- Email: [email protected]
Probability distributions of transport observables in quantum dots: crossover between universal ensembles
Physica A 344 (2004) 677 – 684
www.elsevier.com/locate/physa
Probability distributions of transport observables in quantum dots: crossover between universal ensembles A.M.C. Souzaa;∗ , A.M.S. Macˆedob a Departamento
de Fisica, Universidade Federal de Sergipe, 49100-000 Sao Cristovao-SE, Brazil de Fisica, Universidade Federal de Pernambuco, 50670-901 Recife-PE, Brazil
b Departamento
Available online 4 July 2004
Abstract We study the e-ect on the probability distributions of the conductance and shot-noise power of breaking time-reversal symmetry in quantum dots with ideal contacts and small number of scattering channels. The distributions were found numerically by employing the Landauer–B4uttiker scattering approach and random-matrix theory and are presented as a function of the number of propagating channels and the symmetry breaking parameter. Universal broad distributions are found throughout the crossover region. c 2004 Elsevier B.V. All rights reserved. PACS: 73:23: − b; 73.21.La; 05.45.Mt Keywords: Quantum dot; Random-matrix theory
Ballistic quantum dots are microcavities in which the scattering comes only from specular collisions on the boundary. A quantum dot has sub-micron dimensions, low temperature (¡ 1 K) and a linear dimension that does not exceed the electron mean-free path [1,2]. Experiments have evidenced that the di-erence in the nature of the classical dynamics in chaotic or non-chaotic cavities produces a qualitative di-erence in the transport observables (see e.g. [3]). Many investigations have addressed this problem. The main methods used to study these nanodevices have been the semiclassical trace formula [4], the supersymmetry technique [5,6] and random-matrix theories (for a review, see e.g. [2]). ∗
Corresponding author. E-mail address: [email protected] (A.M.C. Souza).
c 2004 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.06.051
678
A.M.C. Souza, A.M.S. Macˆedo / Physica A 344 (2004) 677 – 684
Quantum Ductuations in quantum dots, imply the existence of probability distributions for the transport observables. These distributions can be classiEed according to the presence, or not, of certain intrinsic symmetries (e.g. time-reversion, spin rotation). The breaking of any of these symmetries leads to a crossover between universality classes. In particular, time-reversal symmetry can be broken by applying an external uniform magnetic Eeld. The associated crossover has attracted much interest of the recent literature due to its experimental relevance (see e.g. [7]). Analytical e-orts on this problem were mainly concentrated on the behaviour of the average and correlation functions of transport observables, such as the Landauer–B4uttiker conductance of the dot [4,8,9]. In this work, we study the e-ect on probability distributions of transport observables of breaking time-reversal symmetry in ballistic quantum dots. The distributions are found numerically by employing the Landauer–Buttiker scattering approach and random-matrix theory. The transport theory is brieDy explained in Section 2. In Section 3, the numerical results are presented. Concluding remarks are given in Section 4.
1. Transport theory A quantum dot consists of a conEned region with a chaotic classical dynamics, coupled to two electron reservoirs, represented by two semi-inEnite perfectly conducting leads, via two tunnel barriers. This system can be described by the Hamiltonian [2] ∗ H= |aEF a| + |H | + (|Wa a| + |aWa |) ; (1) a
a
where {|} is a set of states in the isolated cavity ( = 1; 2; : : : ; M , where M will be taken to inEnity), {|a} is a set of scattering states in the leads (a=1; 2; : : : ; N; =N1 +N2 , with Ni being the number of propagating modes in lead i) at the Fermi energy, EF . W is a nonrandom rectangular (M × N ) matrix describing the coupling between the M states inside the dot and the N modes in the leads, and H is a random matrix describing the states in the cavity. The scattering matrix of the system, associated with H is given by the Mahaux–Weidenm4uller formula S = 1 − 2iW † (EF − H + iWW † )−1 W : It is convenient to write S in the following block structure: r t S= t r
(2)
(3)
in which r and r are random reDection matrices (from left to left and from right to right) and t and t are random transmission matrices (from left to right and from right to left). The transport observables are also random matrices with the same universal † † symmetries. Current conservation implies that the matrices tt † , t t , 1−rr † and 1−r r
