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Commensurability oscillations in a stripe-shaped two-dimensional electron gas
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999 Article No. spmi.1998.0632 Available online at http://www.idealibrary.com on
Commensurability oscillations in a stripe-shaped two-dimensional electron gas† M. T RINDADE DOS S ANTOS , G. M. G USEV, J. R. L EITE Instituto de Fi´ısica da Universidade de S˜ao Paulo, CP 66318, CEP 0513-970, S˜ao Paulo, Brazil A. A. B YKOV, N. T. M OSHEGOV, V. M. K UDRYASHEV, A. I. T OROPOV Y U . V. N ASTAUSHEV Institute of Semiconductor Physics, Novosibirsk, Russia
(Received 26 October 1998) We have studied the commensurability oscillations for a nonplanar, two-dimensional electron gas which is confined to a surface spatially modulated both in the transverse and longitudinal directions. We show, numerically, that coupling the drift in both directions causes the motion to be chaotic. The channeling tori, responsible for conduction along the stripes, are destroyed and the averaged squared drift velocities decrease. c 1999 Academic Press
Key words: two-dimensional electron gas, magnetoresistance, commensurability oscillations.
In the past few years there have been several investigations devoted to lateral superlattice transport problems, particularly on those subjected to periodically modulated magnetic fields. One of the interesting effects is commensurability oscillations of the magnetoresistance as a function of the external field [1]. These oscillations take place due to the interplay between two independent length scales, namely the period of the magnetic modulation, d, and the cyclotron radius, rc . Since these effects are observed at a low magnetic field regime, for high mobility samples (λ F rc , λ F is the Fermi wavelength) a classical theory for the motion of charged particles gives a good description of the dynamics of these systems. One of the important questions arising in this context concerns the connection between the properties of the classical trajectories, such as integrability (determined by the character of the magnetic field), and the properties of the electron transport. The effects of chaotic dynamics on a two-dimensional electron gas (2DEG) in a periodically modulated field have been studied in [2] and more recently in [3]. Low field magnetoresistance, due to channeling of electrons in open orbits, has been reported in [4]. Experimentally, nonuniform and periodically modulated magnetic fields have been achieved by depositing patterned ferromagnetic and superconducting films on top of the sample containing a 2DEG [5]. Recently, new re-growth techniques have been shown to produce a nonplanar 2DEG on a pre-patterned GaAs/AlGaAs † This work was partially supported by FAPESP.
0749–6036/99/010167 + 07
$30.00/0
c 1999 Academic Press
168
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
A
B 10 Bz (x)
δB/B, %
5
Bz (y)
0
–5 –10 0
20 000
40 000 60 000 x, y A
80 000 100 000
Fig. 1. A, 10 µm × 10 µm AFM image of the sample surface after re-growth. B, profile of the normal component of the magnetic field modulation in the directions along the stripes (y) and transverse direction (x).
substrate [6]. Since the electron motion in a 2DEG is sensitive only to the normal component of the magnetic field B, electrons confined to a nonplanar heterojunction experience a magnetic field varying with position. In this case, the modulation of the field is determined by the topography of the 2DEG. In [7] the fabrication of a stripe-shaped 2DEG employing re-growth techniques was reported. This study obtained a pre-patterned substrate with an antidot lattice and, after re-growth, a stripe-shaped surface was achieved. The samples constructed in [7] are shown to exhibit small irregularities in the form of the nonplanar 2DEG, both in the xdirection (transversal to the stripes) and y-direction (z is the growth direction). The surface profile is obtained by employing an atomic force microscopy (AFM) [7], the obtained surface image is shown in Fig. 1A. In Fig. 1B we display the normal component of the magnetic field in transverse, Bz (x), and longitudinal, Bz (y), directions. This figure was constructed by means of a projection of the external field applied normal to the surface. In this work, motivated by these experiments, we have performed numerical calculations for the electron dynamics in such nonplanar 2DEG subjected to a tilted magnetic field, showing the effects of spatially varying normal field components in both the x- and y-directions. Taking into account that such variations couple
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
169
2 0
2
–2 0 –2 0
–2 2
2 0
2
–2 0 –2 0
–2 2
Fig. 2. Typical surfaces modeled by z = A1 cos[k1 (x + A2 cos(βy))]: A, modulated in x-direction A2 = 0; and B, modulated in x- and y-directions A2 6 = 0.
nonlinearly the two degrees of freedom causing the classical trajectories to be altered from regular to chaotic patterns. Therefore, localization and diffusion properties are also modified. We show the averaged squared drift velocity is considerably reduced by considering small deviations of the field from uniform spatial modulation. We have modeled the surface to be of the form z = A1 cos[k1 (x + A2 cos(βy))].
