Magnetospectroscopy of symmetric and anti-symmetric states in double quantum wells

ARTICLE IN PRESS

Physica E 40 (2008) 894–906 www.elsevier.com/locate/physe

Magnetospectroscopy of symmetric and anti-symmetric states in double quantum wells M. Marchewkaa, E.M. Sheregiia,, I. Trallea, D. Plocha, G. Tomakaa, M. Furdaka, A. Kolekb, A. Stadlerb, K. Mleczkob, D. Zakb, W. Strupinskic, A. Jasikc, R. Jakielac a Institute of Physics, University of Rzeszow, Rejtana 16a, 35-310 Rzeszow, Poland Department of Fundamental Electronics, Rzeszow University of Technology, 35-959 Rzeszo´w, W. Pola 2, Poland c Institute of Electronic Materials Technology, Wolczyn´ska 133, 01-919 Warsaw, Poland

b

Received 14 May 2007; received in revised form 5 November 2007; accepted 5 November 2007 Available online 24 November 2007

Abstract The experimental results obtained for magnetotransport in the InGaAs/InAlAs double quantum well (DQW) structures of two different shapes of wells are reported. A beating effect occurring in the Shubnikov–de Haas (SdH) oscillations was observed for both types of structures at low temperatures in the parallel transport when the magnetic field was perpendicular to the layers. An approach for the calculation of the Landau level energies for DQW structures was developed and then applied to the analysis and interpretation of the experimental data related to the beating effect. We also argue that in order to account for the observed magnetotransport phenomena (SdH and integer quantum Hall effect), one should introduce two different quasi-Fermi levels characterizing two electron subsystems regarding the symmetry properties of their states, symmetric and anti-symmetric ones, which are not mixed by electron–electron interaction. r 2007 Elsevier B.V. All rights reserved. PACS: 73.40.Hm Keywords: DQW; Magnetotransport; Symmetric and antisymmetric states

1. Introduction In recent years a great number of publications have appeared, where the experimental studies of electron transport in double quantum well (DQW) structures were reported. The reason for this is the enduring progress in the structure fabrication technology, which allows modification of the shape and width of quantum wells (QWs) or barriers, as well as the progress in measurement techniques, which enables measurement of separate conductivity and Hall response of each layer. These experiments have demonstrated a number of new and interesting phenomena occurring in such structures, which however cannot be observed in a structure with a single two-dimensional-gas (2DEG) layer. Generally, all these phenomena can be Corresponding author. Tel.: +48 17 8625628.

E-mail address: [email protected] (E.M. Sheregii). 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.11.020

divided into two groups, depending on the intensity of interlayer interaction. The phenomenon related to the Coulomb drag effect belongs to the first group, while the phenomenon related to the tunneling effect to the second one. Correspondingly, the phenomena of the first group are accounted for by a weak interaction between the layers, while the phenomena of the second group are attributed to a strong interaction. In order to reveal more clearly the drag effect caused by Coulomb interaction, the tunneling between two QWs has to be negligible. In order to fulfill this requirement, the barrier between two QWs should be adequately wide. Bronsager et al. [1] reported theoretical results concerning the magnetic-field and temperature dependences of the trans-resistivity in DQW layers. Experiments of Ref. [2] confirmed the results obtained in Ref. [1] and have shown also the existence of double-peak structure for each magnetoresistance peak in the samples characterized by

ARTICLE IN PRESS M. Marchewka et al. / Physica E 40 (2008) 894–906

the matched density of states in the layers. Another type of experiments was carried out on samples with unmatched density of states [3]. In this work, the authors demonstrated the existence of negative drag in the integer quantum Hall effect (IQHE) regime; its clear explanation was proposed by Lok et al. [4]. The authors of Ref. [4] argued that the electron spin plays a decisive role in the drag effect. At last, the fractional drag between parallel two-dimensional systems has been observed and measured in a regime of strong interlayer correlations [5], where the Hall drag resistance was observed to be quantized in _/e2. Yet another situation is if one has two non-identical QWs. In this case some of the integer Hall states appear and some others disappear, which depends on the alignment of electron states in two QWs [6]. A lot of new phenomena caused by the tunneling between two 2DEG layers separated by a barrier were also observed in recent years. One of them is the absence of some integer quantum Hall states caused by the splitting of electron states into symmetric and anti-symmetric ones (so-called SAS gap, or DSAS) in DQW systems [7,8]. In a structure where the barrier is thin enough, the SAS gap plays a decisive role. The magnitude of this splitting can be determined directly by means of current–voltage characteristics of such structure; it was proven also that the SAS gap is proportional to the magnetic field [9]. This effect was attributed by Huang and Manasreh [10] to the screening of electron–electron interaction in the DQW structure. In the last few years many new phenomena were observed in structures with strongly correlated 2DEG at very low temperatures: among them the Bose condensation [11], Wigner crystallization [12,13], and the long-time nuclear–lattice relaxation in DQW [14,15]. Some authors [16,17] have considered the Shubnikov–de Haas (SdH) oscillations in a tilted magnetic field with a strong perpendicular B? and small parallel BJ components of the field. The authors of Ref. [17] stated that the small BJ component is responsible for the beating effect in SdH oscillations occurring in DQW structure. If there is no B? component and the magnetic field is parallel to the layers composing DQW, the most important peculiarities of magnetotransport are accounted for by the crossing of Fermi surfaces belonging to two QWs [18–20] and this case is completely different from the two previous ones. Therefore, as far as the electrons in strongly correlated DQW structures are concerned, it is possible to treat them as two separate subsystems, one of them belonging to the symmetric states and the other one belonging to the antisymmetric states. The symmetric and anti-symmetric states differ from each other because the symmetry properties of their wave functions are different. To the best of our knowledge, in the papers published so far, except in our short communications [21,22], the consequences of this essential difference between the symmetry properties of the electron’s wave functions were not treated properly. It is also worth mentioning that the magnetotransport

