{eff}}$ which affects $\beta_{\omega}$ entering ${\mathcal{P}}$ given by Eq. (2). Indeed, the expression we have used is generally valid only when the overall sample thickness greatly exceeds the radiation wavelength, a condition which is not satisfied for the microwave frequency used in our experiment. According to Ref. Dmitriev et al., 2012, a better approximation would be using $\epsilon_{\mathrm{eff}}=1$ which would increase the value of $\beta_{\omega}$ by approximately a factor of 2. However, any increase in $\beta_{\omega}$ would only weaken (or even reverse) the density dependence of ${\mathcal{P}}$ and further increase the disagreement between theory and experiment, both in $\eta$ and $\varphi$.

Finally, we note that by applying the gate voltage we are not only changing the carrier density but also modifying the confinement potential. Numerical simulations show that the 2DEG is pulled away from the top interface towards the center of the quantum well and becomes wider with increasing density. Whether or not such a change of the confinement plays any significant role in the observed enhancement of the MIRO amplitude is unclear at this point and is left for future studies. To investigate this possibility, it would be interesting to perform measurements in different structures, such as heterojunction-insulated gate field-effect transistors, in which confinement becomes stronger with increasing carrier density.

In summary, we have investigated the effect of the carrier density $n_{\rm e}$ on the MIRO amplitude in a high-mobility modulation-doped GaAs/AlGaAs quantum well equipped with an in situ back gate. Our main finding is a significant growth of the MIRO amplitude with increasing density. A Dingle analysis shows that this increase originates primarily from ${\mathcal{P}}\eta$ entering Eq. (1) and not from a slight increase of $\tau_{\rm q}$. This finding is in conflict with theoretical expectations which predict a modest increase of ${\mathcal{P}}$ and a decrease of $\eta$ with increasing density. We further find that the MIRO extrema near the cyclotron resonance move toward each other with increasing $n_{\rm e}$ whereas the theory predicts just the opposite behavior. These findings indicate that our understanding of microwave photoresistance is still lacking and needs further examination.

#Present address: Microsoft Station-Q at Delft University of Technology, 2600 GA Delft, The Netherlands