(1)
In Fig. 2 we show two typical surfaces of this form. The shape of the surface containing the 2DEG produces the inhomogeneity and, therefore, is responsible for the drift of the guiding center (vdrift = α∇B × B, α is a function of B).
170
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
0.008 A/d = 0.02 θ= π/3 θ= π/4 θ= π/6
2 i hVdrift
0.006
0.004
0.002
0.000 0.4
0.6
1.0
0.8 Bz (a.u.)
rc = 2d
1.2
rc = d
Fig. 3. Mean squared drift velocity numerically calculated by means of the classical orbits, for various values of the angle θ, for the case A2 = 0. An average is taken over the phase space. The amplitude of the modulation is set to 2% of its length, d, (A1 /d = 0.02). Commensurability oscillations are shown to have larger amplitudes for parallel external fields. 50.0
2.0
2.0
A
B
C
40.0 0.0
1.0
–2.0
2.0
–4.0
–1.0
y
30.0
20.0
10.0
0.0 –1.0
0.0
1.0 x
2.0
–6.0 –0.5
0.5
1.5
2.5
–2.0 –1.0
x
0.0
1.0
2.0
x
Fig. 4. Typical trajectories on the plane x–y: A, A2 = 0 and parallel magnetic field; B, A2 = 0 and perpendicular magnetic field; and C, A2 6= 0. The surface in this case is x and y dependent.
The electron dynamics is obtained by means of the Lagrangian function 1 X 2 e ri + A · v, L= m 2 c i
(2)
where A = (0, Bz x, Bx y) is the vector potential (B = ∇ × A) and ri = x, y, z are Cartesian coordinates. We
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
171
A/d = 0.02 A2 = 0.0 A2 = 0.2 A2 = 0.3
0.0008
β=k θ = π/3
2 i hVdrift
0.0006
0.0004
0.0002
0.0000 0.0
0.5 rc = 4d
rc = 2d
1.0 Bz (a.u.)
1.5
2.0 rc = d
Fig. 5. Mean squared drift velocity numerically calculated by means of the classical orbits for a given value of the angle θ = π/3, in this case. An average is taken over the phase space. The amplitude of the x-modulation is set to 2% of its length, d, (A1 /d = 0.02), for various amplitudes A2 . The wavenumbers are set to k1 = k2 = 2π .
