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Resonance oscillations of magnetoresistance in double quantum wells

We present experimental and theoretical studies of the magnetoresistance oscillations induced by resonance transitions of electrons between tunnel-coupled states in double quantum wells. The suppression of these oscillations with increasing temperature is irrelevant to the thermal broadening of the Fermi distribution and reflects the temperature dependence of the quantum lifetime of electrons. The gate control of the period and amplitude of the oscillations is demonstrated.


I. INTRODUCTION
The Landau quantization 1 of electron states manifests itself in oscillations of various physical quantities as functions of the applied magnetic field. The magnetoresistance oscillations in solids belong to the most fundamental quantum phenomena of this kind 2 . In particular, the Shubnikov-de Haas oscillations 2 (SdHO) originating from sequential passage of the Landau levels through the Fermi level, are always present in three-dimensional (3D) and two-dimensional (2D) electron system with metallic type of conductivity. Investigation of these oscillations is a powerful source of information about band structure, Fermi surface, and electron interaction mechanisms.
The SdHO are strongly damped with increasing temperature as the thermal broadening of the Fermi distribution exceeds the cyclotron energy. In quasi-2D electron systems, which are realized in quantum wells with two (or more) occupied subbands, the possibility of intersubband transitions of electrons leads to another kind of magnetoresistance oscillations 3 which are not damped considerably when the temperature raises. Such oscillations, known as the magneto-intersubband oscillations (MISO), have been studied in single quantum wells with two populated 2D subbands 3,4,5,6,7,8 and, theoretically, in 2D layers where electron states are split owing to the spin-orbit interaction 9 . The physical mechanism of the MISO is the periodic modulation of the probability of intersubband transitions by the magnetic field as the different Landau levels of the two subbands sequentially come in alignment. Oscillations of similar origin are present in 3D systems such as layered organic conductors with weak coupling between the layers. 10,11,12 As compared to the SdHO, the MISO show a stronger suppression by the disorder, so their observation requires clean (high-mobility) samples. A decrease of the MISO amplitude with increasing temperature reflects temperature dependence of the quantum lifetime of electrons. Experimental studies of the MISO allow one to determine the quantum lifetimes in the region of temperatures where the SdHO disappear completely. Another advantage of the MISO is the possibility of direct and precise determination of the intersubband energy gap, which otherwise can be found from an analysis of the double periodicity of the SdHO observed at low temperatures in various quasi-2D electron systems such as single quantum wells with two populated 2D subbands, 4,6 double quantum wells, 13,14 and 2D systems with spin-orbit splitting. 15,16 The double quantum wells (DQWs), which consist of two quantum wells separated by a barrier, and where the electrons occupy two 2D subbands coupled by tunneling, should be recognized as the most convenient system for studying the MISO phenomenon, owing to the unique possibility to control both intersubband energy gap and probabilities of intersubband transitions in a wide range by using barriers of different widths and biasing the structure by external gates. Nevertheless, no studies of the MISO have been reported so far for these particular systems. In this paper we report on the first observation and systematic investigation of the MISO in symmetric highmobility (µ ≈ 10 6 cm 2 /Vs) GaAs DQWs with different barrier widths. Along with the experimental results, we present a theoretical description of magnetotransport in DQWs, which explains the results of our measurements.
The paper is organized as follows. The experimental results are described in Sec. II. The theoretical description and the discussion of the results are presented in Sec. III. The concluding remarks are given in the last section.

II. EXPERIMENT
The samples are symmetrically doped GaAs DQWs with equal widths d W = 14 nm separated by Al x Ga 1−x As barriers of different thickness d b varied from 1.4 to 5 nm. The samples have high total sheet electron den- sity n s ≈ 9 × 10 11 cm −2 (4.5 × 10 11 cm −2 per one layer). Both layers are shunted by ohmic contacts. The densities in the wells are variable by a gold top gate. The voltage of the gate, V g , changes the density of the well closest to the sample surface, with carrier density in the other well being almost constant. The system is balanced (has equal densities in the wells) at V g = 0. The sample parameters are shown in Table 1. Over a dozen specimens of both the Hall bar and van der Pauw geometries from four wafers have been studied. We measure both longitudinal and Hall resistances at the temperatures T from 0.3 to 50 K and magnetic fields B, directed perpendicular to the well plane, up to 12 T using conventional ac-locking techniques with a bias current of 0.1-1 µA.  Figure 1 shows the low-field part of the magnetoresistance R xx (B) for the sample with d b = 1.4 nm and balanced densities at different temperatures. At T = 0.3 K we see double-periodic SdHO in the form of beating pattern, 13,14 since the energy splitting of 2D subbands in DQWs is small in comparison to the Fermi energy. With the increase of T the small-period SdHO are strongly suppressed and the remaining long-period oscillations can be identified as the MISO, by analyzing their 1/B periodicity and temperature dependence as described below. First, assuming that the magnetoresistance max-ima correspond to the Landau-level alignment condition 5 ∆ T = k ω c , where ∆ T is the intersubband splitting energy (∆ T = ∆ SAS for balanced DQWs), ω c = |e|B/mc is the cyclotron frequency (e is the electron charge, m the effective mass, and c the velocity of light), and k is an integer, we find ∆ T = 4.22 meV, which is close enough to the calculated value of ∆ SAS (see Table 1). Next, the amplitudes of the long-period oscillations, in contrast to the SdHO amplitudes, are saturated at low temperatures. The dependence of R xx (B) in the temperature interval 4.2 < T < 25 K [ Fig. 1 (b)] shows that the oscillations are present at high temperatures. The amplitudes of the oscillations are up to 30% of the total resistance, so these oscillations cannot be attributed to magnetophonon resonances whose relative amplitudes are much smaller. 17 We have checked the 1/B periodicity of the oscillations for the structures with different barriers. Figure  2 shows the magnetoresistance traces versus 1/B, while the results of the fast Fourier transform (FFT) of the magnetoresistance data are shown in the insert. The peak corresponding to the main period shifts towards lower frequencies with increasing d b , and the frequency of the FFT peaks has nearly exponential dependence on d b . Since the splitting energy in the balanced DQWs is known to decrease exponentially with increasing barrier width, this observation is another confirmation of the intersubband origin of the oscillations. Figure 3 shows the low-field part of the magnetoresistance for the sample with d b = 1.4 nm at various fixed gate voltages away from the symmetry point at V g = 0 V. As the absolute value of V g increases, the oscillations are suppressed and their frequency becomes higher. The positive magnetoresistance is clearly seen in these plots.

