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Quantum-chaos control by an applied electric field
Physics Letters A 316 (2003) 336–341 www.elsevier.com/locate/pla
Quantum-chaos control by an applied electric field Ryuichi Ugajin Fusion Domain Laboratory, Sony Corporation, 5-21-15 Higashikojiya, Ota-ku, Tokyo 144-0033, Japan Received 21 May 2003; received in revised form 25 June 2003; accepted 29 June 2003 Communicated by R. Wu
Abstract We investigate the spectral statistics of a quantum particle in coupled double layers, one of which has a high degree of disorder and the other of which is relatively clean. When the transfer between the layers is small, our electronic system shows a transition from quantum chaotic to a regular system driven by the applied electric field threading through the double layers. This effect is governed by an imbalance in the degree of disorder between the GaAs/AlGaAs-based quantum wells, coupled by tunneling transfer. 2003 Elsevier B.V. All rights reserved. PACS: 73.22.-f; 05.45.Mt; 71.30.+h; 71.23.-k
The transformation of a metallic-electron phase to an insulating-electron phase by an external field can be applied to the switching of the electric current. The velocity-modulation transistor [1,2] uses the transformation of an Anderson insulator, in which electrons are localized by random-impurity scattering [3,4], to a normal metal in a heterostructure of compound semiconductors. We have another example, the transformation from a Mott insulator, in which electrons are localized by electron–electron interaction in a lattice [5,6], to a metal of correlated electrons, in an array of quantum dots with asymmetry perpendicular to the array [7,8]. An analysis of the electric-field-induced Mott transition has been carried out using a Hubbard system of double layers, one of which can exhibit a Mott-insulating state if isolated and the other of which can exhibit a metallic state [9,10].
E-mail address: [email protected] (R. Ugajin). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.06.001
Let us consider a particular area of a quasi-twodimensional system in which electrons are influenced by the presence of impurities, as in a disordered quantum dot [11–15]. Let us assume that the diameter of this area is within the coherence length of electrons in the system. While a normal metal of extended states in which wavy electrons are influenced by weak disorder shows quantum chaos, the Anderson insulator of localized states is a typical example of a regular system [16–18]. It may be possible to realize a coupled system of an Anderson insulator and a normal metal using nanostructured materials [19,20]. We consider here a quantum particle in double layers, one of which would be an Anderson insulator with strong disorder and the other of which would be a normal metal with weak disorder if there is no tunneling transfer between the double layers. The tunneling transfer determines to what extent the quantum system is influenced by an applied electric field threading through the double layers. This quantum system is
R. Ugajin / Physics Letters A 316 (2003) 336–341
Fig. 1. Energy-band diagram of GaAs/AlGaAs double quantum wells coupled through a tunneling barrier. Applied electric field E causes a difference in potential φ = Ed between the wells with the effective distance between the wells denoted by d.
usually realized by GaAs/AlGaAs-based double quantum wells, as shown in Fig. 1. Let us introduce coupled double layers with sides L, and whose sites are denoted by rp = (xp , yp ).
(1)
Operator cˆp† creates a quantum particle at lattice site rp of the first layer and operator dˆp† creates a quantum particle at lattice site rp of the second layer. Tightbinding Hamiltonian Hˆ is written as cˆp† cˆq − t2 cˆp† dˆp Hˆ = −t1 dˆp† dˆq − t3 p,q
+
p
vp cˆp† cˆp
p,q
+
p
wp dˆp† dˆp
p
φ † + cˆ cˆp − dˆp† dˆp + h.c., 2 p p
(2)
where p, q denotes the nearest-neighbor sites. Fig. 2 shows a schematic diagram of our double-layered lattice. The first layer has random potential vp : |vp | < Vl /2 and the second layer has random potential wp : |wp | < Vs /2. t1 is the transfer in the first layer and t2 is the transfer in the second layer, while t3 is the transfer between the layers. When t3 = 0, these two layers are decoupled. If Vl /t1 is sufficiently large, the first layer is an Anderson insulator, the spectral statistics of which are characterized by the Poisson distribution of a regular system. If Vs /t2 is sufficiently small, the second layer is a normal metal, the spectral statistics
337
Fig. 2. Schematic diagram of double-layered lattices. t1 and t2 are transfers for the first and second layers, respectively. t3 is the transfer between the two. The pth site of the first layer has potential vp + φ/2, while the qth site of the first layer has potential wq − φ/2, where the random potentials vp and wq are independent.