A.M.C. Souza, A.M.S. Macˆedo / Physica A 344 (2004) 677 – 684
679
have the same set of transmission eigenvalues Tn . As a consequence of the Landauer– B4uttiker scattering theory, the transmission eigenvalues can be used to evaluate transport observables. For instance, the conductance G = limV →0 (I=V ) (I is the current and V is the voltage di-erence between the reservoirs) is written as [2] 2e2 Tn : (4) G= h n It is convenient to deEne the dimensionless conductance g written as the sum n Tn . The discreteness of the electron charge causes time-dependent Ductuation of the current. The zero-frequency limit of the power spectrum of these Ductuations is known as shot-noise power, and is deEned as [2] 4e3 V 4e3 V p: (5) Tn (1 − Tn ) = P= h h n The randomness of H implies the existence of probability distributions for G and P (or g and p). The statistical distributions of physical quantities are found to be consistent with the fundamental symmetries of the Hamiltonian H and comply with the principle of maximum-information entropy. These distributions can be classiEed in accordance with the existence, or not, of certain symmetries (time reversal, spin rotation). Using Cartan’s classiEcation of symmetric spaces, one can infer the existence of ten possible random-matrix theories divided up into three subclasses, viz. three standard Wigner–Dyson classes, three chiral classes and four Bogoliubov–de Gennes classes [11]. This classiEcation, in turn, implies that the properties of many di-erent systems are describable by simple universal laws. The subclasses in each class depend upon the presence or absence of basic symmetries, such as time reversal and spin rotation. The standard classes are appropriated to the study of an electron moving inside a ballistic chaotic cavity, the situation of this work. The breaking of the symmetries may lead to a gradual transition between the universality classes. In particular, we will study the e-ect on the probability distributions of the conductance and shot-noise power of breaking time-reversal symmetry. 2. Numerical results We consider a random Hamiltonian of the form H = H (0) + iH (1) , where the probability distribution of H is M 2 : (6) d(H ) = CM exp − 2 Tr H 4 This choice is known as the Gaussian ensemble [2]. The class with time-reversal symmetry is invariant under orthogonal transformations, it is called Gaussian orthogonal ensemble (GOE). The class without time-reversal symmetry, e.g. system with an external magnetic Eeld, is invariant under unitary transformations and it is called Gaussian unitary ensemble (GUE). The crossover between GOE and GUE is easily obtained by
680
A.M.C. Souza, A.M.S. Macˆedo / Physica A 344 (2004) 677 – 684 4 α 1.00 0.32 0.10 0.032 0.00
d(g)
3
2
1
0 0.0
0.2
0.4
0.6
0.8
1.0
g Fig. 1. Distribution of the conductance probability through a quantum dot with two single-mode contacts (N1 = N2 = 1) for several values of symmetry-breaking parameter .
adjusting the variances following Ref. [10] (Hii(0) )2 = 2 (1 + e−2 )=M ; (Hij(0) )2 = 2 (1 + e−2 )=(2M );
i = j ;
(Hij(1) )2 = 2 (1 − e−2 )=(2M );
i = j ;
(7)
where the parameter produces a gradual transition from one class to another ( = 0 is GOE and → ∞ is GUE). We write H = H ( ) and d = d(H; ) such that (Hij )2 i=j = 2 =M and the average level density is "(E) = (M= ) 1 − (E=2 )2 if M 1, for all . The probability distributions of the conductance and shot-noise power are found numerically by employing Eqs. (2), (4), (5) and (7). They are presented as a function of the number of propagating channels (N1 and N2 ) andthe symmetry breaking parameter . For convenience we deEne the parameter = tanh( ) such that the e-ective parameter controlling the crossover remains in the interval 0 6 6 1. The probability distributions of the dimensionless conductance d(g) are illustrated in Fig. 1 for N1 = N2 = 1 and several values of . The numerical procedure is straightforward but quite demanding, since it requires inversion of large M × M matrices for each realization. We remark that in order to reach the asymptotic behaviour described
A.M.C. Souza, A.M.S. Macˆedo / Physica A 344 (2004) 677 – 684
681
15 α 1.00 0.32 0.10 0.032 0.00
12
d(p)
9
6
3
0 0.00
0.05
0.10
0.15
0.20
0.25
p Fig. 2. Distribution of the shot-noise power probability through a quantum dot with two single-mode contacts (N1 = N2 = 1) for several values of symmetry-breaking parameter .