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phenomena in such structures were not analyzed previously by taking into account the accurately calculated energies of Landau levels (LL). In this paper the parallel magnetotransport in DQW structures composed of two identical QWs separated by a thin barrier is studied. The experimental results are obtained for two different structures: the first one included QWs of quasi-rectangular shape, while the second one included triangular QWs. Both of the structures are based on the InGaAs/InAlAs system and for both of them we observed the beating effect in SdH oscillations in the perpendicular magnetic field arrangement (that is, without BJ component of the field). As will be seen below, in order to interpret the beating effect, which occurs in SdH oscillations in the DQW with matched density of states and in the absence of BJ component of magnetic field, it is necessary to introduce two different quasi-Fermi levels (FLs) for the symmetric and anti-symmetric electron subsystems. 2. Description of the structures The DQWs based on InGaAs/InAlAs/InP structures were produced by means of low-pressure metal organic vapor phase epitaxy (LP-MOVPE) on semi-insulating (1 0 0) InP:Fe substrates at the Institute of Electronic Materials Technology, Warsaw. The method of producing the structures with a single QW, which was reported earlier [23,24], was used also for producing a DQW structure with rectangular, as well as triangular QWs. The structure of the first type (#2506) consists of two In0.65Ga0.35As QW of about 20 nm thickness each, and three In0.52Al0.48As barriers. In each barrier there was the donor d-doping layer (see Fig. 1). The cross-section of #2506 structure is depicted in Fig. 2(a). The second DQW structure (#3181), whose cross-section is shown in Fig. 2(c), also consists of two In0.65Ga0.35As QWs of thickness of about 5 nm and three In0.52Al0.48As barriers. In this case however, the barrier between the QWs was not doped. The corresponding conduction band shape is shown in Fig. 3. For this particular case the 2DEG in each QW can be considered to be placed in a triangular potential well, since, as is shown in Fig. 3, the bottom of the conduction band in each QW makes a triangle with the horizontal line. Both of the structures are symmetric about the middle of central barriers, which means that the QWs are identical for both of them and with great probability their 2DEG densities are matched. Parameters of both types of the structures are listed in Table 1. 3. Experimental results The magnetotransport measurements were performed by means of a superconducting magnet, which gives the possibility to get magnetic fields up to 11 T. The sample was mounted in an anti-cryostat, which enables to change

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InxAl1-xAs X = 0.52

InxGa1-xAs

X = 0.52

20

20 z, nm

InxGa1-xAs

X = 0.52

E, eV

0.5

InxAl1-xAs

InxAl1-xAs

0

20

Fig. 1. Conduction band edge profile of the #2506 structure.

l = 2mm In0.52Al0.48As-barrier, 25 nm

#2506I

R

δ - doping Si layer

l = 2mm

In0.52Al0.48As-spacer, 3 nm

In0.52Al0.48As-barrier, 25 nm

#2506II δ - doping Si layer In0.52Al0.48As-spacer, 3 nm

R = 25 mm

δ - doping Si layer

L/4

L = 2 mm

In0.52Al0.48As-barrier, 5 nm

InxGa1-xAs channel layer

In0.52Al0.48As-spacer, 3 nm

In0.52Al0.48As-spacer, 3 nm

InxGa1-xAs channel layer

δ - doping Si layer

In0.52Al0.48As-spacer, 3 nm

In contact

In contact

In0.52Al0.48As-spacer, 3 nm

InxGa1-xAs channel layer In contact

In contact

InxGa1-xAs channel layer

In0.52Al0.48As-bufor 400 nm InP-buffer 180 nm

δ - doping Si layer

InP-substrate

In0.52Al0.48As-bufor 400 nm InP-buffer 180 nm InP-substrate

Fig. 2. (a) Cross-section of the #2506 structure used in the experiments; (b) upper one: the piece of wafer with its indicated parts from where the #2506I and #2506II was cut out; lower one: view of the sample; (c) cross-section of the #3181 structure.

the temperature ranging from 0.4 to 300 K [25]. External gate voltage was not applied because this field destroys the symmetry of the DQW profiles. The values of Hall (Uxy(B)) and transversal Uxx(B) voltage were recorded for the two opposite directions of magnetic and electric fields. There-

fore, eight records (for magnetic field once going up and then going down) were made and averaged at each single measurement for the transverse magnetoresistance Rxx(B) and Hall resistance Rxy(B) at a given temperature. These curves are shown in Figs. 4(a), (b), 5(a), and (b).

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InxA1l-xAs InxA1l-xAs

InxGa1-xAs

InxGa1-xAs

InxA1l-xAs

E, eV

0.5

0 5

5 z, nm

5

Fig. 3. Conduction band shape for the sample #3181.

Table 1 Parameters of the InGaAs/InAlAs DQW structure Sample

Concentration of InAs in the channels (%)

Thickness of the channels (nm)

QW profile

Number of d-doping layers

Donor concentration of d-doping layers according to technologists (cm2)

#2506 #3181

65 65

20 5

Smooth interface Sharp interface

3 2

3.5  1012 3.5  1012

The pronounced SdH oscillations, which manifest themselves in the Rxx–magnetic field dependence, as well as in IQHE (Rxy(B) curves), are clearly seen. The SdH oscillations also undergo the beating effect, which was observed in our experiments for both types of the structures. We observed four beating-effect nodes for the structure #2506I (see Fig. 4(a)) and three nodes for the sample #3181 (Fig. 5(a)). It is important to mention that no beating effect was observed in the case of the structure with single QW based on the same heterostructures In0.65Ga0.35As/ In0.52Al0.48As [24]. Several plateaus are observed in Rxy(B) curves for the magnetic fields up to 6 T for the sample #2506I, as well as for #3181. It can be seen that the plateaus observed in magnetic fields greater than 6 T, are distorted. There can be two possible explanations for this distortion. The first one accounts for the presence of the additional conducting channels, for instance, due to the d-layers. However, the effect of the additional conducting channels should be more pronounced at 4.2 K, rather than at 0.6 K, while, as is easily seen, the minima of the distorted plateau are deeper at 0.6 K than those at 4.2 K (see Fig. 4(b)). The second explanation of the plateau distortion refers to the QW coupling. The additional maximum on the plateau was already observed earlier (see Ref. [7]). In Fig. 6 the results of another series of measurements of QHE and SdH oscillations are presented. These results were obtained also for the structure #2506, but the sample was cut out from another piece of wafer (see Fig. 2(b)), that is why we denote it as #2506II. It is clearly seen that in this case the QHE plateaus are flat and the Hall resistance Rxy is quantized in _/e2, where the occupation factor is n ¼ (N0+N1)sD. Here N0 is the uppermost LL occupied by electrons in the lowest subband (i ¼ 0), N1 the

