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Wave-function mixing in strongly confined tunnel-coupled quantum point contacts
ARTICLE IN PRESS
Physica E 34 (2006) 526–529 www.elsevier.com/locate/physe
Wave-function mixing in strongly confined tunnel-coupled quantum point contacts G. Apetriia, S.F. Fischera,, U. Kunzea, D. Schuhb,c, G. Abstreiterb a
Lehrstuhl fu¨r Werkstoffe und Nanoelektronik, Ruhr-Universita¨t Bochum, D-44780 Bochum, Germany b Walter Schottky Institut, Technische Universita¨t Mu¨nchen, D-85748 Garching, Germany c Institut fu¨r Angewandte und Experimentelle Physik, Universita¨t Regensburg, 93040 Regensburg, Germany Available online 19 April 2006
Abstract We investigate coupling phenomena in quantum point contacts (QPCs) where a particularly strong lateral confinement leads to large separations of the one-dimensional (1D) subbands (above 10 meV). Closely spaced vertically stacked QPCs were defined by atomic-forcemicroscope nanolithography and wet-chemical etching on a GaAs/AlGaAs double-quantum-well heterostructure with a twodimensional symmetric–antisymmetric energy gap of 4 meV. Top-gate and back-gate voltage variation in combination with a cooling bias technique was used for tuning the energy spectra of the tunnel-coupled QPCs in order to obtain degeneracies of 1D subbands. We observed clear anticrossings caused by mixing of the 1D wave-functions in grey-scale transconductance plots versus top-gate and backgate voltage at 4.2 K. r 2006 Elsevier B.V. All rights reserved. PACS: 73.23.Ad; 73.21.Hb; 73.63.Nm Keywords: Double-quantum-well heterostructure; Quantum point contact; 1D transport; 1D–1D coupling
Coupled one-dimensional (1D) electron systems arouse interest not only as a fundamental problem but also for potential applications in quantum information technology [1,2]. In a vertical-geometry approach closely spaced coupled 1D systems result from stacked two-dimensional electron gases (2DEGs) contained in double-quantum-well (DQW) GaAs/AlGaAs heterostructures [3–6] or, alternatively, from two distinct energy subbands of one 2DEG [7,8]. For systems defined by means of split gates the observation of coupling effects is limited to very low temperatures (mK regime) due to small 1D-subband separations (around 2 meV). An experimental determination of 1D-coupling energies was performed in two cases [4,6]. The highest values reported reach 2.6 meV, while the temperature where coupling was demonstrated was maintained as low as 60 mK [4]. Here, we present clear evidence and evaluation at 4.2 K of 1D coupling in vertically stacked Corresponding author. Tel.: +49 234 3225760; fax: +49 234 3214166.
E-mail address: saskia.fi[email protected] (S.F. Fischer). 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.03.026
quantum point contacts (QPCs), i.e. short ballistic 1D electron systems, with 1D-subband separations in excess of 10 meV. The heterostructure consists of two equally wide (14.5 nm) GaAs layers separated by a 1 nm thick Al0:32 Ga0:68 As barrier. The upper bound of the top QW lies 60 nm below the heterostructure surface. Electrons are provided by two delta-doped supply layers, each on one side of the DQW. A sheet electron density of 4:3 1011 cm2 and a mobility of 2:4 105 cm2 V1 s1 were measured in the dark at 4.2 K for the ungated structure. By dynamic ploughing with an atomic force microscope (AFM) and subsequent wet-chemical etching a 1D constriction was processed in the central channel area of a modulation-doped field-effect transistor [9,10]. The line pattern defining the constriction is shown in the inset of Fig. 3. Proper nanolithography parameters lead to the simultaneous depletion of both 2DEGs of the DQW underneath the etched grooves. In this way two vertically stacked spatially close QPCs with common source and
ARTICLE IN PRESS G. Apetrii et al. / Physica E 34 (2006) 526–529
