Pis'ma v ZhETF, vol. 102, iss. 1, pp. 40-44
© 2015 July 10
Interaction-induced merging of Landau levels in an electron system
of double quantum wells
In memory of V.F.Gantmakher
A. A. Shashkin+1\ V. T. Dolgopolov+, J. W. Clark*, V. R. Shaginyanx, M. V. ZverevoV, V. A. Khodel*0
+Institute of Solid State Physics, 142432 Chernogolovka, Russia * McDonnell Center for the Space Science and Department of Physics, Washington University, MO 63130 St. Louis, USA x Petersburg Nuclear Physics Institute, National Research Center "Kurchatov Institute", 188300 Gatchina, Russia °National Research Centre "Kurchatov Institute", 123182 Moscow, Russia vMoscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia
Submitted 25 May 2015 Resubmitted 27 May 2015
We show that the disappearance of the chemical potential jumps over the range of perpendicular magnetic fields at fixed integer filling factor in a double quantum well with a tunnel barrier is caused by the interaction-induced level merging. The distribution function in the merging regime is special in that the probability to find an electron with energy equal to the chemical potential is different for the two merged levels.
DOI: 10.7868/S0370274X15130081
More than twenty years ago a topological phase transition that is related to the emergence of a flat portion of the single-particle spectrum e(k) at the chemical potential was predicted at T = 0 in strongly correlated Fermi systems [1-5]. In more vivid terms, this transition is associated with the band flattening or swelling of the Fermi surface (for recent reviews, see Refe. [6-8]). The flattening of the single-particle spectrum means that the probability to find a fermion with energy equal to the chemical potential depends on the fermion momentum k. The swelling of the Fermi surface is preceded by an unleashed increase of the quasiparticle effective mass to at the quantum critical point [9, 10].
The topological phase transition characterized by the unusual form of the distribution function is not the only non-trivial manifestation of fermion interactions in strongly correlated Fermi liquids. Another example is the merging of quantum levels in a Fermi system with discrete spectrum in which case the fillings of the two quantum levels at the chemical potential are different [11]. The merging of the spin- and valley-split Landau levels at the chemical potential has been detected near the quantum critical point in a clean strongly-interacting two-dimensional (2D) electron system in (100) Si metal-oxide-semiconductor field-effect
-^e-mail: shashkin@issp.ac.ru
transistors (MOSFETs) [12]. The fact that the merging detected is governed by the effective mass depending on electron density may create the impression that the level merging is a precursor of the swelling of the Fermi surface. As a matter of fact, the two effects are not always related to each other. The diverging effective mass is not necessary for the existence of the effect of level merging.
Here, we show that the disappearance of the chemical potential jumps over the range of perpendicular magnetic fields at fixed integer filling factor in a double quantum well with a tunnel barrier is caused by the interaction-induced level merging. The merging regime corresponds to the special form of the distribution function. In this case the probability to find an electron with energy equal to the chemical potential is different for the two merged levels.
To begin with, we assume that the weakly interacting two-dimensional electron system is subjected to a perpendicular magnetic field B. In the simplest case there are two equidistant ladders of quantum levels for spin up and down directions (see, e.g., Ref. [13]). Both the thermodynamic and kinetic properties of the electron system are determined by the position of the chemical potential relative to the quantum levels, which is in turn determined by the magnetic field and electron density n. The filling factor is equal to v = n/no, where
no = eB/hc is the level degeneracy. When v is fractional, the chemical potential is pinned to the partially filled quantum level. The probability to find an electron at the chemical potential is given by the fractional part of the filling factor and can be varied between zero and one. At integer filling factor there is a jump of the chemical potential. In experiment, the jump manifests itself as a minimum in the longitudinal electrical resistance in the Shubnikov-de Haas effect. The resistance minima in the (B, n) plane correspond to a Landau level fan chart like the one shown in Fig. 1.
