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Classical and quantum dynamics in an array of electron billiards
ARTICLE IN PRESS
Physica E 40 (2008) 1315–1318 www.elsevier.com/locate/physe
Classical and quantum dynamics in an array of electron billiards Roland Brunnera,, Ronald Meiselsa, Friedemar Kuchara, Richard Akisb, David K. Ferryb, Jonathan P. Birdc a Institute of Physics, University of Leoben, A-8700 Leoben, Austria Department of Electrical Engineering, Arizona State University, Tempe, AZ, USA c Department of Electrical Engineering, University at Buffalo, Buffalo, NY, USA
b
Available online 22 September 2007
Abstract We investigate the classical and quantum dynamics in an array of electron billiards within a magnetic field. The resulting modification of the phase space due to the magnetic field can be directly related to the experimentally observed magneto-resistance. We argue that the behaviour of chaos plays an important role in the transport of such ballistic devices. r 2007 Elsevier B.V. All rights reserved. PACS: 05.45.Mt; 73.63.Kv; 75.47.Jn Keywords: Quantum chaos; Magneto-transport; Quantum billiards
1. Introduction The principle of chaotic transport in ballistic systems is of great relevance to the development of nano-electronic devices in the next decade, where miniaturisation of device size will cause the electronic motion to change from diffusive to ballistic. Therefore, a complete knowledge of the chaotic dynamics has to be developed in order to understand how these devices will work. In classical mechanics, one can distinguish between integrable and non-integrable systems [1]. In an integrable conservative system with 2 degrees of freedom, every trajectory lies on a torus constructed by two angular variables which can be associated with two frequencies o1 and o2. Depending on the ratio of the two characteristic frequencies, one either obtains periodic orbits for rational ratios or quasi-periodic orbits for irrational ratios [1]. Nonintegrable systems, however, may be divided into two classes. One class contains the completely chaotic system. In this case, every neighbourhood is traversed in an infinite number of momentum directions [2]. The second class, Corresponding author. Tel.: +43 3842 402 4610; fax: +43 3842 402 4602. E-mail address: [email protected] (R. Brunner).
1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.08.118
most ubiquitous in nature, is that whose phase space is mixed. The Kolmogorov–Arnold–Moser (KAM) theory [1] states that some of the tori, presenting the regular motion survive and others are destroyed if the system is subjected to nonlinear perturbation. The surviving tori are referred as KAM-tori [1]. The KAM theorem is applicable for small and smooth perturbations and for sufficiently irrational ratios (quasi-periodic motion) of the characteristic frequencies o1 and o2. Violating these conditions leads to a so-called non-KAM system. In driven systems, this nonKAM like behaviour has been investigated experimentally under resonant driving conditions [3]. A specific field dealing with the crossover between classical and quantum mechanics is quantum chaos [4], in our case for an ‘‘open’’ system. The understanding of an open system is critical, where a quantum object is affected by its connection to a classical ‘‘measuring’’ environment. One approach that has been proposed to describe the behaviour of such systems is the decoherence theory [5]. According to this, a subset of states of the originally isolated system becomes decoherent through their coupling with the environment, while a specific set of ‘‘pointer states’’ remains robust and eventually correlates with classical regular orbits [5]. Evidence for the pointers states [6] and their one-to-one connection to classical regular
ARTICLE IN PRESS R. Brunner et al. / Physica E 40 (2008) 1315–1318
1316
(1)
R (kΩ)
25
R (k Ω)
20 15 10
(2)
−0.50
−0.4
0.0
0.4
0.8
) V g (V
−0.45
5 0 −0.8
22 20 17 14 12 9 6 4 1
(3)
−0.40
B (T)
Fig. 1. (a) The measured magneto-resistance (white: high resistance, black: low resistance) for the seven-dot array for various gate voltages VG between 0.40 and 0.54 V, at a base temperature T ¼ 10 mK. The maxima and minima are expressed by the ratio of the hybrid frequencies o+/o. Data adapted from Ref. [8]. (b) Left row: classically calculated trajectories for certain initial conditions in the entrance constriction for o+/o ¼ 2, 3 and 4. Right row: quantum-mechanical density probabilities. (yellow: highest probability, black: lowest probability).