need to solve this problem with the constraint that the particle moves on the surface S only, given by eqn (1). For an external field applied in an arbitrary direction, B = (Bx , B y , Bz ), the Lagrangian reads L=
e 1 m(x˙ 2 + y˙ 2 + z˙ 2 ) + (x˙ B y z + y˙ Bz x + z˙ Bx y), 2 c
(3)
where z˙ = −A1 k1 sin(k1 x + A2 cos(k2 y))x˙ + A1 A2 k2 sin(k1 x + A2 cos(k2 y)) sin(k2 y) y˙ , is obtained from eqn (1), A1 , A2 , k1 and k2 are the amplitude and wavenumber of the modulation in the x-and y-directions, respectively. For mass m, electron charge e and light velocity c we have m = e = c = 1. Numerically integrating the respective equations of motion, we have calculated the squared drift velocity averaged over the phase space. First, we consider an external magnetic field applied normal to the surface and tilting away from the normal position (on the plane x–z) to the parallel one (θ = π/2 to θ = 0, Bx = B cos θ ). For the case in which we have A2 = 0, the modulation is one-dimensional and the guiding center drift occurring in the longitudinal direction is uniform (its trajectory is a straight line). For this case, commensurability oscillations are clearly seen in Fig. 3 (the values of Bz for integer multiples of rc are also shown in the figure). We can see that these oscillations have a much larger amplitude for smaller angles θ , close to the parallel position of the external field. In these cases the amplitude modulation of the field is greater. These larger amplitudes, for parallel external fields, are a result of channeling of classical trajectories in open orbits [4]. Figure 4A and B show trajectories for parallel and perpendicular external fields, respectively, where the displacement in one period is much larger in A. Including the modulation in the y-direction (A2 6 = 0) causes the drift to couple the two degrees of freedom and the motion becomes rather complicated. For a given magnetic field, Fig. 4C shows typical trajectories
172
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
A 0.8
y
0.6 0.4 0.2 0.0 B 0.8
y
0.6 0.4 0.2 0.0 C 0.8
y
0.6 0.4 0.2 0.0 –1.0
–0.8
–0.6
–0.4
–0.2
x Fig. 6. Surface of sections on the x–y plane (in arbitrary units) for an external field B = 2.2 (a.u.), applied at an angle θ = π/3. The amplitude of the x-modulation is set to 3% of its length, d. The sections were constructed with x = y = mod(d = 1), px = 0 and p y > 0. The values of A2 are set to 0.1, 0.2 and 0.3 in A, B and C, respectively. For increasing values of A2 , the tori connecting y = 0 and y = 1, responsible for the channeling of classical orbits, are destroyed.
obtained for small values of amplitude modulation. The consequences of this coupling for the transport properties are displayed in Fig. 5. It is clearly seen that the effect of taking into account the y-modulation,
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
173
even for small amplitudes, is to reduce the averaged squared drift velocity, increasing the resistance of the sample. We understand this effect also in terms of the classical orbits. Figure 6 shows the surface of sections (on the x–y plane) for increasing values of the amplitude of the y-modulation. These surfaces were constructed with x and y mod(d), px = 0 and p y > 0. For small values of A2 Fig. 6A exhibits the channeling tori (connecting y = 0 to y = 1), which are responsible for conducting the charged particle along the direction of the stripes. For greater values of A2 , (see Fig. 6B and C), these tori are destroyed and channeling is not possible any more, due to the chaotic nature of the motion. Chaotic diffusion now occurs in both the xand y-directions and the regular drift along the y-direction, associated with commensurability oscillations, is diminished. These results are in qualitative agreement with those of the experiment presented in [7]. For other tests of these theoretical results we are working on a quantitative comparison with the experiments presented in [7] for the low-field magnetoresistance oscillations.
References [1] [2] [3] [4]
}C. W. J. Beenaker, Phys. Rev. Lett. 62, 2020 (1989). }P. Schmelcher and D. L. Shepelyansky, Phys. Rev. B49, 7418 (1994). }Oleg. M. Yevtushenko and Klaus Richter, Phys. Rev. B57, 14 839 (1998). }A. Nogaret, S. Carlton, B. L. Gallagher, P. C. Main, M. Henini, R. Writz, R. Newbury, M. A. Howson, and S. P. Beaumont, Phys. Rev. B55, 24 (1997). [5] }H. A. Carmona, A. K. Geim, A. Nogaret, P. C. Main, T. J. Foster, H. Henini, S. P. Beaumont, and M. G. Blamire, Phys. Rev. Lett. 74, 3009 (1995); D. Weiss, R. R. Gerhardts, M. Seeger, K. von Klitzing, K. Eberl, and H. Nickel, Phys. Rev. Lett. 74, 30013 (1995). [6] }G. M. Gusev, U. Gennser, X. Kleber, D. K. Maude, J. C. Portal, D. I. Lubyshev, P. Basmaji, M. de P. A. Silva, J. C. Rossi, and Yu. V. Nastaushev, Surf. Sci. 361/362, 855 (1996). [7] }G. M. Gusev, A. A. Quivy, J. R. Leite, A. A. Bykov, N. T. Moshegov, V. M. Kudryashev, A. I. Toropov, and Yu. V. Nastaushev, Shubnikov–de Haas Oscillations in a Non-Planar Two dimensional Electron Gas, to be published. Submitted to Appl. Phys. Lett. (1998).