of which are characterized by quantum chaos. When t3 > 0 introduces quantum coherence between the double layers, the Anderson insulator in the first layer interacts with wavy states in the second layer when Vs /t2 is sufficiently small and Vl /t1 is sufficiently large. We have introduced potential difference φ, which is proportional to the strength of the electric field threading through the double layers. We report our investigation of how our electronic system behaves as a function of φ. When t3 is larger than the bandwidth of both layers, it may be useful to introduce a single-particle state for the pth site: |Φp (θ ) = [cˆp† cos θ + dˆp† sin θ ]|0. |Φp (π/4) is a bonding state and |Φp (3π/4) is an antibonding state of two sites sharing the same p. When the double layer is symmetric, single-particle states in the lower subband can be constructed using a superposition of |Φp (π/4), and single-particle states in the upper subband can be constructed using a superposition of |Φp (3π/4). It is clear that both subbands are themselves pure two-dimensional electronic systems. Recall that the quantum correction of conductivity in two dimensions is free from the degree of disorder in a weak localization regime [21]. Therefore, we cannot expect an effective modulation of electronic properties in these subbands even if the effective strength of the random potential in each subband is varied by an applied electric field. As t3 decreases, the bottom of the upper subband and the top of the lower subband become closer and they eventually merge, producing an energy in which the density of states is enhanced from that of a pure two-dimensional electronic system. When t3
338
R. Ugajin / Physics Letters A 316 (2003) 336–341
is sufficiently small, the electronic properties in the energy range of the coupled double layers are free from the restriction of pure two-dimensional behavior of Anderson localization, in which all single-particle states are localized on an infinite plane [22,23]. Let us take t1 = t2 = 1 throughout this Letter. The following investigation will show that a small value of t3 , e.g., 1/2, causes a considerable modulation of electronic properties when an electric field is applied. We take L = 80, so the total number of sites is 2L2 = 12, 800, where each layer has a periodic boundary. Hˆ is numerically diagonalized, yielding eigenvalues m (m = 1, 2, . . . , 12, 800). We used 2,000 levels from 1201 to 3200 in order to calculate the spectral statistics. Before we calculated the spectral statistics, the energy-level sequences were modified by “unfolding” them, so as to give the energy-level sequences a constant density in which the average of nearestneighbor spacing is one [24]. Because we analyze various levels depending on φ, there remain deviations in the average of nearest-neighbor spacing even after the unfolding procedure using the third-order function of energy. However, the deviation has only a small effect on the spectral statistics in our investigation. An item of interest is the nearest-neighbor spacing distribution P (s), which is defined as the probability P (s) ds that εm − εm−1 will be found between s and s + ds. Another item of interest is the ∆3 statistics of Dyson and Mehta [25], which indicate whether the off-diagonal elements of a Hamiltonian matrix, constructed using the eigenfunctions of regular systems, significantly influence the behavior of the system. A quantum system whose classical counterpart is regular, e.g., an Anderson insulator, has spectral statistics characterized by Poisson distribution. The nearest-neighbor spacing distribution PP (s) and the ∆3 -statistics of Poisson distribution are known to be PP (s) = e−s , (3) n ∆3 (n) = . (4) 15 On the other hand, a quantum system whose classical counterpart is chaos, e.g., a normal metal of a disordered medium, has the spectral statistics of a Gaussian orthogonal ensemble (GOE) if the system has timereversal symmetry and is not influenced by spin–orbit interaction. The nearest-neighbor spacing distribution PGOE (s) and the ∆3 -statistics of Poisson distribution
were derived using the random matrix theory as: πs −πs 2 /4 e , (5) 2 1 π2 5 − ∆3 (n) = 2 log(2πn) + γ − + O n−1 , π 8 4 (6)
PGOE (s) =