drain contacts are defined. A top gate covers the constriction, while the back side of the sample serves as a back-gate contact. Two-terminal differential conductance and transconductance measurements at 4.2 K were performed by means of standard lock-in technique. The source-drain excitation voltage was 0.3 mV rms at 433 Hz. Additionally, for transconductance measurements the top-gate voltage was modulated with a 3 mV rms voltage. Fig. 1 depicts the variation of the 2D electron density determined by Shubnikov–de Haas measurements at 2 K as a function of the top-gate voltage V tg . At small V tg only the bottom 2D channel is open exhibiting linear increase of the density nb ¼ n1 with the top-gate voltage. After the onset of the top 2D channel the density nt ¼ n2 increases linearly until the two wells are balanced, while nb remains roughly constant. The region corresponding to balanced QWs is marked by a dotted circle. In this region, the two populations correspond to the split symmetric and antisymmetric states, which extend equally over both QWs. Further increase of V tg destroys the equilibrium and leads to electron localization in the individual wells. The population of the top QW continues to increase linearly, whereas the bottom one maintains constant electron density. Around V tg ¼ 0:3 V both densities saturate. From the minimum separation of the two densities (at the balance point) a 2D-coupling gap of about 4 meV has been estimated [12]. A numerical analysis of the electron states in the DQW was carried out by means of self-consistent calculations [13]. The quantitative agreement between experimentally acquired and calculated electron densities is illustrated in Fig. 1. The maximal absolute deviation of the simulated values from the measured values amounts to 15%. Further
Fig. 1. Filling of the DQW with increasing gate voltage. Full squares represent experimental values determined by Shubnikov–de Haas measurements at 2 K, while the continuous line resulted from self-consistent calculations. n1 and n2 refer to the density of the energetic lower and energetic higher subband, whereas nt and nb denote the population of the top and the bottom QW, respectively. At equilibrium (encircled) the two electron densities correspond to the symmetric and the antisymmetric states.
527
(a)
(c)
(d)
(b)
(e)
Fig. 2. Results of self-consistent calculations on the DQW system. (a) The expectation values of the electrons considered from the heterostructure surface and (b) the spread of the electron wave-function are represented by empty circles for the lower and full circles for the higher 2D subband. (c)–(e) Conduction-band edge E c and probability density of the lower (f21 ) and of the higher (f22 ) 2D subband calculated for three different top-gatevoltage values of (c) 0:15 V, (d) 0.04 V and (e) 0.4 V.
calculation results are shown in Fig. 2. The expectation values hzi i, i.e. the average distances of the electrons in the quantum state i from heterostructure surface, are depicted in Fig. 2(a) as a function of the top-gate voltage. At V tg o0:04 V the electrons of the energetically lower (higher) 2D subband occupy the bottom (top) QW (Fig. 2(c)), while at V tg 40:04 V the opposite situation is found (Fig. 2(e)). The maximal difference between the two expectation values reaches about 15 nm for large positive V tg . The spread ðhz2i i hzi i2 Þ1=2 of the electron wave-functions displayed in Fig. 2(b) is slightly larger for the energetically higher subband, with a maximal difference between the two subbands of 0.9 nm. Fig. 3(a) shows conductance measurements on constrictions with a gap between the etched grooves (geometric width) of 90–170 nm at an etch depth of 50–65 nm. All constrictions show quantized conductance with deviations from a regular increase in multiples of 2e2 =h in form of missing or ill-defined steps. The measured conductance was corrected by series resistances between 800 and 1300 O. The transconductance signal (Fig. 3(b)) consists of minima, which correspond to the conductance plateaus, and maxima associated with the inflection points between successive conductance plateaus. For all fabricated devices the height, width and position of the transconductance peaks show irregular variation with increasing top-gate voltage. The irregular behaviour of both conductance and transconductance proves the existence of two parallel 1D transport channels, resulting from each of the two 2DEGs. In order to investigate coupling effects between 1D subbands of the two QPCs their energy spectra are tuned
ARTICLE IN PRESS G. Apetrii et al. / Physica E 34 (2006) 526–529
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24 20 16 12 8 4 (a) 0
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(b) 0.1
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0.7
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Fig. 3. (a) Conductance at 4.2 K of four constrictions with different geometric widths. The inset shows an AFM image of the groove pattern defining the QPCs. The etch depth amounts to 50–65 nm. (b) Measured transconductance of the 170 nm wide constriction.