corresponding LL for the next subband (i ¼ 1), s ¼ 2 the spin degeneracy, and D ¼ 2 is the symmetric degeneracy (symmetric and anti-symmetric states). At the same time it is necessary to note that together with regularly flat plateaus in QHE for a magnetic fields ranging from 3 T up to 8 T, at magnetic field 8.5 T a small distortion of the plateau is observed. We do not observe the beating effect in SdH oscillations for the sample #2506II (see Fig. 6). In the next section we will prove that the beating effect in SdH oscillations is caused by the SAS gap and the difference in the two samples of structure #2506 is related to the difference in carrier density. The barrier width in the structure #2506 is equal to 20 nm and the initial, that is in zero magnetic field, SAS gap is so small, that it can be completely screened by 2DEG (see Ref. [10] and Section 5 of this paper) in the case of sample #2506II. In the case of sample #2506I the screening is insufficient to cause the collapse of SAS gap. It is even more valid for sample #3181, where the barrier width is 5 nm and the SAS gap is significantly larger. This means that in our case the initial SAS gap, for the two samples of the same #2506 structure, can fluctuate, because the electron density slightly changes along the radius of the wafer. Indeed, the electron density, determined by the slope of Rxy(B) curve in a small magnetic field for the #2506II sample, is greater than for #2506I. It is equal to 4.2  1012 cm2 for the sample #2506II and 3.2  1012 cm2 for #2506I. As is shown in Fig. 2(c), the distance between two contacts for Vxx(B) is about 0.8 mm, while the distance between two samples #2506I and #2506II measured along the radius of wafer is 25 mm. As follows from our experimental data, the difference in carrier concentrations for two samples #2506I and #2506II is about 30%. Since

ARTICLE IN PRESS M. Marchewka et al. / Physica E 40 (2008) 894–906

ρxx arb. unit

ρxx arb. unit

898

#3181 T = 4.2 K

#2506I T = 0.6 K T = 4.2 K 0

2

4

6

8

0

10

2

Magnetic Field, T

6

8

10

8

10

2

0.7 #2506I T = 0.6 K T = 4.2 K

0.6

1.5

Rxy, kΩ

0.5 Rxy, kΩ

4

Magnetic Field, T

0.4

1

0.5

0.3 0.2

0 #3181 T = 4.2 K

0.1 -0.5 0

0 0

2

6 4 Magnetic Field, T

8

10

Fig. 4. (a) SdH curves for the structure #2506I at two temperatures: 0.6 and 4.2 K. (b) Results of QHE measurements for the structure #2506I at temperatures 0.6 and 4.2 K.

the distance between the tips of probes measured along the radius of wafer is about 0.8 mm, the carrier concentration change within each of the samples is less than 1%. Hence, the homogeneity of carrier density is quite satisfactory within the samples used in our experiments. So, for the sample #2506I we observe at the same time the beating effect and distortion of the plateaus in QHE, while for the sample #2506II the beating effect is absent and the plateaus are flat within the experimental error for magnetic fields o8 T. 4. The best-fitting procedure We carried out Fourier analysis of the SdH oscillations for both of the structures, #2506I and #2506II, as well as

2

4 6 Magnetic Field, T

Fig. 5. (a) SdH curves for the structure #3181 at 4.2 K. (b) Results of QHE measurements for the structure #3181 at 4.2 K.

for #3181. It is clearly seen that in the case of sample #2506I (Fig. 7(a)) there is a predominating main harmonic which is split. A similar situation is observed for the sample #3181 (Fig. 7(c)), but in this case the magnitude of splitting is larger. This corresponds to the beating effect observed for these two samples. For the sample #2506II (Fig. 7(b)) there is one strong harmonic too, but it is not as dominant as in the previous two cases. It is possible to assume that the observed number of harmonics suggests that no less than two subbands are occupied by electrons in the case of sample #2506. On the other hand, as was mentioned already, our earlier experiments, with the structures containing a single QW [24], did not show the beating effect which could be attributed to spin splitting. Therefore we can conclude that another kind of electron state splitting at zero magnetic field also exists. In order to reveal this most important

ARTICLE IN PRESS M. Marchewka et al. / Physica E 40 (2008) 894–906

effect, we have used the following method of the experimental SdH curve simulation. The starting point for further analysis is the following expression, which is well known in the theory of SdH [26]:   1 X Dr 5 lP 1=2 bTm0 cosðlpnÞ ¼ r0 2 2B coshðbTm0 =BÞ l¼1  expðlbT D m0 Þ cosð2pðl=PB  1=8  lgÞÞ.

ð1Þ

80

500

70

400

60 300 200

40 30

Rxy, Ω

Rxx, Ω

50

100

20

#2506II

0

T = 1.6 K T = 4.2 K

10

0

2

4

6

8

10

B, T

This formula takes into account the non-parabolicity of conduction band as well as spin splitting. Here Dr is the deviation of r from background resistivity, r0 the zero-field resistivity, B the transverse magnetic field, T the temperature, b ¼ 2p2 kB m0 =_e, P ¼ _e=E f m the SdH period, and, TD the Dingle temperature. The fitting procedure was based on adding only two Fourier harmonics of nearly equal frequencies. According to what was mentioned above, formula (1) can be simplified and reduced to two entries of the Fourier series only. As a result we have h  g pi Dr1xx ¼ exp  1 cos 2ðo1  kBÞ , (2) B B h  g pi (3) Dr2xx ¼ exp  2 cos 2ðo2 þ kBÞ , B B where g1, g2 are coefficients which are equivalent to RbTDm0 in Eq. (1), and o1, o2 are the cyclotron frequencies for two subsystems of LLs. The kB term is included into the argument of cosine in Eqs. (2) and (3) in order to have an additional degree of freedom to get better agreement between the experimental and calculated curves. It turns out that in order to get better agreement between the experimental results and numerical calculations, one should reduce the oscillation frequencies when the magnetic field increases. With this one could confirm the experimental fact that the SAS gap increases if the magnetic field increases and if one supposes that zero magnetic SAS gap causes the beating effect.