where γ is Euler’s constant. Before we investigate the applied-field dependence of the spectral statistics in coupled layers of an Anderson insulator and a normal metal, we examine the spectral statistics of a system in which the two layers are equivalent, i.e., Vs = Vl = V and φ = 0, where we took t3 = 1/2. Note that vp and wp are independent. A transition from localized states to extended states is expected when V changes. The nearest-neighbor spacing distribution P (s) is shown in Fig. 3 and the ∆3 -statistics of Dyson and Mehta are shown in Fig. 4. When V is 18, P (s) is similar to Poisson distribution, i.e., PP (s), where the energylevel sequence is purely random. On the other hand, when V is 3, P (s) is similar to that predicted by GOE, i.e., PGOE (s), where the energy-level sequence is affected by strong repulsion between energy levels. A localized-extended transition in the area of our lattice takes place between v = 3 and v = 18. Let us turn to the ∆3 -statistics of Dyson and Mehta, as shown in Fig. 4. When V = 3, ∆3 (n) is close to GOE. On the other hand, ∆3 (n) when V = 18 closely resembles Poisson distribution. The point of transition of the spectral statistics to Poisson distribution indicates the onset of quantum localization in the double layers. Let us introduce Berry–Robnik distribution to characterize the nearest-neighbor spacing distribution of the crossover region between regular system and quantum chaos: √ π ρs ¯ P2 (s, ρ) = ρ 2 e−ρs erf 2 π ρ¯ 3 s −ρs−π ρ¯ 2 s 2 /4 + 2ρ ρ¯ + (7) e , 2 ∞ 2 where ρ¯ = 1 − ρ and erf(x) = √2π x dτ e−τ is the error function [26]. When ρ = 1, P2 (s, 1) is identical to Poisson distribution, i.e., PP (s). On the other hand, when ρ = 0, P2 (s, 0) is identical to that of GOE, i.e., PGOE (s). The Berry–Robnik parameter ρ measures the difference between the two extreme distributions
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Fig. 3. The nearest-neighbor spacing distribution P (s) in the symmetric double layers of Vs = Vl = V . V is taken to be 3, 6, 9, 12, 15, and 18, while φ = 0, t1 = t2 = 1, and t3 = 1/2. GOE and Poisson distribution are shown by broken lines.
Fig. 4. ∆3 -statistics of Dyson and Mehta in the symmetric double layers of Vs = Vl = V . V is taken to be 3, 6, 9, 12, 15, and 18, while φ = 0, t1 = t2 = 1, and t3 = 1/2. GOE and Poisson distribution are shown by broken lines.
and may be interpreted as the relative phase-space volume of the regular regions [27]. Therefore, the Berry– Robnik parameter ρ can be thought of as proportional to the phase-space volume of states whose quantum counterpart is characterized by localized states. Our
Fig. 5. The Berry–Robnik parameter characterizing the nearest-neighbor spacing distribution of the symmetric double layers as a function of the degree of randomness V = Vs = Vl , where φ = 0, t1 = t2 = 1, and t3 = 1/2.
numerical results of the nearest-neighbor spacing distribution are shown as a histogram in which the values of P (s) are given for each 1/6 of s. When we denote the values of the nearest-neighbor spacing distribution ρ in order to minimize the by P (ex) (sn ), we determine (ex) (s )]2 . deviation Γ (ρ) = 24 j j =1 [P2 (sj , ρ) − P In Fig. 5, we show the Berry–Robnik parameter in the equivalent double layers as a function of V where
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R. Ugajin / Physics Letters A 316 (2003) 336–341
Fig. 6. The nearest-neighbor spacing distribution P (s) when Vs = 2 and Vl = 20. φ is taken to be −4, −2.4, −0.8, 0.8, 2.4, and 4, while t1 = t2 = 1 and t3 = 1/2. GOE and Poisson distribution are shown by broken lines.
the model parameters are the same as in Figs. 3 and 4. When V = 2, the electronic system is characterized by quantum chaos, resulting in ρ = 0. As V increases, the Berry–Robnik parameter increases. When V = 20, ρ approaches 1. The Berry–Robnik parameter rapidly increases from zero in the vicinity of V = 7, where a localized-extended transition of our finite lattice, which has an analog to the Anderson transition of an infinite lattice, occurs. Now we analyze the spectral statistics of asymmetric double layers when Vs = Vl as a function of φ. The nearest-neighbor spacing distribution P (s) is shown in Fig. 6 and the ∆3 -statistics are shown in Fig. 7, when Vs = 2, Vl = 20, and t3 = 1/2. When φ is −4, P (s) is similar to Poisson distribution. On the other hand, when φ is 4, P (s) is similar to GOE. A localizedextended transition, accompanied by a transition of the spectral statistics from those of a regular system to those of quantum chaos, takes place between φ = −4 and φ = 4, as is supported by the ∆3 -statistics shown in Fig. 7. In Fig. 8, we show the Berry–Robnik parameter in order to characterize the nearest-neighbor spacing distribution of our double layers as a function of φ when t3 is taken to be 1/2, 1, 2, 4, and 8. Note that Vs = 2 and Vl = 20 are taken again. Let us focus
Fig. 7. ∆3 -statistics of Dyson and Mehta when Vs = 2 and Vl = 20. φ is taken to be −4, −2.4, −0.8, 0.8, 2.4, and 4, while t1 = t2 = 1 and t3 = 1/2. GOE and Poisson distribution are shown by broken lines.