Commensurability oscillations in a stripe-shaped two-dimensional electron gas† M. T RINDADE DOS S ANTOS , G. M. G USEV, J. R. L EITE Instituto de Fi´ısica da Universidade de S˜ao Paulo, CP 66318, CEP 0513-970, S˜ao Paulo, Brazil A. A. B YKOV, N. T. M OSHEGOV, V. M. K UDRYASHEV, A. I. T OROPOV Y U . V. N ASTAUSHEV Institute of Semiconductor Physics, Novosibirsk, Russia
(Received 26 October 1998) We have studied the commensurability oscillations for a nonplanar, two-dimensional electron gas which is confined to a surface spatially modulated both in the transverse and longitudinal directions. We show, numerically, that coupling the drift in both directions causes the motion to be chaotic. The channeling tori, responsible for conduction along the stripes, are destroyed and the averaged squared drift velocities decrease. c 1999 Academic Press
Key words: two-dimensional electron gas, magnetoresistance, commensurability oscillations.
In the past few years there have been several investigations devoted to lateral superlattice transport problems, particularly on those subjected to periodically modulated magnetic fields. One of the interesting effects is commensurability oscillations of the magnetoresistance as a function of the external field [1]. These oscillations take place due to the interplay between two independent length scales, namely the period of the magnetic modulation, d, and the cyclotron radius, rc . Since these effects are observed at a low magnetic field regime, for high mobility samples (λ F rc , λ F is the Fermi wavelength) a classical theory for the motion of charged particles gives a good description of the dynamics of these systems. One of the important questions arising in this context concerns the connection between the properties of the classical trajectories, such as integrability (determined by the character of the magnetic field), and the properties of the electron transport. The effects of chaotic dynamics on a two-dimensional electron gas (2DEG) in a periodically modulated field have been studied in [2] and more recently in [3]. Low field magnetoresistance, due to channeling of electrons in open orbits, has been reported in [4]. Experimentally, nonuniform and periodically modulated magnetic fields have been achieved by depositing patterned ferromagnetic and superconducting films on top of the sample containing a 2DEG [5]. Recently, new re-growth techniques have been shown to produce a nonplanar 2DEG on a pre-patterned GaAs/AlGaAs † This work was partially supported by FAPESP.
0749–6036/99/010167 + 07
$30.00/0
c 1999 Academic Press
168
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
A
B 10 Bz (x)
δB/B, %
5
Bz (y)
0
–5 –10 0
20 000
40 000 60 000 x, y A
80 000 100 000
Fig. 1. A, 10 µm × 10 µm AFM image of the sample surface after re-growth. B, profile of the normal component of the magnetic field modulation in the directions along the stripes (y) and transverse direction (x).
substrate [6]. Since the electron motion in a 2DEG is sensitive only to the normal component of the magnetic field B, electrons confined to a nonplanar heterojunction experience a magnetic field varying with position. In this case, the modulation of the field is determined by the topography of the 2DEG. In [7] the fabrication of a stripe-shaped 2DEG employing re-growth techniques was reported. This study obtained a pre-patterned substrate with an antidot lattice and, after re-growth, a stripe-shaped surface was achieved. The samples constructed in [7] are shown to exhibit small irregularities in the form of the nonplanar 2DEG, both in the xdirection (transversal to the stripes) and y-direction (z is the growth direction). The surface profile is obtained by employing an atomic force microscopy (AFM) [7], the obtained surface image is shown in Fig. 1A. In Fig. 1B we display the normal component of the magnetic field in transverse, Bz (x), and longitudinal, Bz (y), directions. This figure was constructed by means of a projection of the external field applied normal to the surface. In this work, motivated by these experiments, we have performed numerical calculations for the electron dynamics in such nonplanar 2DEG subjected to a tilted magnetic field, showing the effects of spatially varying normal field components in both the x- and y-directions. Taking into account that such variations couple
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
169
2 0
2
–2 0 –2 0
–2 2
2 0
2
–2 0 –2 0
–2 2
Fig. 2. Typical surfaces modeled by z = A1 cos[k1 (x + A2 cos(βy))]: A, modulated in x-direction A2 = 0; and B, modulated in x- and y-directions A2 6 = 0.
nonlinearly the two degrees of freedom causing the classical trajectories to be altered from regular to chaotic patterns. Therefore, localization and diffusion properties are also modified. We show the averaged squared drift velocity is considerably reduced by considering small deviations of the field from uniform spatial modulation. We have modeled the surface to be of the form z = A1 cos[k1 (x + A2 cos(βy))].