by varying the top-gate voltage, the back-gate voltage and/or by a persistent charging effect obtained by cooling the device under a top-gate bias voltage [10]. Fig. 4 presents grey-scale plots of the transconductance versus top and back-gate voltage for two different constrictions. In the grey-scale plot transconductance maxima are represented in dark, whereas low transconductance values are represented as bright areas. Consequently, the dark traces illustrate the evolution of 1D-subband edges in the QPCs with varying top and back-gate voltage. In Fig. 4(a) two subband edges (denoted as 1b and 2b ) exhibit strong linear variation with the back-gate voltage V bg , while one subband edge (denoted as 1t ) is almost back-gate-voltage independent. The weak variation of the top-QPC subbands with V bg results from a screening of the back-gate field by the electrons in the bottom QPC. In this way, an identification of the 1D subbands was realized by associating each transconductance maximum to the corresponding top or bottom QPC. The clear superposition of the 1t and 2b transconductance peaks indicates a crossing in the energy spectrum, i.e. a degeneracy of the two associated 2D subbands. Unlike the linear back-gatevoltage dependence observed in Fig. 4(a), the maxima traces of Fig. 4(b) exhibit non-linear behaviour with different curvatures, either positive or negative. In this case, the tunnel interaction of the two QPCs causes the lifting of the subband degeneracies leading to anticrossings in the energy spectrum. In the two encircled regions denoted by (1t ; 1b ) and (2t ; 2b ), with minimum separation of transconductance peaks, the detected energy subbands possess mixed 1D wave-functions and split up. The splitting energies were determined directly by energy level
(b)
Fig. 4. Grey-scale representation of the transconductance versus top-gate and back-gate voltage for (a) a 90 nm wide constriction (etch depth: 50 nm) and for (b) a 130 nm wide constriction (etch depth: 65 nm) after cooling under a gate bias voltage of 0.7 V. The indices denote the 1D subbands of the top (1t and 2t ) and of the bottom (1b and 2b ) QPCs. The encircled areas in (b) reflect energy anticrossings due to wave-function mixing. The measurements were performed at 4.2 K.
spectroscopy in the non-linear transport regime [14]. For the (1t ; 1b ) anticrossing a particularly large value of about 5 meV was obtained. Note, in the 2D reservoirs at top-gate voltages above V tg ¼ 0:3 V the energy of the bottom-QW electrons is higher than the energy of the top-QW electrons, with a 2Dsubband separation of 9 meV (Fig. 1 and Ref. [12]). Still, in the constriction the bottom QPC sets on at smaller (Fig. 4(a)) or equal (Fig. 4(b)) top-gate voltage compared to the top QPC. This (over)compensation of the difference in the vertical-confinement energy is induced by the lateral confinement, as the electrons in the top QW are situated 15 nm closer to the heterostructure surface (Fig. 2(a)). In conclusion, strong 1D-coupling effects were demonstrated in AFM-fabricated vertically stacked tunnelcoupled QPCs. The large 1D-subband separations (above 10 meV) and coupling energy gaps (about 5 meV) allow an experimental investigation of coupling properties at liquid helium temperature (4.2 K). G. A. gratefully acknowledges the financial support of the foundation Dr. Isolde Dietrich. Part of this work was supported by the Bundesministerium fu¨r Bildung und Forschung under Grant no. 01BM920. References [1] [2] [3] [4] [5] [6] [7]
A. Bertoni, et al., Phys. Rev. Lett. 84 (2000) 5912. M.J. Gilbert, et al., Appl. Phys. Lett. 81 (2002) 4284. I.M. Castleton, et al., Physica B 249–251 (1998) 157. K.J. Thomas, et al., Phys. Rev. B 59 (1999) 12252. M.A. Blount, et al., Physica E 6 (2000) 689. K.J. Friedland, et al., Physica E 11 (2001) 144. G. Salis, et al., Phys. Rev. B 60 (1999) 7756.
ARTICLE IN PRESS G. Apetrii et al. / Physica E 34 (2006) 526–529 [8] [9] [10] [12]
S.F. Fischer, et al., Phys. Rev. B 71 (2005) 195330. G. Apetrii, et al., Semicond. Sci. Technol. 17 (2002) 735. S.F. Fischer, et al., Appl. Phys. Lett. 81 (2002) 2779. The energy separation DE can be estimated from the difference between the electron densities of the two subbands Dn as DE ¼ ðp_2 =m ÞDn.
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[13] The calculations were performed using a self-consistent simulation program created by M. Rother, Walter Schottky Institute, Technical University of Munich, 1999, computer code AQUILA available at hwww.mathworks.com/matlabcentral/fileexchangei. [14] G. Apetrii, et al., unpublished.