our attention on the Berry–Robnik parameter when t3 = 1/2. When φ = 4, the electronic system is characterized by quantum chaos, resulting in ρ = 0. As φ decreases, the Berry–Robnik parameter rapidly increases. When φ = −4, ρ approaches 1. We have
R. Ugajin / Physics Letters A 316 (2003) 336–341
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double quantum wells with an imbalance in the degree of disorder. This effect enables us to control the quantum diffusion of electrons in the double quantum wells, between which tunneling transfer is small.
References
Fig. 8. The Berry–Robnik parameter characterizing the nearest-neighbor spacing distribution of our double layers as a function of φ with various t3 when Vs = 2 and Vl = 20. We have taken Vs = 2, Vl = 20, and t1 = t2 = 1.
seen a localized-extended transition driven by varying φ, which is thought of as being proportional to the strength of the applied electric field. The dependence of ρ on φ is strong and the transition from quantum chaotic to a regular system is marked because t3 = 1/2 is sufficiently small. As the value of t3 increases, the dependence of ρ on φ becomes less obvious. For example, when t3 = 8, the dependence of ρ on φ is unclear except for the deviations caused by random potentials. As noted previously, as t3 increases, the energy band of our electronic system splits into two subbands, each of which can be thought of as a pure twodimensional electronic system. This split brings about a poor response of the electronic properties to an applied electric field. Our investigation suggests that an applied electric field can control the degree of quantum chaos in the double layers, which can be realized by coupled
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Quantum-chaos control by an applied electric field Ryuichi Ugajin Fusion Domain Laboratory, Sony Corporation, 5-21-15 Higashikojiya, Ota-ku, Tokyo 144-0033, Japan Received 21 May 2003; received in revised form 25 June 2003; accepted 29 June 2003 Communicated by R. Wu
Abstract We investigate the spectral statistics of a quantum particle in coupled double layers, one of which has a high degree of disorder and the other of which is relatively clean. When the transfer between the layers is small, our electronic system shows a transition from quantum chaotic to a regular system driven by the applied electric field threading through the double layers. This effect is governed by an imbalance in the degree of disorder between the GaAs/AlGaAs-based quantum wells, coupled by tunneling transfer. 2003 Elsevier B.V. All rights reserved. PACS: 73.22.-f; 05.45.Mt; 71.30.+h; 71.23.-k
The transformation of a metallic-electron phase to an insulating-electron phase by an external field can be applied to the switching of the electric current. The velocity-modulation transistor [1,2] uses the transformation of an Anderson insulator, in which electrons are localized by random-impurity scattering [3,4], to a normal metal in a heterostructure of compound semiconductors. We have another example, the transformation from a Mott insulator, in which electrons are localized by electron–electron interaction in a lattice [5,6], to a metal of correlated electrons, in an array of quantum dots with asymmetry perpendicular to the array [7,8]. An analysis of the electric-field-induced Mott transition has been carried out using a Hubbard system of double layers, one of which can exhibit a Mott-insulating state if isolated and the other of which can exhibit a metallic state [9,10].
