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Shot noise in a quantum dot system coupled with Majorana bound states

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 315011 (http://iopscience.iop.org/0953-8984/26/31/315011) View the table of contents for this issue, or go to the journal homepage for more

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 315011 (7pp)

doi:10.1088/0953-8984/26/31/315011

Shot noise in a quantum dot system coupled with Majorana bound states Qiao Chen1,2, Ke-Qiu Chen1,4 and Hong-Kang Zhao3 1

  Department of Applied Physics, Hunan University, Changsha 410082, People’s Republic of China   Department of Maths and Physics, Hunan Institute of Engineering, Xiangtan 411104, People’s Republic of China 3   School of Physics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China 2

E-mail: [email protected] Received 17 March 2014, revised 30 May 2014 Accepted for publication 3 June 2014 Published 14 July 2014 Abstract

We investigate the spectral density of shot noise and current for the system of a quantum dot coupled to Majorana bound states (MBS) employing the nonequilibrium Green’s function. The Majorana bound states at the end of the wire strongly affect the shot noise. There are two types of coupling in the system: dot–MBS and MBS–MBS coupling. The curves of shot noise and current versus coupling strength have novel steps owing to the energy-level splitting caused by dot–MBS coupling. The magnitude of these steps increases with the strength of dot–MBS coupling λ but decreases with the strength of MBS–MBS coupling. The steps shift toward the large ∣eV∣ region as λ or ϵM increases. In addition, dot–MBS coupling enhances the shot noise while MBS–MBS coupling suppresses the shot noise. In the absence of MBS–MBS coupling, a sharp jump emerges in the curve of the Fano factor at zero bias owing to the differential conductance being reduced by a factor of 1/2. This provides a novel technique for the detection of Majorana fermions. Keywords: shot noise in a quantum dot system, Majorana bound states, nonequilibrium Green’s function (Some figures may appear in colour only in the online journal)

1. Introduction

two Majorana fermions [2]. Recently, Majorana fermions have attracted a lot of attention owing to their fundamental interest and their potential applications in quantum computation [3–6]. In contrast to ordinary quantum computation, such quantum computation using Majorana fermions does not require quantum error correction since Majorana excitations are immune to local noise by virtue of their nonlocal topological nature [2]. In recent years, several proposals have emerged for the realization of Majorana fermions in a solid state system [7]. Majorana fermions in a solid state system were first perceived as a zero-energy state bound to vortices in p-wave superconductors [8]. Kitaev also introduced a model for a one-dimensional spinless p-wave superconductor, which can host Majorana fermions [9]. From this model we can learn that when we aim to search for Majorana fermions in superconductors, we need to get rid of the spin degree of freedom. It means that an ordinary s-wave superconductor is not suitable and what we need is a p-wave superconductor. However, p-wave pairing does not

Majorana fermions, also referred to as Majorana particles, are fermions that are their own antiparticle. They were hypothesized by Ettore Majorana in 1937 [1]. The difference between Majorana fermions and Dirac fermions can be expressed mathematically in terms of the creation and annihilation operators. For Dirac fermions, the operators c† and c are distinct, they obey fermi statistics and have U (1) symmetry. However, for Majorana fermions, the operators γ = γ†,i.e. the Majorana fermion is its own antiparticle. Therefore, we can define the operators of the Majorana fermion as γ1 = c +  c† and γ2  =  i(c  −  c†) with operators of the Dirac fermion, for which reason the Majorana fermion is a ‘half fermion’, which satisfies non-Abelian statistics. Non-Abelian statistics are essential for topological quantum computation because the quantum gate can be realized by exchanging the position of 4

  Author to whom correspondence should be addressed.

0953-8984/14/315011+7$33.00

1

© 2014 IOP Publishing Ltd  Printed in the UK

Q Chen et al

J. Phys.: Condens. Matter 26 (2014) 315011

This paper is organized as follows. In section  2, we present the Hamiltonian of our system and detailed algebraic expressions for shot noise and current obtained using the nonequilibrium Green function (NGF) and Wick’s theorem. The numerical results are given in section 3 with analyses. Finally, a summary is presented in section 4. 2.  Model and formalism The device under consideration is illustrated in figure  1, in which a QD is coupled with a one-dimensional quantum wire. The quantum wire is taken to have a strong Rashba spin–orbit interaction and proximity induced s-wave superconductivity. The Hamiltonian of the system is

Figure 1.  Sketch of a dot–MBS system: a semiconductor wire on an s-wave superconductor surface, and a magnetic field perpendicular to the surface (z direction). MBSs are at the two ends of the nanowire. The dot couples to one end of the nanowire.

H = HLeads + HDot + HD − L + iϵM η1η2 + λ(d − d †)η1, (1)