Physica E 34 (2006) 526–529 www.elsevier.com/locate/physe
Wave-function mixing in strongly confined tunnel-coupled quantum point contacts G. Apetriia, S.F. Fischera,, U. Kunzea, D. Schuhb,c, G. Abstreiterb a
Lehrstuhl fu¨r Werkstoffe und Nanoelektronik, Ruhr-Universita¨t Bochum, D-44780 Bochum, Germany b Walter Schottky Institut, Technische Universita¨t Mu¨nchen, D-85748 Garching, Germany c Institut fu¨r Angewandte und Experimentelle Physik, Universita¨t Regensburg, 93040 Regensburg, Germany Available online 19 April 2006
Abstract We investigate coupling phenomena in quantum point contacts (QPCs) where a particularly strong lateral confinement leads to large separations of the one-dimensional (1D) subbands (above 10 meV). Closely spaced vertically stacked QPCs were defined by atomic-forcemicroscope nanolithography and wet-chemical etching on a GaAs/AlGaAs double-quantum-well heterostructure with a twodimensional symmetric–antisymmetric energy gap of 4 meV. Top-gate and back-gate voltage variation in combination with a cooling bias technique was used for tuning the energy spectra of the tunnel-coupled QPCs in order to obtain degeneracies of 1D subbands. We observed clear anticrossings caused by mixing of the 1D wave-functions in grey-scale transconductance plots versus top-gate and backgate voltage at 4.2 K. r 2006 Elsevier B.V. All rights reserved. PACS: 73.23.Ad; 73.21.Hb; 73.63.Nm Keywords: Double-quantum-well heterostructure; Quantum point contact; 1D transport; 1D–1D coupling
Coupled one-dimensional (1D) electron systems arouse interest not only as a fundamental problem but also for potential applications in quantum information technology [1,2]. In a vertical-geometry approach closely spaced coupled 1D systems result from stacked two-dimensional electron gases (2DEGs) contained in double-quantum-well (DQW) GaAs/AlGaAs heterostructures [3–6] or, alternatively, from two distinct energy subbands of one 2DEG [7,8]. For systems defined by means of split gates the observation of coupling effects is limited to very low temperatures (mK regime) due to small 1D-subband separations (around 2 meV). An experimental determination of 1D-coupling energies was performed in two cases [4,6]. The highest values reported reach 2.6 meV, while the temperature where coupling was demonstrated was maintained as low as 60 mK [4]. Here, we present clear evidence and evaluation at 4.2 K of 1D coupling in vertically stacked Corresponding author. Tel.: +49 234 3225760; fax: +49 234 3214166.
E-mail address: saskia.fi[email protected] (S.F. Fischer). 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.03.026
quantum point contacts (QPCs), i.e. short ballistic 1D electron systems, with 1D-subband separations in excess of 10 meV. The heterostructure consists of two equally wide (14.5 nm) GaAs layers separated by a 1 nm thick Al0:32 Ga0:68 As barrier. The upper bound of the top QW lies 60 nm below the heterostructure surface. Electrons are provided by two delta-doped supply layers, each on one side of the DQW. A sheet electron density of 4:3 1011 cm2 and a mobility of 2:4 105 cm2 V1 s1 were measured in the dark at 4.2 K for the ungated structure. By dynamic ploughing with an atomic force microscope (AFM) and subsequent wet-chemical etching a 1D constriction was processed in the central channel area of a modulation-doped field-effect transistor [9,10]. The line pattern defining the constriction is shown in the inset of Fig. 3. Proper nanolithography parameters lead to the simultaneous depletion of both 2DEGs of the DQW underneath the etched grooves. In this way two vertically stacked spatially close QPCs with common source and
ARTICLE IN PRESS G. Apetrii et al. / Physica E 34 (2006) 526–529
drain contacts are defined. A top gate covers the constriction, while the back side of the sample serves as a back-gate contact. Two-terminal differential conductance and transconductance measurements at 4.2 K were performed by means of standard lock-in technique. The source-drain excitation voltage was 0.3 mV rms at 433 Hz. Additionally, for transconductance measurements the top-gate voltage was modulated with a 3 mV rms voltage. Fig. 1 depicts the variation of the 2D electron density determined by Shubnikov–de Haas measurements at 2 K as a function of the top-gate voltage V tg . At small V tg only the bottom 2D channel is open exhibiting linear increase of the density nb ¼ n1 