E-mail address: [email protected] (R. Ugajin). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.06.001
Let us consider a particular area of a quasi-twodimensional system in which electrons are influenced by the presence of impurities, as in a disordered quantum dot [11–15]. Let us assume that the diameter of this area is within the coherence length of electrons in the system. While a normal metal of extended states in which wavy electrons are influenced by weak disorder shows quantum chaos, the Anderson insulator of localized states is a typical example of a regular system [16–18]. It may be possible to realize a coupled system of an Anderson insulator and a normal metal using nanostructured materials [19,20]. We consider here a quantum particle in double layers, one of which would be an Anderson insulator with strong disorder and the other of which would be a normal metal with weak disorder if there is no tunneling transfer between the double layers. The tunneling transfer determines to what extent the quantum system is influenced by an applied electric field threading through the double layers. This quantum system is
R. Ugajin / Physics Letters A 316 (2003) 336–341
Fig. 1. Energy-band diagram of GaAs/AlGaAs double quantum wells coupled through a tunneling barrier. Applied electric field E causes a difference in potential φ = Ed between the wells with the effective distance between the wells denoted by d.
usually realized by GaAs/AlGaAs-based double quantum wells, as shown in Fig. 1. Let us introduce coupled double layers with sides L, and whose sites are denoted by rp = (xp , yp ).
(1)
Operator cˆp† creates a quantum particle at lattice site rp of the first layer and operator dˆp† creates a quantum particle at lattice site rp of the second layer. Tightbinding Hamiltonian Hˆ is written as cˆp† cˆq − t2 cˆp† dˆp Hˆ = −t1 dˆp† dˆq − t3 p,q
+
p
vp cˆp† cˆp
p,q
+
p
wp dˆp† dˆp
p
φ † + cˆ cˆp − dˆp† dˆp + h.c., 2 p p
(2)
where p, q denotes the nearest-neighbor sites. Fig. 2 shows a schematic diagram of our double-layered lattice. The first layer has random potential vp : |vp | < Vl /2 and the second layer has random potential wp : |wp | < Vs /2. t1 is the transfer in the first layer and t2 is the transfer in the second layer, while t3 is the transfer between the layers. When t3 = 0, these two layers are decoupled. If Vl /t1 is sufficiently large, the first layer is an Anderson insulator, the spectral statistics of which are characterized by the Poisson distribution of a regular system. If Vs /t2 is sufficiently small, the second layer is a normal metal, the spectral statistics
337
Fig. 2. Schematic diagram of double-layered lattices. t1 and t2 are transfers for the first and second layers, respectively. t3 is the transfer between the two. The pth site of the first layer has potential vp + φ/2, while the qth site of the first layer has potential wq − φ/2, where the random potentials vp and wq are independent.
of which are characterized by quantum chaos. When t3 > 0 introduces quantum coherence between the double layers, the Anderson insulator in the first layer interacts with wavy states in the second layer when Vs /t2 is sufficiently small and Vl /t1 is sufficiently large. We have introduced potential difference φ, which is proportional to the strength of the electric field threading through the double layers. We report our investigation of how our electronic system behaves as a function of φ. When t3 is larger than the bandwidth of both layers, it may be useful to introduce a single-particle state for the pth site: |Φp (θ ) = [cˆp† cos θ + dˆp† sin θ ]|0. |Φp (π/4) is a bonding state and |Φp (3π/4) is an antibonding state of two sites sharing the same p. When the double layer is symmetric, single-particle states in the lower subband can be constructed using a superposition of |Φp (π/4), and single-particle states in the upper subband can be constructed using a superposition of |Φp (3π/4). It is clear that both subbands are themselves pure two-dimensional electronic systems. Recall that the quantum correction of conductivity in two dimensions is free from the degree of disorder in a weak localization regime [21]. Therefore, we cannot expect an effective modulation of electronic properties in these subbands even if the effective strength of the random potential in each subband is varied by an applied electric field. As t3 decreases, the bottom of the upper subband and the top of the lower subband become closer and they eventually merge, producing an energy in which the density of states is enhanced from that of a pure two-dimensional electronic system. When t3
338
R. Ugajin / Physics Letters A 316 (2003) 336–341