(1)
In Fig. 2 we show two typical surfaces of this form. The shape of the surface containing the 2DEG produces the inhomogeneity and, therefore, is responsible for the drift of the guiding center (vdrift = α∇B × B, α is a function of B).
170
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
0.008 A/d = 0.02 θ= π/3 θ= π/4 θ= π/6
2 i hVdrift
0.006
0.004
0.002
0.000 0.4
0.6
1.0
0.8 Bz (a.u.)
rc = 2d
1.2
rc = d
Fig. 3. Mean squared drift velocity numerically calculated by means of the classical orbits, for various values of the angle θ, for the case A2 = 0. An average is taken over the phase space. The amplitude of the modulation is set to 2% of its length, d, (A1 /d = 0.02). Commensurability oscillations are shown to have larger amplitudes for parallel external fields. 50.0
2.0
2.0
A
B
C
40.0 0.0
1.0
–2.0
2.0
–4.0
–1.0
y
30.0
20.0
10.0
0.0 –1.0
0.0
1.0 x
2.0
–6.0 –0.5
0.5
1.5
2.5
–2.0 –1.0
x
0.0
1.0
2.0
x
Fig. 4. Typical trajectories on the plane x–y: A, A2 = 0 and parallel magnetic field; B, A2 = 0 and perpendicular magnetic field; and C, A2 6= 0. The surface in this case is x and y dependent.
The electron dynamics is obtained by means of the Lagrangian function 1 X 2 e ri + A · v, L= m 2 c i
(2)
where A = (0, Bz x, Bx y) is the vector potential (B = ∇ × A) and ri = x, y, z are Cartesian coordinates. We
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
171
A/d = 0.02 A2 = 0.0 A2 = 0.2 A2 = 0.3
0.0008
β=k θ = π/3
2 i hVdrift
0.0006
0.0004
0.0002
0.0000 0.0
0.5 rc = 4d
rc = 2d
1.0 Bz (a.u.)
1.5
2.0 rc = d
Fig. 5. Mean squared drift velocity numerically calculated by means of the classical orbits for a given value of the angle θ = π/3, in this case. An average is taken over the phase space. The amplitude of the x-modulation is set to 2% of its length, d, (A1 /d = 0.02), for various amplitudes A2 . The wavenumbers are set to k1 = k2 = 2π .
need to solve this problem with the constraint that the particle moves on the surface S only, given by eqn (1). For an external field applied in an arbitrary direction, B = (Bx , B y , Bz ), the Lagrangian reads L=
e 1 m(x˙ 2 + y˙ 2 + z˙ 2 ) + (x˙ B y z + y˙ Bz x + z˙ Bx y), 2 c
(3)
where z˙ = −A1 k1 sin(k1 x + A2 cos(k2 y))x˙ + A1 A2 k2 sin(k1 x + A2 cos(k2 y)) sin(k2 y) y˙ , is obtained from eqn (1), A1 , A2 , k1 and k2 are the amplitude and wavenumber of the modulation in the x-and y-directions, respectively. For mass m, electron charge e and light velocity c we have m = e = c = 1. Numerically integrating the respective equations of motion, we have calculated the squared drift velocity averaged over the phase space. First, we consider an external magnetic field applied normal to the surface and tilting away from the normal position (on the plane x–z) to the parallel one (θ = π/2 to θ = 0, Bx = B cos θ ). For the case in which we have A2 = 0, the modulation is one-dimensional and the guiding center drift occurring in the longitudinal direction is uniform (its trajectory is a straight line). For this case, commensurability oscillations are clearly seen in Fig. 3 (the values of Bz for integer multiples of rc are also shown in the figure). We can see that these oscillations have a much larger amplitude for smaller angles θ , close to the parallel position of the external field. In these cases the amplitude modulation of the field is greater. These larger amplitudes, for parallel external fields, are a result of channeling of classical trajectories in open orbits [4]. Figure 4A and B show trajectories for parallel and perpendicular external fields, respectively, where the displacement in one period is much larger in A. Including the modulation in the y-direction (A2 6 = 0) causes the drift to couple the two degrees of freedom and the motion becomes rather complicated. For a given magnetic field, Fig. 4C shows typical trajectories