with the top-gate voltage. After the onset of the top 2D channel the density nt ¼ n2 increases linearly until the two wells are balanced, while nb remains roughly constant. The region corresponding to balanced QWs is marked by a dotted circle. In this region, the two populations correspond to the split symmetric and antisymmetric states, which extend equally over both QWs. Further increase of V tg destroys the equilibrium and leads to electron localization in the individual wells. The population of the top QW continues to increase linearly, whereas the bottom one maintains constant electron density. Around V tg ¼ 0:3 V both densities saturate. From the minimum separation of the two densities (at the balance point) a 2D-coupling gap of about 4 meV has been estimated [12]. A numerical analysis of the electron states in the DQW was carried out by means of self-consistent calculations [13]. The quantitative agreement between experimentally acquired and calculated electron densities is illustrated in Fig. 1. The maximal absolute deviation of the simulated values from the measured values amounts to 15%. Further
Fig. 1. Filling of the DQW with increasing gate voltage. Full squares represent experimental values determined by Shubnikov–de Haas measurements at 2 K, while the continuous line resulted from self-consistent calculations. n1 and n2 refer to the density of the energetic lower and energetic higher subband, whereas nt and nb denote the population of the top and the bottom QW, respectively. At equilibrium (encircled) the two electron densities correspond to the symmetric and the antisymmetric states.
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(a)
(c)
(d)
(b)
(e)
Fig. 2. Results of self-consistent calculations on the DQW system. (a) The expectation values of the electrons considered from the heterostructure surface and (b) the spread of the electron wave-function are represented by empty circles for the lower and full circles for the higher 2D subband. (c)–(e) Conduction-band edge E c and probability density of the lower (f21 ) and of the higher (f22 ) 2D subband calculated for three different top-gatevoltage values of (c) 0:15 V, (d) 0.04 V and (e) 0.4 V.
calculation results are shown in Fig. 2. The expectation values hzi i, i.e. the average distances of the electrons in the quantum state i from heterostructure surface, are depicted in Fig. 2(a) as a function of the top-gate voltage. At V tg o0:04 V the electrons of the energetically lower (higher) 2D subband occupy the bottom (top) QW (Fig. 2(c)), while at V tg 40:04 V the opposite situation is found (Fig. 2(e)). The maximal difference between the two expectation values reaches about 15 nm for large positive V tg . The spread ðhz2i i hzi i2 Þ1=2 of the electron wave-functions displayed in Fig. 2(b) is slightly larger for the energetically higher subband, with a maximal difference between the two subbands of 0.9 nm. Fig. 3(a) shows conductance measurements on constrictions with a gap between the etched grooves (geometric width) of 90–170 nm at an etch depth of 50–65 nm. All constrictions show quantized conductance with deviations from a regular increase in multiples of 2e2 =h in form of missing or ill-defined steps. The measured conductance was corrected by series resistances between 800 and 1300 O. The transconductance signal (Fig. 3(b)) consists of minima, which correspond to the conductance plateaus, and maxima associated with the inflection points between successive conductance plateaus. For all fabricated devices the height, width and position of the transconductance peaks show irregular variation with increasing top-gate voltage. The irregular behaviour of both conductance and transconductance proves the existence of two parallel 1D transport channels, resulting from each of the two 2DEGs. In order to investigate coupling effects between 1D subbands of the two QPCs their energy spectra are tuned
ARTICLE IN PRESS G. Apetrii et al. / Physica E 34 (2006) 526–529
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24 20 16 12 8 4 (a) 0
(a)
(b) 0.1
0.2
0.3
0.4
0.5
0.6
0.7
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Fig. 3. (a) Conductance at 4.2 K of four constrictions with different geometric widths. The inset shows an AFM image of the groove pattern defining the QPCs. The etch depth amounts to 50–65 nm. (b) Measured transconductance of the 170 nm wide constriction.