is sufficiently small, the electronic properties in the energy range of the coupled double layers are free from the restriction of pure two-dimensional behavior of Anderson localization, in which all single-particle states are localized on an infinite plane [22,23]. Let us take t1 = t2 = 1 throughout this Letter. The following investigation will show that a small value of t3 , e.g., 1/2, causes a considerable modulation of electronic properties when an electric field is applied. We take L = 80, so the total number of sites is 2L2 = 12, 800, where each layer has a periodic boundary. Hˆ is numerically diagonalized, yielding eigenvalues m (m = 1, 2, . . . , 12, 800). We used 2,000 levels from 1201 to 3200 in order to calculate the spectral statistics. Before we calculated the spectral statistics, the energy-level sequences were modified by “unfolding” them, so as to give the energy-level sequences a constant density in which the average of nearestneighbor spacing is one [24]. Because we analyze various levels depending on φ, there remain deviations in the average of nearest-neighbor spacing even after the unfolding procedure using the third-order function of energy. However, the deviation has only a small effect on the spectral statistics in our investigation. An item of interest is the nearest-neighbor spacing distribution P (s), which is defined as the probability P (s) ds that εm − εm−1 will be found between s and s + ds. Another item of interest is the ∆3 statistics of Dyson and Mehta [25], which indicate whether the off-diagonal elements of a Hamiltonian matrix, constructed using the eigenfunctions of regular systems, significantly influence the behavior of the system. A quantum system whose classical counterpart is regular, e.g., an Anderson insulator, has spectral statistics characterized by Poisson distribution. The nearest-neighbor spacing distribution PP (s) and the ∆3 -statistics of Poisson distribution are known to be PP (s) = e−s , (3) n ∆3 (n) = . (4) 15 On the other hand, a quantum system whose classical counterpart is chaos, e.g., a normal metal of a disordered medium, has the spectral statistics of a Gaussian orthogonal ensemble (GOE) if the system has timereversal symmetry and is not influenced by spin–orbit interaction. The nearest-neighbor spacing distribution PGOE (s) and the ∆3 -statistics of Poisson distribution
were derived using the random matrix theory as: πs −πs 2 /4 e , (5) 2 1 π2 5 − ∆3 (n) = 2 log(2πn) + γ − + O n−1 , π 8 4 (6)
PGOE (s) =
where γ is Euler’s constant. Before we investigate the applied-field dependence of the spectral statistics in coupled layers of an Anderson insulator and a normal metal, we examine the spectral statistics of a system in which the two layers are equivalent, i.e., Vs = Vl = V and φ = 0, where we took t3 = 1/2. Note that vp and wp are independent. A transition from localized states to extended states is expected when V changes. The nearest-neighbor spacing distribution P (s) is shown in Fig. 3 and the ∆3 -statistics of Dyson and Mehta are shown in Fig. 4. When V is 18, P (s) is similar to Poisson distribution, i.e., PP (s), where the energylevel sequence is purely random. On the other hand, when V is 3, P (s) is similar to that predicted by GOE, i.e., PGOE (s), where the energy-level sequence is affected by strong repulsion between energy levels. A localized-extended transition in the area of our lattice takes place between v = 3 and v = 18. Let us turn to the ∆3 -statistics of Dyson and Mehta, as shown in Fig. 4. When V = 3, ∆3 (n) is close to GOE. On the other hand, ∆3 (n) when V = 18 closely resembles Poisson distribution. The point of transition of the spectral statistics to Poisson distribution indicates the onset of quantum localization in the double layers. Let us introduce Berry–Robnik distribution to characterize the nearest-neighbor spacing distribution of the crossover region between regular system and quantum chaos: √ π ρs ¯ P2 (s, ρ) = ρ 2 e−ρs erf 2 π ρ¯ 3 s −ρs−π ρ¯ 2 s 2 /4 + 2ρ ρ¯ + (7) e , 2 ∞ 2 where ρ¯ = 1 − ρ and erf(x) = √2π x dτ e−τ is the error function [26]. When ρ = 1, P2 (s, 1) is identical to Poisson distribution, i.e., PP (s). On the other hand, when ρ = 0, P2 (s, 0) is identical to that of GOE, i.e., PGOE (s). The Berry–Robnik parameter ρ measures the difference between the two extreme distributions
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Fig. 3. The nearest-neighbor spacing distribution P (s) in the symmetric double layers of Vs = Vl = V . V is taken to be 3, 6, 9, 12, 15, and 18, while φ = 0, t1 = t2 = 1, and t3 = 1/2. GOE and Poisson distribution are shown by broken lines.