in Eq. (1), one needs a large number, M 1. Note that for the GOE and GUE limits, the numerical results agree to the analytical expressions of the known cases of the literature [12]. In Fig. 2, we show the probability distributions d(p) of the dimensionless shot-noise power for N1 = N2 = 1 and several values of . The case N1 = N2 = 2 and N1 = N2 = 4 are shown in Figs. 3 and 4, respectively. Observe that when Ni increases, the distributions tends towards Gaussian distributions. The extreme example of deviations from a Gaussian is N1 = N2 = 1 as shown in Fig. 1. In this case, we observed that a power-law Et, d(g) = a(c + g)b , where a, b and c are free parameters, agrees well with our numerical data. Finally, using the curve of the average conductance as a function of the symmetry breaking parameter, we can estimate the point at which the transition is e-ectively complete. From the point of inDection of this curve we obtain c ≈ 1=M , in agreement with perturbation theory estimates [10]. 3. Conclusions In summary, we presented a thorough numerical analysis of the behaviour of the probability distributions of the conductance and shot-noise power upon breaking timereversal symmetry in ballistic quantum dots with ideal contacts. The distributions reproduce the analytical expressions available in the literature in some limiting cases.
682
A.M.C. Souza, A.M.S. Macˆedo / Physica A 344 (2004) 677 – 684
2.0
d (g)
1.5
1.0
0.5
0.0 0.0
0.5
1.0 g
(a)
1.5
2.0
7 6
d (p)
5 4 3 2 1 0
(b)
0.0
0.1
0.2
p
0.3
0.4
0.5
Fig. 3. Distribution of the conductance (a) and shot-noise power (b) probability through a quantum dot with two double-mode contacts (N1 = N2 = 2) for several values of associated with the examples presented in Fig. 1.
The most notable generic feature of our data is the presence of broad, albeit universal, distributions throughout the whole crossover region for systems with a small number of scattering channels. A detailed analytical description of this property is highly desirable, in view of potential applications of these devices in quantum information/communication theory. We pose this as a challenge to current methods of quantum transport theories.
A.M.C. Souza, A.M.S. Macˆedo / Physica A 344 (2004) 677 – 684
683
0.07 0.06
d (g)
0.05 0.04 0.03 0.02 0.01 0.00 0.0
(a)
0.5
1.0
1.5
2.0 g
2.5
3.0
3.5
4.0
0.25
0.20
d (p)
0.15
0.10
0.05
0.00 0.0
0.2
0.4
(b)
0.6
0.8
1.0
p
Fig. 4. Distribution of the conductance (a) and shot-noise power (b) probabilities through a quantum dot with two mode contacts where N1 = N2 = 4 for several values of associated with the examples presented in Fig. 1.
Acknowledgements This work was partially supported by CNPq, FAP-SE and FINEP (Brazilian Agencies). References [1] M. Switkes, et al., Appl. Phys. Lett. 72 (1998) 471. [2] C.W.J. Beenakker, Rev. Mod. Phys. 69 (1997) 731.
684 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
A.M.C. Souza, A.M.S. Macˆedo / Physica A 344 (2004) 677 – 684 A.M. Chang, et al., Phys. Rev. Lett. 76 (1996) 1695. H.U. Baranger, R.A. Jalabert, A.D. Stone, Phys. Rev. Lett. 70 (1993) 3876. K.B. Efetov, Supersymmetry in Disorder and Chaos, Cambridge University Press, Cambridge, 1997. J.E.F. Araujo, A.M.S. Macˆedo, Phys. Rev. B 58 (1998) R13379. C.M. Marcus, et al., Phys. Rev. Lett. 69 (1992) 506. Z. Pluhar, H.A. Weidenm4uller, J.A. Zuk, C.H. Lewenkopf, Phys. Rev. Lett. 73 (1994) 2115. K.B. Efetov, Phys. Rev. Lett. 74 (1995) 2299. G. Lenz, F. Haake, Phys. Rev. Lett. 65 (1990) 2325. M. Zirnbauer, J. Math. Phys. 37 (1996) 4986. P.A. Mello, H.U. Baranger, Waves Random Media 9 (1999) 105.