Amplitude, arb. unit

Amplitude, arb. unit

Fig. 6. SdH oscillations and QHE for the second sample #2506II of the structure #2056.

899

10

20

30

40

50

60

70

10

20

30

40

50

60

70

Bf, T

Amplitude, arb. unit

Bf, T

0

10

20

30 40 Bf, T

50

60

70

Fig. 7. SdH-oscillations’ Fourier transform plotted versus Bf ¼ ½Dð1=BÞ1 : (a) for the #2506I structure, (b) for the #2506II structure, and (c) for the #3181 structure.

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900

#3181 T = 4.2 K simulation

ρxx arb. unit

ρxx arb. unit

#2506I T = 0.6 K T = 4.2 K simulation

0

2

4

6

8

10

0

Magnetic Field, T

2

4

6

8

10

Magnetic Field, T

Fig. 8. (a) Upper curve represents the experimental data for SdH oscillations (#2506I) and the lower one represents the results of the calculations performed according to the method described in Section 4. (b) Corresponding quantities for the structure #3181.

The results of the calculations are shown in Figs. 8(a) and (b) for the samples #2506I and #3181, respectively. The fitting parameters, used in the calculations, were chosen to be: g1 ¼ 8.8, g2 ¼ 8, o1 ¼ 34.9, o2 ¼ 40.27, k ¼ 0.095, and g1 ¼ 9, g2 ¼ 8.8, o1 ¼ 30.8, o2 ¼ 23.8, k ¼ 0.095 for these two samples, respectively. In this way we have obtained two oscillation components for each SdH curve. These curves are shown in Figs. 9(a), (b) and 10(a), (b). They should be interpreted using the LL energy calculation carried out for the symmetric and antisymmetric states separately, by means of the best-fit procedure. 5. Calculation of Landau-level energy In our calculations we have used the model proposed in Ref. [10], where symmetric DQWs separated by a middle barrier were considered. An external magnetic field was applied perpendicular to the planes of QWs. The electron’s behavior was described by the Schro¨dinger equation which included the DQW potential, as well as the self-consistent Hartree potentials. The in-plane movement in each of the subbands of the DQW was supposed to be Landau quantized. The total electron energy, which includes the contributions of the in-plane as well as vertical electron motion, can be written as   E ni ¼ n þ 12 _oc þ E i þ V Fni , (4) where V Fni is the exchange energy (which involves screening) given by Eqs. (12)–(18) of Ref. [10], for DQW based on the GaAs well-material and Al0.3Ga0.7As barrier-material with a well width of 14 nm and a barrier width of 3 nm. The authors of Ref. [10] have found an approximately linear B dependence of the tunneling gap for magnetic fields 0oBo9 T. One can conclude, therefore, that the

total splitting Dt of energy states of the two QWs, consists of two parts: a constant DSAS, determined by the overlapping of the electron wave functions of two QWs, and the changeable tunneling term V Fni , involving the screening factor: Dt ¼ DSAS þ V Fni .

(5)

The last term, which is strongly reduced when B increases up to the values for which the quantum limit occurs, can be written as V Fni ¼ ðK 0  kBÞ,

(6)

where K0 is the maximum screening factor in zero magnetic field depending mainly on the electron densities and (kB) is the changeable screening factor, decreasing with B up to the value when the first LL in two QWs starts getting occupied. As was observed experimentally [10], the total energy gap Dt between symmetric and anti-symmetric states in DQW is proportional to the magnetic field. We adapted this model for our particular case of symmetric In0.65Ga0.35As/In0.52Al0.48As DQW structure (#2506I), assuming rectangularity of QWs. The last one enables the calculation of LL energies according to the formula [27] ðE 0  E ? ÞðE g þ E 0 þ E ? Þ _2 p2 ði þ 1Þ2 , ¼ Eg 2mc a2 r Eg Eg E? ¼  þ 2 2

(7)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  

4mB B m0 1 1  1þ f1  nþ  g0 f 2 , Eg 2 2 mc (8)

f2 ¼

E g þ ð2=3ÞD , E ? þ E g þ ð2=3ÞD

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400

350

350

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300

E, meV

E, meV

M. Marchewka et al. / Physica E 40 (2008) 894–906

250 EF

200

901

250 200

150

150

100

100

50

50

EF

arb. unit

arb. unit

0

2

0 0

2

4 6 Magnetic Field, T

8

10

6 4 Magnetic Field, T

8

10

400

350

350

300

300

E, meV

E, meV

450 400

250

250

200 EF

200

150

100

100

50

EF

150

arb. unit

arb. unit

50

0

0

2

4 6 Magnetic Field, T

8

10

Fig. 9. Best fit of the Fermi level energy corresponding to the (a) symmetric part of the SdH oscillations and (b) anti-symmetric part of the SdH oscillations, for the case of structure #2506I.

and   ðE g þ DÞ E ? þ E g þ ð2=3ÞD  , (9) f1 ¼  E g þ ð2=3ÞD ðE ? þ E g þ DÞ   where E 0 ¼ E  DsSAS þ kE ? , Eg is the energy gap (between the top of valence band and the edge of conduction band), D the spin–orbit splitting, m0 the electron mass in vacuum, mc the effective electron mass

2

4 6 Magnetic Field, T

8

10

Fig. 10. Best fit of the Fermi level energy corresponding to the (a) symmetric part of the SdH oscillations and (b) anti-symmetric part of the SdH oscillations, for the case of structure #3181.

corresponding to the edge of the conduction band, mB the Bohr magneton, n ¼ 0,1,2,y stands for the LLs, i ¼ 0,1,2,y the number of subbands, E? the LL energy for bulk semiconductor (we use here the notation of Zawadzki [27]), and E the unknown energy of the n LLs for the ith subband in QW. The parameter DsSAS is the energy gap determined by the overlapping of the wave functions, which is diminished by the screening at zero magnetic field, as follows from Eqs. (5) and (6): Dt ¼ ðDSAS  K 0 Þ þ k0 B  DsSAS þ k0 B.