III. THEORY AND DISCUSSION
The mechanism responsible for the oscillations observed relies on intersubband scattering, resonantly modulated by Landau quantization. A quantitative description of magnetotransport in DQWs taking into account elastic scattering of electrons is based on the Hamiltonian written as the following 2 × 2 matrix in the single-well eigenstate basis: 18 where π = −i ∇ r − (e/c)A r is the kinematic momentum operator, A r is the vector potential describing the magnetic field, V i r are the random potentials of impurities and other inhomogeneities in the upper (i = u) and lower (i = l) well, and r = (x, y) is the in-plane coordinate. Next, ∆ SAS denotes the subband splitting owing to the tunnel coupling, and ∆ is the splitting in the absence of tunneling. In symmetric (balanced) DQWs ∆ = 0. According to Eq. (1), the free-electron energy spectrum is given by two sets of Landau levels of the upper (+) and lower (−) subbands: ε ± n = ±∆ T /2 + ε n , where ε n = ω c (n + 1/2) and ∆ T = ∆ 2 SAS + ∆ 2 is the total intersubband gap. In symmetric DQWs ∆ T = ∆ SAS is the energy gap between the states with symmetric and antisymmetric wave functions ψ (−) z and ψ (+) z . Since we consider weak enough magnetic fields, the Zeeman splitting is neglected.
Calculation of the diagonal (σ d ) and non-diagonal (σ ⊥ ) components of conductivity tensor uses the Kubo formalism and Green's function technique in the Landau-level representation. 19 In the model of short-range symmetric scattering potential characterized by the correlator ≃ wδ ii ′ δ(r − r ′ ), and in the limit of weak disorder, when the scattering rate 1/τ = mw/ 3 is small in comparison to ∆ T / , one has and σ ⊥ = |e|cn s /B + δσ ⊥ with where f ε is the equilibrium Fermi distribution and The quantities Σ (±) ε are the self-energies for + and − states. Their real (Σ (±) ′ ε ) and imaginary (Σ (±) ′′ ε ) parts can be found using the self-consistent Born approximation 20 generalized for DQWs: where δ 2 = (∆/∆ T ) 2 . The energies ±∆ T /2 are included in the self-energies for notational convenience. Equations and leads to the following expression for the resistivity where ρ 0 = m/e 2 n s τ is the zero-field Drude resistivity, and The Fermi energy ε F is counted from the middle point between the subbands, and the function T = (2π 2 T / ω c )/ sinh(2π 2 T / ω c ) describes thermal suppression of the resistivity oscillations. The second (∝ e −α ) term in Eq. (6) describes usual SdHO with beatings, while the last (∝ e −2α ) term contains the MISO contribution which does not have the thermal suppression (the temperature-dependent corrections to this term are neglected because they are small in comparison to the second term). The expression for the coefficients a ± can be rewritten in terms of intersubband and intrasubband scattering times in DQWs, which are given by τ inter = 2τ /(1 − δ 2 ) and τ intra = 2τ /(1 + δ 2 ), respectively, so that τ −1 = τ −1 intra + τ −1 inter . This representation shows that the MISO contribution (at the coefficient a − ) is proportional to the intersubband scattering rate τ −1 inter . As the temperature raises, the second term in the righthand side of Eq. (6) vanishes, and only the MISO remain. These oscillations are periodic in 1/B and have maxima under the condition ∆ T = k ω c . In balanced DQWs (δ = 0 and a + = a − ), when the electron density distributions |ψ (−) z | 2 and |ψ (+) z | 2 for − and + states are equal, the probability of intersubband scattering is high. As the system is driven out of the balance by the gates (δ 2 increases), the wave functions for − and + subbands are localized in the different wells, and intersubband scattering is suppressed, so the MISO amplitude (coefficient a − ) is reduced, while the non-oscillating positive manetoresistance (coefficient a + ) becomes larger. This behavior is seen in experiment (Fig. 3), along with enhanced frequency of the oscillations owing to larger intersubband separation. Equation (6) shows that the resistivity in the minima, when ∆ T = (k + 1/2) ω c , is equal to ρ 0 for balanced DQWs in low magnetic fields. This feature, also seen in our experiment, is described by the fact that the total density of states, (mω c /π 2 )[S (+) ′′ ε +S (−) ′′ ε ], given by a superposition of the densities of states in + and − subbands, does not oscillate with energy under the condition ∆ T = (k + 1/2) ω c , so the magnetic field has no effect on the scattering and resistivity under this condition. For a more detailed comparison of theory and experiment, we have carried out a fit of the magnetoresistance (6) to the data obtained for the balanced DQWs with d b = 1.4 nm at T = 4.2 K. In order to fit the MISO frequency at low B, we have taken ∆ T = 4.22 meV, while the amplitude fit has been done assuming that the ratio of the transport time τ tr = 37 ps determined from the mobility to the quantum lifetime τ standing in the Dingle factor is equal to 12 (large ratios of τ tr to τ are normally met in modulation-doped heterostructures). With these parameters used, we obtain an excellent agreement of theory and experiment in the region of magnetic fields up to 0.4 T; see Fig. 4. A good agreement is also obtained for lower temperatures, when the SdHO are present. Above 0.4 T, the observed MISO frequency becomes slightly smaller than expected from the theory. We do not know exact origin of this discrepancy, and assume that it is possibly related to modification of the tunnel coupling by the magnetic field. 21 The amplitude fit of the MISO in a wide region of temperatures allows us to plot the temperature dependence of the quantum lifetime of electrons. We have found that τ is constant at T < 1 K and decreases at higher temperatures. This decrease cannot be attributed to quasielastic scattering of electrons by acoustic phonons, because the acoustic-phonon scattering rates in GaAs quantum wells are more than one order smaller in the described temperature region. On the other hand, the consideration of electron-electron scattering in DQWs 18,22 gives a reasonable explanation of the observed temperature behavior. To prove this, we have compared the experimental dependence shown in the insert to Fig. 4 with the theoretically predicted dependence 18,22 where T 0 = ε F /τ (0) and λ is a numerical coefficient of the order of unity, considered here as a single fitting parameter. Using the experimentally determined values ε F ≃ 17 meV and /τ (0) ≃ 0.2 meV, we have found a good agreement between experiment and theory in the whole temperature region for λ = 3.5.

