Magnetization of bilayer two-dimensional electron systems

We have measured the magnetization of strongly coupled bilayer two-dimensional electron systems. When sweeping the magnetic field steps appear in the magnetization whenever a transition between discrete energy levels takes place. At magnetic fields for which the Landau-level splitting dominates, the steps occurring at total filling factors νT = 4j are related to transitions between adjacent Landau levels j − 1 and j; such steps can also be observed in single layers. Additional magnetization steps showing up in bilayers at half Landau-level filling νT = 4j + 2 are associated with transitions from a symmetric to an antisymmetric state inside the same Landau level j. The observed size of the Landau-level steps in bilayers is considerably lower than expected theoretically. These findings are explained using a model with a large background density of states.


Introduction
The magnetization of an electron system is one of the few physical properties which can directly access its ground state. In particular, magnetization experiments give an insight into the single-particle energies as well as correlation effects. In semiconductor-based two-dimensional electron systems (2DESs), the first efforts of measuring the magnetization were focussed on single layers [1]- [5]. Only recently the magnetization of more complex systems also exhibiting electron-correlation effects were investigated [6]- [8]. In this paper, we present experiments on the magnetization of strongly coupled bilayer 2DESs. We will show that we can visualize transitions between Landau levels as well as intra-Landau-level electronic transitions from a symmetric to an anti-symmetric state as steps in the magnetic-field dependence of the magnetization. Using a model involving a finite background density-of-states (DOS) [9] we will explain the low magnetization steps for Landau-level transitions.

Magnetization and torque
In most general terms the magnetization M of an electron system with constant particle number N is defined from its free energy F as dF M = ------dB .
(1) N This magnetization leads to a torque on the sample given by 3 Institute of P h y s ic s ^DEUTSCHE PHYSIKALISCHE GESELLSCHAFT The first term, M x B, represents the torque of an anisotropic magnetic moment with a component perpendicular to the applied magnetic field. The second term arises from a magnetic force V (M ■ B) on the sample; it is only present when the centre of mass is situated at a distance d from the torque axis and when the sample is placed in a field gradient.

Magnetization of a two-dimensional (2D) electron system
A quantizing magnetic field Bn perpendicular to the 2DES leads to the formation of Landau levels at discrete energies, Ej = (j + 1 )fiwc, where Mc = eBn/m * is the cyclotron frequency for electrons with an effective mass m*. In a bilayer, each Landau level is split into a symmetric and an antisymmetric state with a symmetric-antisymmetric energy splitting Asas. In GaAs the additional spin splitting of these levels is so small that it will be neglected for simplicity in the following. When increasing the magnetic field, the levels are successively depopulated. At transition points where one level becomes totally empty the field dependence of the free energy changes leading to a step in the magnetization perpendicular to the 2DES. The step size AM± is given by the Maxwell relation N AM± = -A,x.
(3) Bn Here A/x is the change of the chemical potential at the transition point and N is the total number of electrons.
For a bilayer 2DES with an additional degree of freedom into the third dimension the free energy also depends on the in-plane magnetic field By. With equation (1) this leads to an additional in-plane magnetization My. However, since the Landau-level occupancy only depends on the perpendicular field component, the in-plane magnetization does not display any discontinuity.
In our experiments on 2DESs with a strongly anisotropic magnetization the sample is placed symmetrically onto the torque wire in the centre of the field. Due to the magnetization anisotropy a finite contribution M T x B T can be measured. The torque resulting from it can be written in two components as: where ft is the angle between the 2DES normal and the applied magnetic field. The first term describes the magnetization perpendicular to the 2DES; it contains both magnetization steps arising from discontinuities in the chemical potential as well as a continuous background. The second term, originating from magnetization components in the 2DES-plane, is fully continuous.

Torque magnetometer with optical detection
The precise working principle of our magnetometer is illustrated in figure 1 [10]. A 2DES sample is mounted onto a torque wire with a finite angle ft between the sample normal and the direction of the magnetic field. As described in equation (4) a magnetization of the sample leads to a torque which results into a small rotation of the sample around the torque axis. The sample rotation is detected optically by focusing a laser beam reflected on the backside of the sample onto 4 Institute of P h y s ic s ^DEUTSCHE PHYSIKALISCHE GESELLSCHAFT a quadrant detector. In order to keep the sample at a fixed angle we counteract the torque using the magnetic moment Mfb = IA of a feedback coil glued on the back of the sample (I is the current through the feedback coil and A its cross-sectional area). Discontinuities in M± will directly show up as steps in Mfb, AM ± = -AMfb. Additionally, Mfb contains a continuous contribution arising from the continuous part in M±, from My and from the substrate magnetization. Operation in feedback mode also allows to damp mechanical noise in the set-up by means of an active PIDcontrol which considerably enhances the experimental sensitivity.