orbits within the mixed phase space [7] has been provided in open quantum dots (electron billiards). In this paper, we address the problem of the connection between the classical and quantum description of transport in an array of seven dots to an experimentally observed quantity, namely the magneto-resistance. In the experiments [8], transport occurs in a twodimensional electron gas (2DEG) situated some 60 nm below the surface of an AlGaAs/GaAs heterostructure. The electron concentration is ns ¼ 2.38 1011 cm2, which corresponds to a Fermi energy of 8.5 meV (vF ¼ 2.1 105 m/s). The mobility is m ¼ 1.24 106 cm2/V s at 10 mK, corresponding to an elastic mean free path of lel9.7 mm, which is much larger than the dot size (lithographic size of each dot: 0.4 0.7 mm), indicating that ballistic transport should be dominant within the dot array. In Fig. 1a, we reproduce the key point of the magneto-resistance, which is a large peak occurring at BE70.2 T, with subsidiary peaks, e.g. at BE70.5 T. 2. Calculations In the simulations of the seven-dot array, parabolic confining potentials are used to reproduce the individual dot profiles. The array of billiards extending in the xdirection is defined by dot and constriction regions. The constriction induces the chaotic behaviour within the array which is modified by the magnetic field. The parabolic potential has been adjusted to match what one expects from fully self-consistent calculations [9]. In the classical calculation, the ballistic motion of the electrons is solved analytically in the magnetic field within each region (dot or
constriction) [10]. In the presence of the magnetic field, the transport of the classical particles is governed by two hybrid frequencies o+ and o. We relate these frequencies to the frequencies o1 and o2 constructing the torus within each billiard [7]. To investigate states in the phase space of the dot array, Poincare´ sections (PS) are computed at the centre line of the dots (y ¼ 0). Trajectories are calculated on the Fermi surface by choosing a starting angle and starting position situated within the constriction or the dot. In the quantum mechanical calculations of the same system, the transport is computed by a recursive scattering matrix formulation with the conductance found from the Landauer equation [11]. For comparison, it is useful to introduce a quantum-phase space portrait corresponding to the PS. This is done by introducing the Husimi distributions (HP) projected in the vx–x plane [12]. As with the Poincare´ plot, we plot along the y ¼ 0 axis of the dots and the quantum mechanical phase space probability is obtained by transforming the wave function into a coherent state. Within the PS–HP plot, the PS is then projected on the HP to allow direct comparison between the classical and quantum mechanical phase space [7]. 3. Results and discussion The peaks at BE70.2 and 70.5 T in Fig. 1(a) correspond to trajectories with two and four bounces at the ‘‘wall’’ of each dot. These back-scattered trajectories can be expressed as even integer ratios of o+/o, as shown by the arrows in Fig. 1(a). The minimum at BE0.4 T corresponds to a transmitted trajectory which can be expressed by an odd integer ratio of o+/o ¼ 3. Examples
ARTICLE IN PRESS R. Brunner et al. / Physica E 40 (2008) 1315–1318
Vy (m/s)
(1)
1317
(2)
0.0
0.4 0.8 X (μ m)
(1)
0.0
1.2
)
/s
Vx
(m
(2)
Vx/VF
1.0 0.5 0.0 −0.5 −1.0
0.4
0.8
1.2
1.6 X (μm)
2.0
2.4
2.8
Fig. 2. Phase space analysis for o+/o ¼ 1.6269 (BE0.16 T). (a) Three-dimensional PS shows the mixed phase space on a spheroid in the vy–vx–x space for the first three dots of the seven-billiard array. The phase space is characterised by large chaotic regions and two types of KAM-islands symbolised by (1) and (2). (b) PS–HP plot: projection of the upper hemispheroid (vy40) in vx–x where vx is normalised to vF ¼ 2.1 105 m/s; the PS is presented by the white points; colour plot represents the quantum-mechanical phase space probability from low (dark) to high (bright).