(10)

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Table 2 Band-structure parameters used for the LL energy calculation, Eg is the energy gap, mc =m0 the ratio of electron mass to effective electron mass, and D the spin–orbital splitting x

Eg (eV)

mc =m0

gc

D (eV)

0.65

0.723

0.0229

–9.909

0.355

Thus, the SAS gap is proportional to the magnetic field B. The coefficient of proportionality k0 can be considered as a fitting parameter and is equal to 2k ¼ 0.19, as was determined in the previous section. We assume a triangular potential shape in the case of asymmetric DQW (#3181) with one donor d-layer in the barrier on one side of each QW. This means that Eq. (4) has to be modified for QW with the triangular potential (see Ref. [27]):    1=2  a1=2   2 b 1=2 1=2 ða þ bÞa b þ ða  bÞ ln   ðb  aÞ1=2   1=2 Eg ¼  4eF _pði þ 3=4Þ, ð11Þ 2mc  where a ¼ E 0  E ? ; b ¼ E g þ E 0 þ E ? ; E 0 ¼ E  DsSAS þ kE ? Þ; and F is the electric field caused by the existence of the interface which is determined by the potential U ¼ eFz. The magnitude of E? is determined in the same way as before, by means of Eqs. (8) and (9). Eqs. (7) and (11), together with Eqs. (8) and (9), allow the calculation of LL energies in rectangular and triangular DQW with the variation of spin-splitting and tunneling gaps in a magnetic field taken into account. We then used the results of these calculations to account for the SdH oscillations and to find the parameters of 2DEG for both of the structures (the FL, carrier density, etc.). The calculations of the LLs for the sample #2506I were performed using the band parameters, determined previously for the sample with a single quasi-rectangular QW [24]. These parameters are listed in Table 2, they are the same for both of the samples. As for the values, DsSAS ¼ 1 meV for the #2506I structure and DsSAS ¼ 5:5 meV for the #3181 structure.

FLs in both of them are also different if the tunneling effect is negligible. As a result of this difference, the beating effect could be observed. It is worth mentioning however, that according to Ref. [6] where the mismatching of electron densities was created in a controllable way, the beating effect was not observed. We believe that in our case the beating effect is not the result of difference in carrier densities in two QWs, but is the result of the existence of SAS gap in zero magnetic field. There are three evidences for this. The first one is the symmetry of the structures in question about the center of the middle barrier: both of the QWs were produced in a way as to make them identical. The second one is that the width of the central barrier is small enough in order to equalize the carrier densities in both of the QWs of the structure #2506, as well as in #3181. Note that in the case of #3181 structure the barrier width is equal to 5 nm and in the case of #2506 structure it is 20 nm. If the beating effect were caused by the difference in electron densities, it would be more pronounced in the case of #2506 structure than in #3181 and the period of beating (versus 1/B) would be smaller for the structure #2506, while in our experiment it was just the opposite. And at last the third evidence. In the two samples of the same structure (#2506I and #2506II) the electron densities are different, but the difference is comparatively small. If the beating effect were caused by the difference in electron densities in two QWs, it would be impossible to explain why the beating effect disappears when the electron densities increase slightly in one of the samples (#2506II). On the contrary, the disappearing of beating in #2506II can be very naturally explained, if we attribute it to the existence of SAS gap in zero magnetic field, because the small increase in electron density leads to the collapse of small SAS gap due to screening. Therefore, the non-zero SAS gap causes the SdH oscillations at two different frequencies (versus 1/B) and leads to the beating effect. In the next subsection we will show that our experimental data related to beating effect can be sufficiently well interpreted, if one takes into account the accurately calculated LL energies and introduces two quasi-FLs, characterizing two electron subsystems, which belong to the symmetric and anti-symmetric states in a DQW structure.

6. Discussion

6.2. Two quasi-Fermi levels

6.1. The role of SAS gap

The calculations of LL energies were performed according to Eq. (7) for the sample #2506I and according to Eq. (11) for the sample #3181 using the parameters shown in Table 2. The results of our calculations for the sample #2506I are shown in Fig. 9(a) (for the symmetric states) and in Fig. 9(b) for the anti-symmetric ones. The calculated energies are plotted against magnetic field for two subbands (i ¼ 1, 2). Usually the FL position is determined by LLs and FL crossings corresponding to the maxima of two components of SdH oscillations, acquired by simulation of beating effect. It is worth mentioning that

As was mentioned above, since no beating effect in SdH oscillations was observed in the case of a structure with a single QW [24], we presume that this effect cannot be attributed to the spin splitting of LLs. There are two alternative explanations of this effect. The first one is due to the existence of the SAS gap at zero magnetic field. The second one is due to different carrier densities in two QWs (the so-called mismatching of carrier densities). It is well known that for different carrier densities in two QWs, the