Samples
In this paper, we present the magnetization of two bilayer 2DESs, one with a weak and one with a strong interlayer coupling. The samples consist of two 10 nm wide GaAs quantum wells embedded in Ga0.65Al0.35As. Electrons are provided by modulation doping the left and right barriers symmetrically with Si. Magnetotransport experiments on reference bilayer samples with similar layer sequence and doping and equipped with a top gate, have shown that the layers of our ungated sample are well balanced. In the weakly coupled bilayer a 4 nm barrier separates the two quantum wells, the total electron concentration is n = 9.2 x 1011 cm-2 and the Hall mobility 4.2 x 105cm2 Vs-1. The sample area is 0.37 cm2. The strongly coupled bilayer has a 2.5 nm barrier, a total electron concentration of n = 7.4 x 1011cm-2, Hall mobility of 1.9 x 105cm2Vs-1 and sample area 0.50 cm2. The concentrations and mobilities were determined at the actual experimental conditions in zero field, i.e. at low temperature after illumination with an infrared light-emitting diode until saturation was reached. Transport experiments were performed to confirm that the laser light from the optical read-out scheme has no effect on the 2D electron system. Self-consistent calculations yield estimates of the symmetric-antisymmetric splitting in the balanced bilayers: Asas ^ 1.5 meV for the weakly coupled system and Asas ^ 3 meV for the strongly coupled bilayer. As a reference we compare the bilayer magnetization to that of a high-mobility single-layer 2DES in a GaAs-AlGaAs heterojunction with an electron concentration n = 4.8 x 1011cm 2, mobility 2.2 x 106cm2Vs-1 and sample area 0.56 cm2 [11]. Transitions of the chemical potential between two Landau levels show up as magnetization steps at filling factors vT = 4 j for the bilayers and vs = 2j for the single layer. For the bilayer additional magnetization steps related to transitions between the symmetric and antisymmetric state within the same Landau level appear at vT = 4 j + 2. In the weakly coupled bilayer Asas is just strong enough to visualize these transitions as shoulders in the magnetization signal; in the strongly coupled system clear steps are visible. These observations clearly show that it is indeed possible to observe discontinuities in the thermodynamic properties of an electronic state as measurable magnetization steps.
For the single layer the step size of the Landau-level steps saturates at the theoretically expected AM = 2NxB for low filling factors (see also [12]). For higher filling factors the The lines indicate the step sizes as theoretically expected for these filling factors. 6 experimentally observed step sizes are reduced due to finite-temperature effects and level broadening. In contrast, the corresponding step sizes for the bilayer 2DESs saturate at a considerably smaller value, AM < NiB (an additional peak-feature appearing around v = 4 will be discussed further on). Part of this reduction can be explained by the fact that the transition takes place from the symmetric state of a higher Landau level to the antisymmetric state of the Landau level below. The jump in the chemical potential is not hwc, but rather hmc -Asas which is expected to lead to a correspondingly reduced magnetization step AM = 2NiB(hMc -A SAS)/hmc. In particular, for the strongly coupled bilayer we observe a disappearance of the Landau-level step at v = 16. When supposing hmc = Asas at this particular filling factor this yields indeed a quite realistic value of 3.3 meV for ASAS.At higher filling factors Asas exceeds hwc, which complicates the energy-level scheme: steps across the symmetricantisymmetric gap (v = 20, 24 and 28) are now associated with an energy gap of Asas -hwc, and the Landau-level steps (v = 18, 22 and 26) with A/x = 2hmc -Asas. The magnetization steps between two Landau levels at lower filling factors v = 12, 8, and 4 are much stronger reduced than expected from AM = 2NiB(hMc -ASAS)/h Mc (broken line in figure 3). This observation is summarized in figure 3 where we compare the experimentally measured magnetization steps at Landau-level transitions for the strongly coupled bilayer and the single layer to the theoretically expected behaviour as described above. The experimental step size at filling factor v is extracted from the data by extrapolating the sloping magnetization traces before and after the step and measuring their difference AM at the position of v. For the single layer the experimental values for low enough filling factors, i.e. v = 2,4 indeed fall on the (solid) line; for the bilayer the step sizes are clearly smaller than naively expected: the reduction becomes even more pronounced for lower filling factors. We have verified that the step sizes for the lowest two Landau level steps no longer depend on temperature for T < 2 K; thermal broadening can therefore be excluded to be responsible for the observed reduction.