of the trajectories at these integer values, and the corresponding quantum probabilities are shown in Fig. 1(b). At both maxima and minima, where o+/o takes integer values, the system is defined as non-KAM-like [7]. In the flank of the peak, we obtain KAM-like systems, as shown in Fig. 2, for o+/o ¼ 1.6269 (BE0.16 T). In order to illustrate the carrier dynamics in greater detail, we compute the PS which shows the mixed phase space on a spheroid in the vx–vy–x space (Fig. 2(a)). To define a trajectory as either chaotic or regular, the long-time behaviour of the system has to be studied. Therefore, the dot is closed after the trajectory has started, allowing the classical particle to bounce around in the dot for a long time (ms or several orders larger than the characteristic frequencies o+ and o) [7]. The classical phase space of the plots in Fig. 2(a) and (b) shows large chaotic regions and two different types of periodically arranged KAM-islands, which are centred at vx ¼ 0 and marked as (1) and (2). The two islands arise due to the applied magnetic field. The centre orbits in the KAM islands correspond to either the o+ (1) or the o (2) solution. In Fig. 2(b) the PS–HP plot in the vx–x plane is shown. Highest phase space probability is found in the periodically arranged type-(2) KAM islands. Since the KAM islands are classically inaccessible in phase space, the transport through the array in the KAM system is mainly attributed to phase space tunnelling by connecting the regular orbits of the KAM-island (pointer states) with the sea of chaos. In the open system, the chaotic-like regions
correspond to the current-currying states, states which are strongly connected to the environment. 4. Conclusion In conclusion, for arrays of electron billiards, we have found excellent agreement between classical and quantum mechanical calculations of corresponding quantities like electron trajectories and density probabilities. The calculated results are related to structure in the experimental low-field magneto-resistance and special values of the ratio of the hybrid frequencies characterising the confined electron system in a magnetic field. We have, furthermore, shown that chaotic dynamics plays an important role in the transport characteristics through the electron billiards. Generally, electron billiards have proven to be very suitable for the investigation of the crossover between the classical and the quantum mechanical world. Acknowledgement This work was partly supported by FWF, Austria, project no. 15513. References [1] L.E. Reichl, The Transition to Chaos, Springer, New York, 2004. [2] A.D. Stone, Physics Today, August 2005.
ARTICLE IN PRESS 1318
R. Brunner et al. / Physica E 40 (2008) 1315–1318
[3] T.M. Fromhold, et al., Nature 428 (2004) 726. [4] M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, New York, 1990. [5] W.H. Zurek, Rev. Mod. Phys. 75 (2003) 715. [6] D.K. Ferry, et al., Phys. Rev. Lett. 93 (2004) 026803. [7] R. Brunner, et al., Phys. Rev. Lett. 98 (2007) 204101. [8] R. Brunner, et al., Physica E 21 (2004) 491; M. Elhassan, et al., Phys. Rev. B 70 (2004) 205341.
[9] R. Akis, et al., Electron Transport in Quantum Dots, Kluwer, Boston, 2003 (Chapter 6). [10] R. Brunner, et al., J. Comput. Electron. 6 (2007) 93. [11] R. Landauer, IBM J. Res. Dev. 1 (1957) 223. [12] B. Crespi, et al., Phys. Rev. E 47 (1993) 986; K. Husimi, Proc. Phys. Math. Soc. Jpn. 22 (1940) 246.
Physica E 40 (2008) 1315–1318 www.elsevier.com/locate/physe
Classical and quantum dynamics in an array of electron billiards Roland Brunnera,, Ronald Meiselsa, Friedemar Kuchara, Richard Akisb, David K. Ferryb, Jonathan P. Birdc a Institute of Physics, University of Leoben, A-8700 Leoben, Austria Department of Electrical Engineering, Arizona State University, Tempe, AZ, USA c Department of Electrical Engineering, University at Buffalo, Buffalo, NY, USA
b
Available online 22 September 2007
Abstract We investigate the classical and quantum dynamics in an array of electron billiards within a magnetic field. The resulting modification of the phase space due to the magnetic field can be directly related to the experimentally observed magneto-resistance. We argue that the behaviour of chaos plays an important role in the transport of such ballistic devices. r 2007 Elsevier B.V. All rights reserved. PACS: 05.45.Mt; 73.63.Kv; 75.47.Jn Keywords: Quantum chaos; Magneto-transport; Quantum billiards
1. Introduction The principle of chaotic transport in ballistic systems is of great relevance to the development of nano-electronic devices in the next decade, where miniaturisation of device size will cause the electronic motion to change from diffusive to ballistic. Therefore, a complete knowledge of the chaotic dynamics has to be developed in order to understand how these devices will work. In classical mechanics, one can distinguish between integrable and non-integrable systems [1]. In an integrable conservative system with 2 degrees of freedom, every trajectory lies on a torus constructed by two angular variables which can be associated with two frequencies o1 and o2. Depending on the ratio of the two characteristic frequencies, one either obtains periodic orbits for rational ratios or quasi-periodic orbits for irrational ratios [1]. Nonintegrable systems, however, may be divided into two classes. One class contains the completely chaotic system. In this case, every neighbourhood is traversed in an infinite number of momentum directions [2]. The second class, Corresponding author. Tel.: +43 3842 402 4610; fax: +43 3842 402 4602. E-mail address: [email protected] (R. Brunner).