www.elsevier.com/locate/physa
Probability distributions of transport observables in quantum dots: crossover between universal ensembles A.M.C. Souzaa;∗ , A.M.S. Macˆedob a Departamento
de Fisica, Universidade Federal de Sergipe, 49100-000 Sao Cristovao-SE, Brazil de Fisica, Universidade Federal de Pernambuco, 50670-901 Recife-PE, Brazil
b Departamento
Available online 4 July 2004
Abstract We study the e-ect on the probability distributions of the conductance and shot-noise power of breaking time-reversal symmetry in quantum dots with ideal contacts and small number of scattering channels. The distributions were found numerically by employing the Landauer–B4uttiker scattering approach and random-matrix theory and are presented as a function of the number of propagating channels and the symmetry breaking parameter. Universal broad distributions are found throughout the crossover region. c 2004 Elsevier B.V. All rights reserved. PACS: 73:23: − b; 73.21.La; 05.45.Mt Keywords: Quantum dot; Random-matrix theory
Ballistic quantum dots are microcavities in which the scattering comes only from specular collisions on the boundary. A quantum dot has sub-micron dimensions, low temperature (¡ 1 K) and a linear dimension that does not exceed the electron mean-free path [1,2]. Experiments have evidenced that the di-erence in the nature of the classical dynamics in chaotic or non-chaotic cavities produces a qualitative di-erence in the transport observables (see e.g. [3]). Many investigations have addressed this problem. The main methods used to study these nanodevices have been the semiclassical trace formula [4], the supersymmetry technique [5,6] and random-matrix theories (for a review, see e.g. [2]). ∗
Corresponding author. E-mail address: [email protected] (A.M.C. Souza).
c 2004 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.06.051
678
A.M.C. Souza, A.M.S. Macˆedo / Physica A 344 (2004) 677 – 684
Quantum Ductuations in quantum dots, imply the existence of probability distributions for the transport observables. These distributions can be classiEed according to the presence, or not, of certain intrinsic symmetries (e.g. time-reversion, spin rotation). The breaking of any of these symmetries leads to a crossover between universality classes. In particular, time-reversal symmetry can be broken by applying an external uniform magnetic Eeld. The associated crossover has attracted much interest of the recent literature due to its experimental relevance (see e.g. [7]). Analytical e-orts on this problem were mainly concentrated on the behaviour of the average and correlation functions of transport observables, such as the Landauer–B4uttiker conductance of the dot [4,8,9]. In this work, we study the e-ect on probability distributions of transport observables of breaking time-reversal symmetry in ballistic quantum dots. The distributions are found numerically by employing the Landauer–Buttiker scattering approach and random-matrix theory. The transport theory is brieDy explained in Section 2. In Section 3, the numerical results are presented. Concluding remarks are given in Section 4.
1. Transport theory A quantum dot consists of a conEned region with a chaotic classical dynamics, coupled to two electron reservoirs, represented by two semi-inEnite perfectly conducting leads, via two tunnel barriers. This system can be described by the Hamiltonian [2] ∗ H= |aEF a| + |H | + (|Wa a| + |aWa |) ; (1) a
a
where {|} is a set of states in the isolated cavity ( = 1; 2; : : : ; M , where M will be taken to inEnity), {|a} is a set of scattering states in the leads (a=1; 2; : : : ; N; =N1 +N2 , with Ni being the number of propagating modes in lead i) at the Fermi energy, EF . W is a nonrandom rectangular (M × N ) matrix describing the coupling between the M states inside the dot and the N modes in the leads, and H is a random matrix describing the states in the cavity. The scattering matrix of the system, associated with H is given by the Mahaux–Weidenm4uller formula S = 1 − 2iW † (EF − H + iWW † )−1 W : It is convenient to write S in the following block structure: r t S= t r
(2)
(3)
in which r and r are random reDection matrices (from left to left and from right to right) and t and t are random transmission matrices (from left to right and from right to left). The transport observables are also random matrices with the same universal † † symmetries. Current conservation implies that the matrices tt † , t t , 1−rr † and 1−r r