ARTICLE IN PRESS M. Marchewka et al. / Physica E 40 (2008) 894–906

as is seen in Figs. 9(a) and (b), the FL-positions are perfectly well fitted to the oscillation curves and are unique, since they correspond to the regularly spaced crossings of LLs, equidistant with respect to the inverse magnetic field. The FL displacement, up or down, destroys this regularity immediately, due to the superposition of the two space-quantized subbands. We should admit that we could not achieve a satisfactory agreement between our experimental data and the theoretical predictions about the SdH oscillation peaks positions, if we tried to use the single FL common for both types of the states, symmetric and anti-symmetric ones. As is clearly seen in Figs. 9(a) and (b), the FLs for the two oscillation components are different, and the difference is equal to 12 meV. A similar picture is true for the sample #3181. In the case of this sample only one subband is occupied, therefore determining FL is simpler. In spite of this, it was impossible to find a single FL which would be common for both subsystems, symmetric and anti-symmetric. It turns out that the difference between the two FLs for the symmetric and anti-symmetric states in this case is equal to 32 meV (see Figs. 10(a) and (b), respectively). It is interesting to note that, in the case of sample #2506I, the value of total splitting Dt is approximately equal to 15 meV for n ¼ 0, i ¼ 0, in a magnetic field of about B ¼ 10 T, and should increase if the field increases up to about 30 T, where the predicted quantum limit should occur. The obtained values of the FL energies give us the possibility to estimate the density of carriers for both structures. These values are 3.8  1012 cm2 for the sample #2506I (good agreement with date obtained by the slope of Rxy(B) curve in small magnetic field) and 2.5  1012 cm2 for the sample #3181. In the first case this value is slightly greater than what the technologists set it up to be, and in the second case it is slightly smaller. So, our experimental data force us to introduce the concept of two quasi-FLs, by means of which two electron subsystems can be characterized. These two subsystems are electrons belonging to the two different types of states, symmetric and anti-symmetric ones. One can encounter the concept of quasi-FLs in different contexts, for instance, in semiconductor physics. Quasi-FLs for electrons and holes are very often used; quasi-FLs for the electron states with k+ and k (k are for the electron wave vectors along say, +x- and x-axis) are introduced to explain the peculiarities of IQHE [28,29], and so on. Generally speaking this concept simply means that one can treat two subsystems as weakly interacting and having their own rates of establishing the equilibrium state. Therefore, our next step is to argue that indeed, we could interpret our data, introducing the two quasi-FLs which characterize two separate electron subsystems. 6.3. Electron–electron interaction for the two subsystems: symmetric and anti-symmetric states Suppose now that the 2DEG in the DQW structure under consideration is off the thermodynamic equilibrium

903

state and the deviations from the equilibrium are small enough. Obviously, these deviations are caused by the current which flows through the structure. Then the electron distribution functions, one of which corresponds to the symmetric state f(a) and the other to anti-symmetric one f(b), can be represented as follows:     ~ þ df ða;bÞ k~ , f ða;bÞ ¼ f ða;bÞ k (12) 0 where     df ðaÞ k~ ¼ dk~k~0 df ðaÞ k~0 , 0     ðbÞ ~ k ¼ dk~k~0 df ðbÞ k~0 , df

ð13Þ

0

Here the superscript a stand for the symmetric and b for the anti-symmetric states (see Appendix A, where the wave function for these states are considered). Denote the electrons belonging to subsystem a by means of ca, and those belonging to subsystem b as cb and define the rate of establishing the equilibrium state in each subsystem as  0 i   o Xn ða;bÞ h 1 ~ þ f ða;bÞ k~0 W ða;bÞ  ¼ k W ~ ~0 1  f ða;bÞ 0 0 0 k!k k~ !k~ 0 tða;bÞ k~ k~   ða;bÞ ~0 k X ða;bÞ 1  f 0  , ð14Þ ¼ W ~ ~0 k!k 0 1  f ða;bÞ k~ 0 k~ where W ða;bÞ ~ ~0

0

~ k!k ðk~ !kÞ

are the probabilities for the electrons

belonging to the subsystem a or b to be scattered from the 0 state k~ to the state k~ and vice versa (see Fig. 11). These probabilities are determined by the next formula:  E2    2p  ða;bÞ ða;bÞ ^ ða;bÞ d E ða;bÞ  E ða;bÞ , C W i!f ¼ j V jC (15) f i i f  _   E  E   ða;bÞ , where Cða;bÞ , C are the initial and final states, E ða;bÞ  i f i their energies, respectively, and V^ is the operator E ða;bÞ f

E

k2 k1

k`2 k`1 k1

k2 k`2

k`1

ΔSAS k

Fig. 11. Diagram representing electron–electron scattering for the case of two subbands, symmetric and anti-symmetric.

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904

corresponding to the perturbation responsible for the transition. It is well known that sometimes mainly the electron–electron (ee) collisions lead to the establishing of equilibrium states within the electron gas in a semiconductor at low temperatures [30]. This means that whatever the initial distribution function is, the final distribution can be described by a displaced Fermi (or displaced Maxwellian) distribution function. However, for establishing such distribution, an important condition has to be satisfied: ee collisions should be more frequent than the electron– phonon (eph) collisions. In other words, tee 5teph , where tee and teph are the ee and eph scattering times, respectively. Since the temperatures at which our experiments were carried out were very low, we can suppose the last condition to be fulfilled. Now in order to calculate tee ¼ tða;bÞ for the two subsystems, we have to calculate W ða;bÞ , the corresponding transition probabilities. There~ ~0 k!k

fore, we should analyze the probabilities for two electrons  ða;bÞ to the states to transit from the states l 1 k~1 ; l 2 k~2  0 ða;bÞ 0 l 01 k~1 ; l 02 k~2 , where l 1 ; l 01 ; l 2 ; l 02 stand for the quantum ~ Notice that the subscripts 1,2 stand numbers, other than k. for the electrons 1 and 2 in a pair, while the superscripts (a, b) are to distinguish the symmetric and anti-symmetric states of electrons in the DQW structure. Then W ða;bÞ ¼ 1;2!10 ;20

2 2p  ða;bÞ  M 1;2!10 ;20  _ 

ða;bÞ

 d ða;bÞ þ ða;bÞ  0 1 1 2

ða;bÞ

 0 2

 .

ð16Þ

Here we introduced the following notations: l1k11, ð1;2Þ 0 ð1;2Þ 0 ð1;2Þ l 01 k01  10 etc.; ð1;2Þ 1 , 2 ,  1 ,  2 , are the electron energies in the initial and final states, respectively. It is necessary to calculate the matrix element M ða;bÞ in ð1;2Þ!ð10 ;20 Þ Eq. (16) to show the different rates of establishing of quasiequilibrium states in the two subsystems. The calculation of two elements M ðaÞ and M ðbÞ is carried out ð1;2Þ!ð10 ;20 Þ ð1;2Þ!ð10 ;20 Þ in Appendix B. On inspecting Eqs. (B.3) and (B.4), even without evaluating these integrals, one can easily see that M ðaÞ aM ðbÞ and hence, the same is true ð1;2Þ!ð10 ;20 Þ ð1;2Þ!ð10 ;20 Þ ðaÞ ðbÞ for tða;bÞ ee : tee atee . That is, the rates of establishing of quasi-equilibrium state under e–e scattering in the two subsystems, corresponding to the symmetric and antisymmetric subbands, are different. The same arguments are also valid, if we consider the e–ph interaction. Moreover, in the case of e–ph interaction, the formulae analogous to (B.3) and (B.4) are even simpler, because in this case the initial as well the final electron states are one-particle states, but not the two-particle ones, as in the case of e–e scattering. Anyway, one can state that both tða;bÞ and tða;bÞ ee eph , in the two subsystems corresponding to the symmetric and anti-symmetric states, are different: ðaÞ ðbÞ ðbÞ tðaÞ ee atee ; teph ateph .