1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.08.118
most ubiquitous in nature, is that whose phase space is mixed. The Kolmogorov–Arnold–Moser (KAM) theory [1] states that some of the tori, presenting the regular motion survive and others are destroyed if the system is subjected to nonlinear perturbation. The surviving tori are referred as KAM-tori [1]. The KAM theorem is applicable for small and smooth perturbations and for sufficiently irrational ratios (quasi-periodic motion) of the characteristic frequencies o1 and o2. Violating these conditions leads to a so-called non-KAM system. In driven systems, this nonKAM like behaviour has been investigated experimentally under resonant driving conditions [3]. A specific field dealing with the crossover between classical and quantum mechanics is quantum chaos [4], in our case for an ‘‘open’’ system. The understanding of an open system is critical, where a quantum object is affected by its connection to a classical ‘‘measuring’’ environment. One approach that has been proposed to describe the behaviour of such systems is the decoherence theory [5]. According to this, a subset of states of the originally isolated system becomes decoherent through their coupling with the environment, while a specific set of ‘‘pointer states’’ remains robust and eventually correlates with classical regular orbits [5]. Evidence for the pointers states [6] and their one-to-one connection to classical regular
ARTICLE IN PRESS R. Brunner et al. / Physica E 40 (2008) 1315–1318
1316
(1)
R (kΩ)
25
R (k Ω)
20 15 10
(2)
−0.50
−0.4
0.0
0.4
0.8
) V g (V
−0.45
5 0 −0.8
22 20 17 14 12 9 6 4 1
(3)
−0.40
B (T)
Fig. 1. (a) The measured magneto-resistance (white: high resistance, black: low resistance) for the seven-dot array for various gate voltages VG between 0.40 and 0.54 V, at a base temperature T ¼ 10 mK. The maxima and minima are expressed by the ratio of the hybrid frequencies o+/o. Data adapted from Ref. [8]. (b) Left row: classically calculated trajectories for certain initial conditions in the entrance constriction for o+/o ¼ 2, 3 and 4. Right row: quantum-mechanical density probabilities. (yellow: highest probability, black: lowest probability).
orbits within the mixed phase space [7] has been provided in open quantum dots (electron billiards). In this paper, we address the problem of the connection between the classical and quantum description of transport in an array of seven dots to an experimentally observed quantity, namely the magneto-resistance. In the experiments [8], transport occurs in a twodimensional electron gas (2DEG) situated some 60 nm below the surface of an AlGaAs/GaAs heterostructure. The electron concentration is ns ¼ 2.38 1011 cm2, which corresponds to a Fermi energy of 8.5 meV (vF ¼ 2.1 105 m/s). The mobility is m ¼ 1.24 106 cm2/V s at 10 mK, corresponding to an elastic mean free path of lel9.7 mm, which is much larger than the dot size (lithographic size of each dot: 0.4 0.7 mm), indicating that ballistic transport should be dominant within the dot array. In Fig. 1a, we reproduce the key point of the magneto-resistance, which is a large peak occurring at BE70.2 T, with subsidiary peaks, e.g. at BE70.5 T. 2. Calculations In the simulations of the seven-dot array, parabolic confining potentials are used to reproduce the individual dot profiles. The array of billiards extending in the xdirection is defined by dot and constriction regions. The constriction induces the chaotic behaviour within the array which is modified by the magnetic field. The parabolic potential has been adjusted to match what one expects from fully self-consistent calculations [9]. In the classical calculation, the ballistic motion of the electrons is solved analytically in the magnetic field within each region (dot or
constriction) [10]. In the presence of the magnetic field, the transport of the classical particles is governed by two hybrid frequencies o+ and o. We relate these frequencies to the frequencies o1 and o2 constructing the torus within each billiard [7]. To investigate states in the phase space of the dot array, Poincare´ sections (PS) are computed at the centre line of the dots (y ¼ 0). Trajectories are calculated on the Fermi surface by choosing a starting angle and starting position situated within the constriction or the dot. In the quantum mechanical calculations of the same system, the transport is computed by a recursive scattering matrix formulation with the conductance found from the Landauer equation [11]. For comparison, it is useful to introduce a quantum-phase space portrait corresponding to the PS. This is done by introducing the Husimi distributions (HP) projected in the vx–x plane [12]. As with the Poincare´ plot, we plot along the y ¼ 0 axis of the dots and the quantum mechanical phase space probability is obtained by transforming the wave function into a coherent state. Within the PS–HP plot, the PS is then projected on the HP to allow direct comparison between the classical and quantum mechanical phase space [7]. 3. Results and discussion The peaks at BE70.2 and 70.5 T in Fig. 1(a) correspond to trajectories with two and four bounces at the ‘‘wall’’ of each dot. These back-scattered trajectories can be expressed as even integer ratios of o+/o, as shown by the arrows in Fig. 1(a). The minimum at BE0.4 T corresponds to a transmitted trajectory which can be expressed by an odd integer ratio of o+/o ¼ 3. Examples
ARTICLE IN PRESS R. Brunner et al. / Physica E 40 (2008) 1315–1318
Vy (m/s)
(1)
1317
(2)
0.0
0.4 0.8 X (μ m)
(1)
0.0
1.2
)
/s
Vx
(m
(2)
Vx/VF
1.0 0.5 0.0 −0.5 −1.0
0.4
0.8
1.2
1.6 X (μm)
2.0
2.4
2.8
Fig. 2. Phase space analysis for o+/o ¼ 1.6269 (BE0.16 T). (a) Three-dimensional PS shows the mixed phase space on a spheroid in the vy–vx–x space for the first three dots of the seven-billiard array. The phase space is characterised by large chaotic regions and two types of KAM-islands symbolised by (1) and (2). (b) PS–HP plot: projection of the upper hemispheroid (vy40) in vx–x where vx is normalised to vF ¼ 2.1 105 m/s; the PS is presented by the white points; colour plot represents the quantum-mechanical phase space probability from low (dark) to high (bright).