IV. CONCLUSIONS
In conclusion, we have observed and investigated the magnetoresistance oscillations caused by resonant modulation of elastic intersubband scattering in DQWs by Landau quantization. These oscillations survive at high temperatures and represent a prominent and well-reproducible feature of magnetotransport in highmobility DQWs.
The oscillations we observe are caused by the same physical mechanism as the magneto-intersubband oscillations (MISO) previously studied in single quantum wells with two populated subbands. However, there are qualitatively different features in manifestation of the MISO phenomenon in DQWs. First, since the Fermi energy in DQWs is typically much larger than the subband separation, the MISO have a large period as compared to the SdHO period, and they are clearly identifiable without an additional Fourier analysis. For the same reason, one can observe all intersubband resonances up to the last one, when the cyclotron energy equals the subband separation, in relatively weak magnetic fields (see Fig. 1 b). Next, since the intesubband scattering is DQWs is easily controllable by modifying the wave functions of electrons via the gate, the MISO appear to be very sensitive to the gate voltage. We also point out that observation of the large-amplitude MISO requires large quantum lifetimes and high probability of intersubband scattering. These two requirements are difficult to satisfy simultaneously in single quantum wells with two populated subbands, because large quantum lifetimes are usually attainable in the modulation-doped systems, where the scattering probability strongly decreases with increasing momentum transfer, and the intersubband scattering, which requires a large momentum transfer, is suppressed. 5 In contrast, the intersubband scattering in DQWs does not require a large momentum transfer, and its probability is comparable to the probability of intrasubband scattering.
Theoretical consideration of the magnetotransport in DQWs leads to an analytical expression for the oscillating resistivity which survives at high temperatures, the last term in Eq. (6). This expression, as far as we know, has not been derived previously. Its simple form is a consequence of symmetric scattering, when the subbands are characterized by a single quantum lifetime τ , common for both subbands. The quantum lifetime can be extracted from the SdHO amplitudes at low temperatures, while the measurements of the MISO amplitudes allow us to find its temperature dependence in a wide region of temperatures. This dependence becomes an independent confirmation of the influence of electron-electron scattering on the quantum lifetime of 2D electrons.