The only thing that also has to be proven is that these two subsystems are weakly interacting. To this end let us where consider the transition matrix element M ðcÞ ð1;2Þ!ð10 ;20 Þ (1,2) stands for the initial state which includes the symmetric state cai , and (10 ,20 ) stands for the final state which includes the anti-symmetric one, cbf . Then, under the integral sign in Eq. (B.3), one should have the product of two functions ja ðz1 Þ and jb ðz1 Þ, one is even and the other one is odd. Notice also that Uee is an even function of its variables. Hence, M ðcÞ should be very nearly zero due to the ð1;2Þ!ð10 ;20 Þ parity selection rule, and this means that the e–e scattering does not mix the states of different parities. Of course, e–ph interaction can, and usually do, mix them. However, in our experiments the temperature was such that kB T5_oLO , where oLO is the optical phonon frequency. Then the electron energy relaxation time can be determined by the electron scattering on acoustic phonons. This time can be estimated to be about 1 ns [30], while the time flight across the structure is about a few ps, and hence, first, the electron transport on such structure can be considered as ballistic with respect to the scattering by acoustic phonons and second, the quasi-equilibrium state in each of the subsystems corresponding to the symmetric and anti-symmetric states is established by e–e scattering.1 This is the reason why we strongly believe that the two electron subsystems, symmetric and anti-symmetric ones, in the case of current flow and lack of thermal equilibrium can be characterized by two quasi-FLs, because, first, the main mechanism of electron relaxation, namely the e–e scattering, does not mix the states of different parities and second, the equilibrium between generation and relaxation process is established at different levels due to the current flow in the system. This situation is not unique: a very similar situation arises in spin injection from a ferromagnetic metal to diffusive semiconductor [31]. It is assumed that the rate of scattering events without spin flips of spin-up (m) and spindown (k) electrons is much larger than the spin-flip rate. This implies that under non-equilibrium conditions two electrochemical potentials mm and mk may be defined at any point and they need not be equal (see also Ref. [32]).

7. Summary We have studied the parallel magnetotransport in DQW structures of two different potential shapes (quasi-rectangular and quasi-triangular) and found an important contribution of the symmetry properties of the charge carrier eigenstates to the magnetotransport phenomena. The beating effect that occurred in SdH oscillations was observed for both types of DQW. To explain this effect, we 1 The scattering by the neutral defects on interfaces, of course also occurs, but the high electron mobility by which our samples are characterized (2.5  105 cm2 V1 s1) suggests that the contribution of this mechanism to the electron relaxation is not significant.

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developed a special scheme for the LL energy calculations based essentially on the model proposed in Ref. [10] and carried out necessary simulations of the beating effect. In order to obtain agreement between our experimental data and the theoretical predictions concerning SdH oscillations, we introduced two different quasi-FLs, which characterize subsystems of symmetric and anti-symmetric states in DQWs. The existence of two different quasi FLs simply means that in DQWs consisting of two identical InGaAs QWs, separated by InAlAs barriers no wider than 20 nm, the total electron system can be considered as divided into two subsystems characterized by symmetric and anti-symmetric wave functions, respectively. One can treat then these two subsystems as weakly interacting in the sense that e–e scattering does not mix them and that they are described by their own distribution functions characterized by their own quasi-FLs. Appendix A The quantum magnetotransport in DQW can be treated in the framework of the effective mass approximation [29]. The main equation of the effective mass model is of the form 2 3  2 ~ i_r  e A 6 7 þ UðzÞ5cð~ rÞ ¼ Ecð~ rÞ, (A.1) 4E s þ 2m where Es ¼ Ec+e1, Ec stands for the bottom of the conduction band, e1 is the energy of the first subband due to space quantization along z-axis, and U(z) stands for the potential shape along z-axis, as in the case of Figs. 1 or 3. In order to take into account the magnetic field arrangement (BJez, ez is the unit vector of z-axis, and B?j, where j is the current density) choose the next vector potential gauge: Ax=By, Ay=0, Az=0. Look again at Figs. 1 and 3; what is peculiar about these pictures is that regardless of the shape of QWs, rectangular or triangular, the structures are symmetric about the center of the barrier, which is in the middle of the structure. The Hamiltonian, which describes an electron moving in the potential like that of Figs. 1 and 3, is invariant under inversion z-z and hence, its eigenfunctions are doubly degenerate. However, due to the tunneling effect this degeneracy is removed and the lowest energy state in each QW becomes split into two, bounding and anti-bounding states represented by even and odd functions, respectively. We termed these states also as symmetric and anti-symmetric ones. Then the eigenfunctions of the Hamiltonian (A.1) are of the form: 1 caðbÞ ð~ rÞ ¼ pffiffiffiffiffiffi expðikx xÞwm ðyÞjaðbÞ ðzÞ. Lx

(A.2)

In the last expression kx is the wave vector along x-axis, wm(y) the wave function corresponding to the mth LL, Lx the structure length along x-axis, and jaðbÞ ðzÞ stands for the anti-symmetric (symmetric) state corresponding to the symmetric structure of Figs. 1 or 3.