172
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
A 0.8
y
0.6 0.4 0.2 0.0 B 0.8
y
0.6 0.4 0.2 0.0 C 0.8
y
0.6 0.4 0.2 0.0 –1.0
–0.8
–0.6
–0.4
–0.2
x Fig. 6. Surface of sections on the x–y plane (in arbitrary units) for an external field B = 2.2 (a.u.), applied at an angle θ = π/3. The amplitude of the x-modulation is set to 3% of its length, d. The sections were constructed with x = y = mod(d = 1), px = 0 and p y > 0. The values of A2 are set to 0.1, 0.2 and 0.3 in A, B and C, respectively. For increasing values of A2 , the tori connecting y = 0 and y = 1, responsible for the channeling of classical orbits, are destroyed.
obtained for small values of amplitude modulation. The consequences of this coupling for the transport properties are displayed in Fig. 5. It is clearly seen that the effect of taking into account the y-modulation,
Superlattices and Microstructures, Vol. 25, No. 1/2, 1999
173
even for small amplitudes, is to reduce the averaged squared drift velocity, increasing the resistance of the sample. We understand this effect also in terms of the classical orbits. Figure 6 shows the surface of sections (on the x–y plane) for increasing values of the amplitude of the y-modulation. These surfaces were constructed with x and y mod(d), px = 0 and p y > 0. For small values of A2 Fig. 6A exhibits the channeling tori (connecting y = 0 to y = 1), which are responsible for conducting the charged particle along the direction of the stripes. For greater values of A2 , (see Fig. 6B and C), these tori are destroyed and channeling is not possible any more, due to the chaotic nature of the motion. Chaotic diffusion now occurs in both the xand y-directions and the regular drift along the y-direction, associated with commensurability oscillations, is diminished. These results are in qualitative agreement with those of the experiment presented in [7]. For other tests of these theoretical results we are working on a quantitative comparison with the experiments presented in [7] for the low-field magnetoresistance oscillations.
References [1] [2] [3] [4]
}C. W. J. Beenaker, Phys. Rev. Lett. 62, 2020 (1989). }P. Schmelcher and D. L. Shepelyansky, Phys. Rev. B49, 7418 (1994). }Oleg. M. Yevtushenko and Klaus Richter, Phys. Rev. B57, 14 839 (1998). }A. Nogaret, S. Carlton, B. L. Gallagher, P. C. Main, M. Henini, R. Writz, R. Newbury, M. A. Howson, and S. P. Beaumont, Phys. Rev. B55, 24 (1997). [5] }H. A. Carmona, A. K. Geim, A. Nogaret, P. C. Main, T. J. Foster, H. Henini, S. P. Beaumont, and M. G. Blamire, Phys. Rev. Lett. 74, 3009 (1995); D. Weiss, R. R. Gerhardts, M. Seeger, K. von Klitzing, K. Eberl, and H. Nickel, Phys. Rev. Lett. 74, 30013 (1995). [6] }G. M. Gusev, U. Gennser, X. Kleber, D. K. Maude, J. C. Portal, D. I. Lubyshev, P. Basmaji, M. de P. A. Silva, J. C. Rossi, and Yu. V. Nastaushev, Surf. Sci. 361/362, 855 (1996). [7] }G. M. Gusev, A. A. Quivy, J. R. Leite, A. A. Bykov, N. T. Moshegov, V. M. Kudryashev, A. I. Toropov, and Yu. V. Nastaushev, Shubnikov–de Haas Oscillations in a Non-Planar Two dimensional Electron Gas, to be published. Submitted to Appl. Phys. Lett. (1998).