of the trajectories at these integer values, and the corresponding quantum probabilities are shown in Fig. 1(b). At both maxima and minima, where o+/o takes integer values, the system is defined as non-KAM-like [7]. In the flank of the peak, we obtain KAM-like systems, as shown in Fig. 2, for o+/o ¼ 1.6269 (BE0.16 T). In order to illustrate the carrier dynamics in greater detail, we compute the PS which shows the mixed phase space on a spheroid in the vx–vy–x space (Fig. 2(a)). To define a trajectory as either chaotic or regular, the long-time behaviour of the system has to be studied. Therefore, the dot is closed after the trajectory has started, allowing the classical particle to bounce around in the dot for a long time (ms or several orders larger than the characteristic frequencies o+ and o) [7]. The classical phase space of the plots in Fig. 2(a) and (b) shows large chaotic regions and two different types of periodically arranged KAM-islands, which are centred at vx ¼ 0 and marked as (1) and (2). The two islands arise due to the applied magnetic field. The centre orbits in the KAM islands correspond to either the o+ (1) or the o (2) solution. In Fig. 2(b) the PS–HP plot in the vx–x plane is shown. Highest phase space probability is found in the periodically arranged type-(2) KAM islands. Since the KAM islands are classically inaccessible in phase space, the transport through the array in the KAM system is mainly attributed to phase space tunnelling by connecting the regular orbits of the KAM-island (pointer states) with the sea of chaos. In the open system, the chaotic-like regions
correspond to the current-currying states, states which are strongly connected to the environment. 4. Conclusion In conclusion, for arrays of electron billiards, we have found excellent agreement between classical and quantum mechanical calculations of corresponding quantities like electron trajectories and density probabilities. The calculated results are related to structure in the experimental low-field magneto-resistance and special values of the ratio of the hybrid frequencies characterising the confined electron system in a magnetic field. We have, furthermore, shown that chaotic dynamics plays an important role in the transport characteristics through the electron billiards. Generally, electron billiards have proven to be very suitable for the investigation of the crossover between the classical and the quantum mechanical world. Acknowledgement This work was partly supported by FWF, Austria, project no. 15513. References [1] L.E. Reichl, The Transition to Chaos, Springer, New York, 2004. [2] A.D. Stone, Physics Today, August 2005.
ARTICLE IN PRESS 1318
R. Brunner et al. / Physica E 40 (2008) 1315–1318
[3] T.M. Fromhold, et al., Nature 428 (2004) 726. [4] M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, New York, 1990. [5] W.H. Zurek, Rev. Mod. Phys. 75 (2003) 715. [6] D.K. Ferry, et al., Phys. Rev. Lett. 93 (2004) 026803. [7] R. Brunner, et al., Phys. Rev. Lett. 98 (2007) 204101. [8] R. Brunner, et al., Physica E 21 (2004) 491; M. Elhassan, et al., Phys. Rev. B 70 (2004) 205341.
[9] R. Akis, et al., Electron Transport in Quantum Dots, Kluwer, Boston, 2003 (Chapter 6). [10] R. Brunner, et al., J. Comput. Electron. 6 (2007) 93. [11] R. Landauer, IBM J. Res. Dev. 1 (1957) 223. [12] B. Crespi, et al., Phys. Rev. E 47 (1993) 986; K. Husimi, Proc. Phys. Math. Soc. Jpn. 22 (1940) 246.