A.M.C. Souza, A.M.S. Macˆedo / Physica A 344 (2004) 677 – 684
679
have the same set of transmission eigenvalues Tn . As a consequence of the Landauer– B4uttiker scattering theory, the transmission eigenvalues can be used to evaluate transport observables. For instance, the conductance G = limV →0 (I=V ) (I is the current and V is the voltage di-erence between the reservoirs) is written as [2] 2e2 Tn : (4) G= h n It is convenient to deEne the dimensionless conductance g written as the sum n Tn . The discreteness of the electron charge causes time-dependent Ductuation of the current. The zero-frequency limit of the power spectrum of these Ductuations is known as shot-noise power, and is deEned as [2] 4e3 V 4e3 V p: (5) Tn (1 − Tn ) = P= h h n The randomness of H implies the existence of probability distributions for G and P (or g and p). The statistical distributions of physical quantities are found to be consistent with the fundamental symmetries of the Hamiltonian H and comply with the principle of maximum-information entropy. These distributions can be classiEed in accordance with the existence, or not, of certain symmetries (time reversal, spin rotation). Using Cartan’s classiEcation of symmetric spaces, one can infer the existence of ten possible random-matrix theories divided up into three subclasses, viz. three standard Wigner–Dyson classes, three chiral classes and four Bogoliubov–de Gennes classes [11]. This classiEcation, in turn, implies that the properties of many di-erent systems are describable by simple universal laws. The subclasses in each class depend upon the presence or absence of basic symmetries, such as time reversal and spin rotation. The standard classes are appropriated to the study of an electron moving inside a ballistic chaotic cavity, the situation of this work. The breaking of the symmetries may lead to a gradual transition between the universality classes. In particular, we will study the e-ect on the probability distributions of the conductance and shot-noise power of breaking time-reversal symmetry. 2. Numerical results We consider a random Hamiltonian of the form H = H (0) + iH (1) , where the probability distribution of H is M 2 : (6) d(H ) = CM exp − 2 Tr H 4 This choice is known as the Gaussian ensemble [2]. The class with time-reversal symmetry is invariant under orthogonal transformations, it is called Gaussian orthogonal ensemble (GOE). The class without time-reversal symmetry, e.g. system with an external magnetic Eeld, is invariant under unitary transformations and it is called Gaussian unitary ensemble (GUE). The crossover between GOE and GUE is easily obtained by
680
A.M.C. Souza, A.M.S. Macˆedo / Physica A 344 (2004) 677 – 684 4 α 1.00 0.32 0.10 0.032 0.00
d(g)
3
2
1
0 0.0
0.2
0.4
0.6
0.8
1.0
g Fig. 1. Distribution of the conductance probability through a quantum dot with two single-mode contacts (N1 = N2 = 1) for several values of symmetry-breaking parameter .
adjusting the variances following Ref. [10] (Hii(0) )2 = 2 (1 + e−2 )=M ; (Hij(0) )2 = 2 (1 + e−2 )=(2M );
i = j ;
(Hij(1) )2 = 2 (1 − e−2 )=(2M );
i = j ;
(7)
where the parameter produces a gradual transition from one class to another ( = 0 is GOE and → ∞ is GUE). We write H = H ( ) and d = d(H; ) such that (Hij )2 i=j = 2 =M and the average level density is "(E) = (M= ) 1 − (E=2 )2 if M 1, for all . The probability distributions of the conductance and shot-noise power are found numerically by employing Eqs. (2), (4), (5) and (7). They are presented as a function of the number of propagating channels (N1 and N2 ) andthe symmetry breaking parameter . For convenience we deEne the parameter = tanh( ) such that the e-ective parameter controlling the crossover remains in the interval 0 6 6 1. The probability distributions of the dimensionless conductance d(g) are illustrated in Fig. 1 for N1 = N2 = 1 and several values of . The numerical procedure is straightforward but quite demanding, since it requires inversion of large M × M matrices for each realization. We remark that in order to reach the asymptotic behaviour described
A.M.C. Souza, A.M.S. Macˆedo / Physica A 344 (2004) 677 – 684
681
15 α 1.00 0.32 0.10 0.032 0.00
12
d(p)
9
6
3
0 0.00
0.05
0.10
0.15
0.20
0.25
p Fig. 2. Distribution of the shot-noise power probability through a quantum dot with two single-mode contacts (N1 = N2 = 1) for several values of symmetry-breaking parameter .