-0.2
9
-0.4
-0.6
*bal /: 'I'4 - // i * 7 V /7 A''^ / r y* / : .V
• - ; / / y • i ' i 5 ' o • / /• i / y / # h» " Hi// / . 1 r ! / ~ / tr y* i i i i / /• ' " \¡¡///y till/ ' -Ujj; / ~ ' 1¡¡¿// / ' w//s' 1/ thl * : 1
B (T)
Fig. 1. Landau level fan chart for the double quantum well shown in Fig. 2. Positions of the longitudinal resistance minima in the (B, Vg) plane are marked by the dots. The filling factor v for the double layer electron system as well as the filling factor i>i (1/2) for the back (front) layer are indicated. Over the shaded ctrects, the merging of quantum levels in perpendicular magnetic fields is impossible. In the regions marked by the ovals, no resistance minima are observed in a perpendicular magnetic field, whereas these appear in a tilted magnetic field
If the magnetic field is tilted by an angle /3, the spacing between the quantum levels in each of the spin ladders is equal to hivc = heB cos(/3)/mc, and the shift between the ladders equals gii&B, where the Lande factor g is assumed to be isotropic, ^b = eh/2mec is the Bohr magneton, and me is the free electron mass. Increasing the tilt angle leads to crossing the quantum levels of the two ladders. The crossing happens for the first time at an angle /31 that satisfies the condition
COs(/?i
gm 2 mP
(1)
At /? = /?!, the chemical potential jumps at even filling factors and the corresponding fan chart lines will disappear. This effect is well known and is used for the experimental determination of the product gm [14-16]. Note that in experiment, the chemical potential jumps should be absent at tilt angles in the vicinity of depending on the sample quality and temperature.
We now take into account the interaction between the electrons of neighboring quantum levels and increase the tilt angle in the vicinity of /?i. Tentatively, the quantum level filled before crossing should have got emptied with increasing /3. However, if the single-particle energy of electrons on the emptying level decreases due to the electron interaction, both levels remain pinned to the chemical potential over a wide range of angles A/?i that is determined by the interaction strength. The probability to find an electron at the chemical potential is different for opposite spin orientations, being dependent on the external parameter which is the tilt angle. Such a behavior is known as the merging of quantum levels.
In the above hypothetical consideration the crossing or merging of quantum levels is controlled by the tilt angle of the magnetic field. In the experiments on a strongly-interacting 2D electron system in (100) Si MOSFETs, the disappearance of the longitudinal resistance minima is analyzed when changing both the perpendicular magnetic field and electron density at fixed filling factor z/ = 4(i + l), where i is an integer. In this case the level merging occurs near the quantum critical point, as controlled by the effective mass depending on electron density [12]. One might think that the level merging is a precursor of the Fermi surface swelling. In fact, the two effects are not necessarily related to each other. Below, we demonstrate that the effect of level merging occurs in a bilayer 2D electron system with a tunnel barrier between the electron layers. Note that although the Shubnikov-de Haas effect in similar double layer electron systems was investigated in a number of publications [17-21], only the level crossing was observed in Refs. [17, 20, 21].
The samples used contain a parabolic quantum well grown on a GaAs substrate, as shown schematically in Fig. 2. The width of the parabolic part of the well, limited by vertical walls, is about 760 A. At the center of the well there is a narrow tunnel barrier that consists of three monolayers of AlxGai_xAs (x = 0.3). The symmetrically doped structure is capped by 600 A AlGaAs and 40 A GaAs layers over which a metallic gate is evaporated. The presence of the tunnel barrier leads to a splitting of each subband bottom caused by quantization in the z direction. At the point of the symmetric electron density distribution, the splitting energy is
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A. A. Shashkin, V. T. Dolgopolov, J. W. Clark et al.
1000
z(A)
Fig. 2. (Color online) Schematic diagram of the bottom of the conduction band for the AlGaAs double quantum well in the absence of electrons. The parabolic part of the well is grown when varying the A1 content from zero at the center to 0.1 on the edge of the well. The tunnel barrier at the center is created by three monolayers of Al^Gai-^As (x = 0.3). The thick blue lines correspond to the silicon doped layers
equal to 1.3 meV. The structure of the quantum levels in the bilayer 2D electron system in perpendicular magnetic fields is similar to that in the 2D electron system in (100) Si MOSFETs, where the spin and valley splittings are present, with a distinction that the spin splitting in accessible magnetic fields is the smallest (Fig. 3a).
Fig. 3. (a) - Layout and filling of the quantum levels in the bilayer electron system in the merging regime at filling factor v = 3. (b) - The distribution function of the electrons in the merging regime at v = 3
Applying a voltage Vg between the gate and the contact to the quantum well makes it possible to tune the electron density. The electrons appear in the back part of the quantum well when the gate voltage is above Vthi ~ —
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