905

Appendix B Two electrons interacting in the ee scattering are indistinguishable; hence, the two electron states (1,2) and (10 ,20 ), before and after scattering event, should be described by the Hartree–Fock determinants as follows:  1 c1 ð~ (B.1) r1 Þc2 ð~ r2 Þ ! pffiffiffi c1 ð~ r1 Þc2 ð~ r2 Þ  c1 ð~ r2 Þc2 ð~ r1 Þ . 2 In order to calculate the matrix elements entering the   in Eq. (14) using the previous expressions for W ða;bÞ ~ ~0 ~0 ~ k!k k !k

expression and Eq. (A.2), one should substitute j1 ð~ r1 Þ, j2 ð~ r2 Þ (i, j ¼ 1, 2) by the functions 1 ri Þ ¼ pffiffiffiffiffiffi expðikjx xi Þwmj ðyi Þjb ðzi Þ cjb ð~ Lx 1 ¼ pffiffiffiffiffiffi jkj ~ ðxi Þwmj ðyi Þjb ðzi Þ, Lx 1 cja ð~ ri Þ ¼ pffiffiffiffiffiffi expðikjx xi Þwmj ðyi Þja ðzi Þ Lx 1 ¼ pffiffiffiffiffiffi jkj ~ ðxi Þwmj ðyi Þja ðzi Þ, Lx

ðB:2Þ

where ja(zi) and jb(zi) correspond to the symmetric and anti-symmetric states, respectively. Here in the last expressions kjkjx is the wave vector corresponding to the electron movement along x-axis perpendicular to z-axis. Introducing the Fourier transform of the screened Coulomb potential as Z d3~ q ee U ee ðrÞ ¼ U ~q expði~ q~ rÞ d~ q, ð2pÞ3 the corresponding matrix elements can be written down as follows: M ðaÞ ¼ I 1  I 2, 1;2!10 ;20 Z Z Z 1 1 p ffiffi ffi dq dy dy2 dz2 wm0 1 ðy1 Þja ðz1 Þ dz I1 ¼ 1 ? 1 ð2pÞ3 2  wm0 2 ðy2 Þja ðz2 ÞU ee q? wm1 ðy1 Þja ðz1 Þwm2 ðy2 Þja ðz2 Þ

   exp i~ q?~ r? dqx; k01 þk1 dqx; k02 þk2; qx , Z Z Z 1 1 p ffiffi ffi dz dq dy dy2 dz2 wm0 1 ðy1 Þja ðz1 Þ I2 ¼ 1 ? 1 ð2pÞ3 2  wm0 2 ðy2 Þja ðz2 ÞU ee q? wm2 ðy1 Þja ðz1 Þwm1 ðy2 Þ

  0 0 ja ðz2 Þ exp i~ q?~ r? dqx; k2 k1 dk1 k2 ;qx , ðB:3Þ M ðbÞ ¼ I 01  I 02 , 1;2!10 ;20 Z Z Z 1 1 p ffiffi ffi dq dy dy2 dz2 wm0 1 ðy1 Þjb ðz1 Þ dz I 01 ¼ 1 ? 1 ð2pÞ3 2  wm0 2 ðy2 Þjb ðz2 ÞU ee q? wm1 ðy1 Þjb ðz1 Þwm2 ðy2 Þjb ðz2 Þ

   exp i~ q?~ r? dqx ;k01 þk1 dqx ;k02 þk2 ;qx ,

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Z Z Z 1 1 pffiffiffi dq? dy1 dz1 dy2 dz2 wm0 1 ðy1 Þjb ðz1 Þ ¼ ð2pÞ3 2  wm0 2 ðy2 Þjb ðz2 ÞU ee q? wm2 ðy1 Þjb ðz1 Þwm1 ðy2 Þjb ðz2 Þ

   exp i~ q?~ r? dqx ;k2 k01 dk1 k02 ;qx , ðB:4Þ

where U ee q? is the two-dimensional Fourier transform of the screened Coulomb potential, q? ¼ ðqx ; qy Þ; and r? ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðy1  y2 Þ2 þ ðz1  z2 Þ2 . Note that regardless of the form of screening that occurs, the corresponding potential qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Uee(r), r ¼ ðx1  x2 Þ þ ðy1  y2 Þ2 þ ðz1  z2 Þ2 is an even function of its variables. References [1] M.C. Bronsager, K. Flensberg, B. Yu-Kuang Hu, A.-P. Jauho, Phys. Rev. Lett. 77 (1996) 1366. [2] H. Rubel, A. Fischer, W. Dietsche, K. von Klitzing, K. Eberl, Phys. Rev. Lett. 78 (1996) 1763. [3] X.G. Feng, S. Zelakiewicz, H. Noh, T.J. Ragucci, T.J. Gramila, Phys. Rev. Lett. 81 (1998) 3219. [4] J.G.S. Lok, S. Kraus, M. Pohlt, W. Dietsche, K. von Klitzing, W. Wegscheider, M. Bichler, Phys. Rev. B. 63 (2001) 041305(R). [5] M. Kellog, I.B. Spielman, J.P. Eisenstein, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 88 (2002) 126804. [6] H.S. Fresser, H. Frey, F.E. Prins, D.A. Haram, D.P. Kern, J. Bottcher, H. Kunzel, Semicond. Sci. Technol. 15 (2000) 242. [7] G.S. Boebinger, H.W. Jiang, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 64 (1990) 1793. [8] K. Ensslin, A. Wixforth, M. Sundaram, P.F. Hopkins, J.H. English, A.C. Gossard, et al., Phys. Rev. B. 47 (1992) 1366. [9] K.M. Brown, N. Turner, J.T. Nicholls, E.H. Linfield, M. Pepper, D.A. Ritchie, G.A.C. Jones, Phys. Rev. B 50 (1994) 15465. [10] D. Huang, M.O. Manasreh, Phys. Rev. B 54 (1996) 2044. [11] J.P. Eisenstein, A.H. MacDonald, Nature 432 (2004) 691. [12] R.M. Lewis, P.D. Ye, L.W. Engel, D.C. Tsui, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 89 (2002) 136804-1. [13] R.M. Lewis, Y.P. Chen, L.W. Engel, D.C. Tsui, L.N. Pfeiffer, K.W. West, Phys. Rev. B. 71 (2005) 081301(R). [14] G. Gervais, H.L. Stormer, D.C. Tsui, L.W. Engel, P.L. Kuhns, W.G. Moulton, A.P. Reyes, L.N. Pfeiffer, K.W. Baldwin, K.W. West, Phys. Rev. B. 72 (2005) 041310(R).

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