in Eq. (1), one needs a large number, M 1. Note that for the GOE and GUE limits, the numerical results agree to the analytical expressions of the known cases of the literature [12]. In Fig. 2, we show the probability distributions d(p) of the dimensionless shot-noise power for N1 = N2 = 1 and several values of . The case N1 = N2 = 2 and N1 = N2 = 4 are shown in Figs. 3 and 4, respectively. Observe that when Ni increases, the distributions tends towards Gaussian distributions. The extreme example of deviations from a Gaussian is N1 = N2 = 1 as shown in Fig. 1. In this case, we observed that a power-law Et, d(g) = a(c + g)b , where a, b and c are free parameters, agrees well with our numerical data. Finally, using the curve of the average conductance as a function of the symmetry breaking parameter, we can estimate the point at which the transition is e-ectively complete. From the point of inDection of this curve we obtain c ≈ 1=M , in agreement with perturbation theory estimates [10]. 3. Conclusions In summary, we presented a thorough numerical analysis of the behaviour of the probability distributions of the conductance and shot-noise power upon breaking timereversal symmetry in ballistic quantum dots with ideal contacts. The distributions reproduce the analytical expressions available in the literature in some limiting cases.
682
A.M.C. Souza, A.M.S. Macˆedo / Physica A 344 (2004) 677 – 684
2.0
d (g)
1.5
1.0
0.5
0.0 0.0
0.5
1.0 g
(a)
1.5
2.0
7 6
d (p)
5 4 3 2 1 0
(b)
0.0
0.1
0.2
p
0.3
0.4
0.5
Fig. 3. Distribution of the conductance (a) and shot-noise power (b) probability through a quantum dot with two double-mode contacts (N1 = N2 = 2) for several values of associated with the examples presented in Fig. 1.
The most notable generic feature of our data is the presence of broad, albeit universal, distributions throughout the whole crossover region for systems with a small number of scattering channels. A detailed analytical description of this property is highly desirable, in view of potential applications of these devices in quantum information/communication theory. We pose this as a challenge to current methods of quantum transport theories.
A.M.C. Souza, A.M.S. Macˆedo / Physica A 344 (2004) 677 – 684
683
0.07 0.06
d (g)
0.05 0.04 0.03 0.02 0.01 0.00 0.0
(a)
0.5
1.0
1.5
2.0 g
2.5
3.0
3.5
4.0
0.25
0.20
d (p)
0.15
0.10
0.05
0.00 0.0
0.2
0.4
(b)
0.6
0.8
1.0
p
Fig. 4. Distribution of the conductance (a) and shot-noise power (b) probabilities through a quantum dot with two mode contacts where N1 = N2 = 4 for several values of associated with the examples presented in Fig. 1.
Acknowledgements This work was partially supported by CNPq, FAP-SE and FINEP (Brazilian Agencies). References [1] M. Switkes, et al., Appl. Phys. Lett. 72 (1998) 471. [2] C.W.J. Beenakker, Rev. Mod. Phys. 69 (1997) 731.
684 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
A.M.C. Souza, A.M.S. Macˆedo / Physica A 344 (2004) 677 – 684 A.M. Chang, et al., Phys. Rev. Lett. 76 (1996) 1695. H.U. Baranger, R.A. Jalabert, A.D. Stone, Phys. Rev. Lett. 70 (1993) 3876. K.B. Efetov, Supersymmetry in Disorder and Chaos, Cambridge University Press, Cambridge, 1997. J.E.F. Araujo, A.M.S. Macˆedo, Phys. Rev. B 58 (1998) R13379. C.M. Marcus, et al., Phys. Rev. Lett. 69 (1992) 506. Z. Pluhar, H.A. Weidenm4uller, J.A. Zuk, C.H. Lewenkopf, Phys. Rev. Lett. 73 (1994) 2115. K.B. Efetov, Phys. Rev. Lett. 74 (1995) 2299. G. Lenz, F. Haake, Phys. Rev. Lett. 65 (1990) 2325. M. Zirnbauer, J. Math. Phys. 37 (1996) 4986. P.A. Mello, H.U. Baranger, Waves Random Media 9 (1999) 105.