Model calculations
In order to explain this considerable reduction of experimentally observed magnetization steps at Landau-level transitions, we use a phenomenological DOS model with Gaussianbroadened Landau levels and a finite background DOS [9] assuming balanced layers. As model parameters for sample 2 we use Asas = 3.3 meV (see above), a constant Landau-level broadening r = 0.2 meV and a filling-factor dependent background DOS Db = (0.20 + 0.016vT)D0 (D0 is the DOS at zero magnetic field for one symmetry state including the two-fold degeneracy of spin). In figure 4 this background DOS model is plotted together with the DOS values derived from the width Av and size AM of the observed magnetization steps following the procedure described in [9]. For simplicity, we chose to model the background DOS by a straight line; to improve the agreement between the magnetization data and the calculations we had to position this line at slightly larger background DOS values than would be expected from the data points (■) in figure 4. In figure 5, we compare our magnetization calculations to the experimental results and to an ideal bilayer with no level broadening and no background DOS. The expectation for an ideal bilayer predicts clearly far too large Landau-level steps in the magnetization. Our simple model calculation, however, reasonably reproduces the experimentally observed results, although not in detail.

Discussion
We have shown that it is necessary to include a large background DOS in order to explain the strongly reduced magnetization steps at Landau-level transitions in bilayers. As pointed out by e.g. [9], the reduction of magnetization steps by the presence of a background DOS can be related to the sample quality: a higher mobility sample is expected to have a lower background DOS. Since in coupled bilayer 2DESs electron-electron correlations might induce new effects, like those observed recently in the two-valley system of Si/SiGe heterostructures [8], it is tempting to propose that inter-layer or inter-valley coupling naturally introduces a considerable background DOS in the equivalent single-particle spectrum: due to the large coupling between different energy levels a number of electrons can no more be assigned to a specific level and participate in a large background DOS. Further experiments with samples of higher mobility and while varying the inter-layer coupling by an adjustable in-plane magnetic field may help to address this interesting question more definitely.

Correlation effects at v = 4?
As briefly mentioned in section 4.1 an additional peak in the magnetization appears around vT = 4 when the j = 1 Landau level is depopulated. Here the chemical potential jumps from a symmetric spin-up state of the first Landau level to an antisymmetric spin-down state in the lowest Landau level. The appearance of this peak is shown in more detail in figure 6. For high temperatures T > 4 K solely a magnetization step of 1 NiB is visible. As discussed above, the 9 Institute of P h y s ic s ^DEUTSCHE PHYSIKALISCHE GESELLSCHAFT e (T) step size is reduced considerably by DOS effects. When cooling down to lower temperatures an additional peak in the magnetization signal appears just above vT = 4. The peak does not depend on the direction of the magnetic-field sweep and, therefore, it cannot be due to eddy currents [9,13]. Furthermore, because the bilayer is well balanced, also recharging effects [14] are improbable to explain the occurrence of the structure. The drastic dependence on temperature, however, hints to a correlation-induced cause for the appearance of this peak.
The final total step size reaches a value around 1.5 N/i-B which is indeed very close to the theoretically expected value for an ideal bilayer with no background DOS and broadening. This observation strongly proposes that the background DOS disappears just before the vT = 4 transition. In other words, the coupling of the few electrons remaining in the first Landau levels weakens and the background DOS is reduced.
More correlation effects may play a role and require further experiments with adjustable inter-layer coupling.

Conclusion
In conclusion, we have directly visualized the complex energy-level structure of strongly coupled bilayer 2DESs by measuring their magnetization. Magnetization steps were related to intra Landau-level transitions of the chemical potential between bilayer states of different symmetry and to transitions between two different Landau-levels, intuitively showing the fundamental thermodynamic nature of magnetization.
In contrast to theoretical expectations for a bilayer, the magnetization steps at inter-Landaulevel transitions are reduced considerably which we were able to explain by involving a large background DOS. At the transition to the lowest Landau level this background DOS seemed to disappear resulting into an additional peak observed in the magnetization.