by varying the top-gate voltage, the back-gate voltage and/or by a persistent charging effect obtained by cooling the device under a top-gate bias voltage [10]. Fig. 4 presents grey-scale plots of the transconductance versus top and back-gate voltage for two different constrictions. In the grey-scale plot transconductance maxima are represented in dark, whereas low transconductance values are represented as bright areas. Consequently, the dark traces illustrate the evolution of 1D-subband edges in the QPCs with varying top and back-gate voltage. In Fig. 4(a) two subband edges (denoted as 1b and 2b ) exhibit strong linear variation with the back-gate voltage V bg , while one subband edge (denoted as 1t ) is almost back-gate-voltage independent. The weak variation of the top-QPC subbands with V bg results from a screening of the back-gate field by the electrons in the bottom QPC. In this way, an identification of the 1D subbands was realized by associating each transconductance maximum to the corresponding top or bottom QPC. The clear superposition of the 1t and 2b transconductance peaks indicates a crossing in the energy spectrum, i.e. a degeneracy of the two associated 2D subbands. Unlike the linear back-gatevoltage dependence observed in Fig. 4(a), the maxima traces of Fig. 4(b) exhibit non-linear behaviour with different curvatures, either positive or negative. In this case, the tunnel interaction of the two QPCs causes the lifting of the subband degeneracies leading to anticrossings in the energy spectrum. In the two encircled regions denoted by (1t ; 1b ) and (2t ; 2b ), with minimum separation of transconductance peaks, the detected energy subbands possess mixed 1D wave-functions and split up. The splitting energies were determined directly by energy level
(b)
Fig. 4. Grey-scale representation of the transconductance versus top-gate and back-gate voltage for (a) a 90 nm wide constriction (etch depth: 50 nm) and for (b) a 130 nm wide constriction (etch depth: 65 nm) after cooling under a gate bias voltage of 0.7 V. The indices denote the 1D subbands of the top (1t and 2t ) and of the bottom (1b and 2b ) QPCs. The encircled areas in (b) reflect energy anticrossings due to wave-function mixing. The measurements were performed at 4.2 K.
spectroscopy in the non-linear transport regime [14]. For the (1t ; 1b ) anticrossing a particularly large value of about 5 meV was obtained. Note, in the 2D reservoirs at top-gate voltages above V tg ¼ 0:3 V the energy of the bottom-QW electrons is higher than the energy of the top-QW electrons, with a 2Dsubband separation of 9 meV (Fig. 1 and Ref. [12]). Still, in the constriction the bottom QPC sets on at smaller (Fig. 4(a)) or equal (Fig. 4(b)) top-gate voltage compared to the top QPC. This (over)compensation of the difference in the vertical-confinement energy is induced by the lateral confinement, as the electrons in the top QW are situated 15 nm closer to the heterostructure surface (Fig. 2(a)). In conclusion, strong 1D-coupling effects were demonstrated in AFM-fabricated vertically stacked tunnelcoupled QPCs. The large 1D-subband separations (above 10 meV) and coupling energy gaps (about 5 meV) allow an experimental investigation of coupling properties at liquid helium temperature (4.2 K). G. A. gratefully acknowledges the financial support of the foundation Dr. Isolde Dietrich. Part of this work was supported by the Bundesministerium fu¨r Bildung und Forschung under Grant no. 01BM920. References [1] [2] [3] [4] [5] [6] [7]
A. Bertoni, et al., Phys. Rev. Lett. 84 (2000) 5912. M.J. Gilbert, et al., Appl. Phys. Lett. 81 (2002) 4284. I.M. Castleton, et al., Physica B 249–251 (1998) 157. K.J. Thomas, et al., Phys. Rev. B 59 (1999) 12252. M.A. Blount, et al., Physica E 6 (2000) 689. K.J. Friedland, et al., Physica E 11 (2001) 144. G. Salis, et al., Phys. Rev. B 60 (1999) 7756.
ARTICLE IN PRESS G. Apetrii et al. / Physica E 34 (2006) 526–529 [8] [9] [10] [12]
S.F. Fischer, et al., Phys. Rev. B 71 (2005) 195330. G. Apetrii, et al., Semicond. Sci. Technol. 17 (2002) 735. S.F. Fischer, et al., Appl. Phys. Lett. 81 (2002) 2779. The energy separation DE can be estimated from the difference between the electron densities of the two subbands Dn as DE ¼ ðp_2 =m ÞDn.
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[13] The calculations were performed using a self-consistent simulation program created by M. Rother, Walter Schottky Institute, Technical University of Munich, 1999, computer code AQUILA available at hwww.mathworks.com/matlabcentral/fileexchangei. [14] G. Apetrii, et al., unpublished.