Fig. 4. ∆3 -statistics of Dyson and Mehta in the symmetric double layers of Vs = Vl = V . V is taken to be 3, 6, 9, 12, 15, and 18, while φ = 0, t1 = t2 = 1, and t3 = 1/2. GOE and Poisson distribution are shown by broken lines.
and may be interpreted as the relative phase-space volume of the regular regions [27]. Therefore, the Berry– Robnik parameter ρ can be thought of as proportional to the phase-space volume of states whose quantum counterpart is characterized by localized states. Our
Fig. 5. The Berry–Robnik parameter characterizing the nearest-neighbor spacing distribution of the symmetric double layers as a function of the degree of randomness V = Vs = Vl , where φ = 0, t1 = t2 = 1, and t3 = 1/2.
numerical results of the nearest-neighbor spacing distribution are shown as a histogram in which the values of P (s) are given for each 1/6 of s. When we denote the values of the nearest-neighbor spacing distribution ρ in order to minimize the by P (ex) (sn ), we determine (ex) (s )]2 . deviation Γ (ρ) = 24 j j =1 [P2 (sj , ρ) − P In Fig. 5, we show the Berry–Robnik parameter in the equivalent double layers as a function of V where
340
R. Ugajin / Physics Letters A 316 (2003) 336–341
Fig. 6. The nearest-neighbor spacing distribution P (s) when Vs = 2 and Vl = 20. φ is taken to be −4, −2.4, −0.8, 0.8, 2.4, and 4, while t1 = t2 = 1 and t3 = 1/2. GOE and Poisson distribution are shown by broken lines.
the model parameters are the same as in Figs. 3 and 4. When V = 2, the electronic system is characterized by quantum chaos, resulting in ρ = 0. As V increases, the Berry–Robnik parameter increases. When V = 20, ρ approaches 1. The Berry–Robnik parameter rapidly increases from zero in the vicinity of V = 7, where a localized-extended transition of our finite lattice, which has an analog to the Anderson transition of an infinite lattice, occurs. Now we analyze the spectral statistics of asymmetric double layers when Vs = Vl as a function of φ. The nearest-neighbor spacing distribution P (s) is shown in Fig. 6 and the ∆3 -statistics are shown in Fig. 7, when Vs = 2, Vl = 20, and t3 = 1/2. When φ is −4, P (s) is similar to Poisson distribution. On the other hand, when φ is 4, P (s) is similar to GOE. A localizedextended transition, accompanied by a transition of the spectral statistics from those of a regular system to those of quantum chaos, takes place between φ = −4 and φ = 4, as is supported by the ∆3 -statistics shown in Fig. 7. In Fig. 8, we show the Berry–Robnik parameter in order to characterize the nearest-neighbor spacing distribution of our double layers as a function of φ when t3 is taken to be 1/2, 1, 2, 4, and 8. Note that Vs = 2 and Vl = 20 are taken again. Let us focus
Fig. 7. ∆3 -statistics of Dyson and Mehta when Vs = 2 and Vl = 20. φ is taken to be −4, −2.4, −0.8, 0.8, 2.4, and 4, while t1 = t2 = 1 and t3 = 1/2. GOE and Poisson distribution are shown by broken lines.
our attention on the Berry–Robnik parameter when t3 = 1/2. When φ = 4, the electronic system is characterized by quantum chaos, resulting in ρ = 0. As φ decreases, the Berry–Robnik parameter rapidly increases. When φ = −4, ρ approaches 1. We have
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double quantum wells with an imbalance in the degree of disorder. This effect enables us to control the quantum diffusion of electrons in the double quantum wells, between which tunneling transfer is small.
References
Fig. 8. The Berry–Robnik parameter characterizing the nearest-neighbor spacing distribution of our double layers as a function of φ with various t3 when Vs = 2 and Vl = 20. We have taken Vs = 2, Vl = 20, and t1 = t2 = 1.
seen a localized-extended transition driven by varying φ, which is thought of as being proportional to the strength of the applied electric field. The dependence of ρ on φ is strong and the transition from quantum chaotic to a regular system is marked because t3 = 1/2 is sufficiently small. As the value of t3 increases, the dependence of ρ on φ becomes less obvious. For example, when t3 = 8, the dependence of ρ on φ is unclear except for the deviations caused by random potentials. As noted previously, as t3 increases, the energy band of our electronic system splits into two subbands, each of which can be thought of as a pure twodimensional electronic system. This split brings about a poor response of the electronic properties to an applied electric field. Our investigation suggests that an applied electric field can control the degree of quantum chaos in the double layers, which can be realized by coupled
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