come about naturally. Several proposals employ half-metals in proximity to superconductors [10, 11], spin-orbit-coupled quantum wells [12–14], nanowires [15–18] and topological insulators [19–21] to realize Majorana fermions in a solid state context. In view of these proposals, detecting unambiguously the existence of MBSs is now the foremost issue. Various suggestions have been made to detect and verify the existence of MBSs, including measurements of noise [22–24], resonant Andreev reflection with a scanning tunneling miscroscope [25] and periodic Majorana–Josephson currents [15, 16]. Recently, Flensberg studied the tunneling properties in a quantum dot (QD) coupled to two MBSs (MBS–dot–MBS) [26]. The QDs can be used for the readout of the state of a Majorana system via charge measurement. The transport properties of the dot–MBS coupled system have also been studied by other researchers [28, 29]. The transport properties are sensitive to the coupling configurations. Shot noise is a low-temperature physical phenomenon that occurs owing to temporal fluctuations of electrical current flowing through a conductor or semiconductor. It is a fundamental physical signature and thus provides information concerning transport additional to conductance and current. Recently, Cao et al investigated shot noise through a quantum dot under finite bias voltage, and they paid particular attention to the Majorana dynamic aspect [27]. They defined a parameter λ1 to describe the U (1) symmetry breaking interaction. A spectral dip together with a pronounced zerofrequency noise enhancement emerged due to the Majorana coherent oscillation dynamics between the nanowire and the quantum dot. Motivated by the work of Cao et al [27] and Liu et al [28], in the present work, we investigate the zero frequency shot noise of a QD–MBS coupled system, where there is both QD–MBS and MBS–MBS coupling. The two kinds of coupling strongly affect shot noise and current. The shot noise and current have novel steps arising from the energy-level splitting caused by the dot–MBS coupling. The magnitude of the step increases with the dot–MBS coupling strength λ, while it decreases with the MBS–MBS coupling strength ϵM. The steps shift toward the region of large bias as λ or ϵM increases. In addition, the dot–MBS coupling enhances the shot noise, while the MBS–MBS coupling suppresses the shot noise.

where HLeads = ∑ ∑k ϵk ck†αckα describes the left and α = L, R right metallic leads, HDot = ϵdd†d describes the dot with energy level ϵd, and HD − L = ∑ ∑k Vkα ( ck†αd + H . c .) describes α = L, R the coupling between the dot and leads. η1 and η2 are Majorana fermion zero modes at the two ends of the quantum wire, and ϵM ∼ e−L/ξ is the coupling between the Majorana bound states, where L is the length of the wire and ξ is the superconducting coherence length. The last part of H describes the coupling between the dot and MBS, and λ is the coupling strength. It is useful to change the Majorana fermion representation to an ordinary fermion representation by defining η1 = ( f + f † ) / 2 and η2 = i( f − f † ) / 2 , where f †( f ) creates (annihilates) a fermion and f † f = 0, 1 counts the occupation of the corresponding states. The last three terms then become 1 HMBS = ϵM ( f † f − ) + λ ( d − d † ) ( f + f † ) / 2 . (2) 2

The spectral density of shot noise is defined by the Fourier transformation of the current correlation: (3) Πγγ ′ ( t , t ′ ) = < δI^γ ( t ) δI^γ ′ ( t ′ ) > + < δI^γ ′ ( t ′ ) δI^γ ( t ) > , where δI^ ( t ) = I^ ( t ) −< I^ ( t ) >. The symbol   in the above γ

γ

γ

formula denotes the quantum expectation value of the electron state and the ensemble average over the system. The current operator can be obtained by considering the continuity equation and the Heisenberg equation, as employed by Jauho, Wingreen, and Meir via an NGF technique [30]. The current operator for tunneling from the γth lead into the QD at time t is found to be

ie I^γ ( t ) = ∑ [ Vkγck†γ ( t ) d ( t ) − V k*γd † ( t ) ckγ ( t ) ] . (4) ℏ k

Substituting the current operator equation  (4) into the correlation function equation  (3), we encounter the correlation functions of four electron operators related to the electrons in the leads ck†α and ckα and electrons in the QD d†,d. Since our system is related to normal electron tunneling, the correlation functions can be expressed by the correlation products with an annihilation electron operator and a creation electron operator employing the Wick theorem. The correlation 2

Q Chen et al

J. Phys.: Condens. Matter 26 (2014) 315011

0.1

I/I

0

0.05 0

−0.05

Increasing λ

(a)

0.06

S/S

0

0.04 Increasing λ 0.02 (b) 0 −0.8

−0.6

−0.4

−0.2

0 eV

0.2

0.4

0.6

0.8

Figure 2.  Current and shot noise versus source-drain bias eV. The parameters used are ΓL = ΓR = 0.05, ϵM = 0.1, ϵd = 0, and λ = 0, 0.03,

0.07, 0.1, 0.2, 0.3.

function can then be expressed by Green functions, and we solve these Green functions using the equation  of motion and Langreth relations [31]. Applying the Fourier transformation over the two times t and t′, and using the relation 1 Sγγ ′( Ω ) δ ( Ω + Ω ′ ) = Πγγ ′ ( Ω , Ω ′ ), we obtain the shot 2π noise of self-correlation S = SLL(0) in the left terminal as S=−

2e2 h

K͠ ( ϵ ) =

∫ d ϵ[Gr(ϵ)ΣL(ϵ)

iΓ 2

−∣ λ ∣2 K ( ϵ )

.

∼ where Σ ( ϵ ) = (1 + ∣ λ ∣4 ∣ K͠ ( ϵ ) ∣2 ) Σ ( ϵ ) and Σ (ϵ) = ±i∑γ Γγf γ (ϵ). We define fγ(ϵ)  =  1  −  fγ(ϵ), where fγ(ϵ) is the Fermi distribution function of the γth lead. Substituting all Green functions and self-energies into equation (5), the equation reduces to

+ G(ϵ)ΣLr (ϵ) + (ΣLr (ϵ) − ΣLa(ϵ))G(ϵ)Ga(ϵ) + G(ϵ) + G >(ϵ)ΣL(ϵ)G

Shot noise in a quantum dot system coupled with Majorana bound states.

We investigate the spectral density of shot noise and current for the system of a quantum dot coupled to Majorana bound states (MBS) employing the non...
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