Journal of Magnetism and Magnetic Materials 140144 (1995) 12071208
mm
~a
Journalof magnetism and magnetic
JR
materials
ELSEVIER
The kondo effect to nonlinear transport through a quantum dot Fumiko Yamaguchi *, Masahito Yamaga, Kiyoshi Kawamura Department of Physics, Faculty of Science and Technology, Keio Unicersity, Yokohama 223, Japan
Abstract The Kondo effect is possibly observed in electronic transport through quantum dots in semiconducting devices. Current and its fluctuation depend logarithmically on bias voltage when voltage is higher than the Kondo temperature. At low voltage regime, the dependence is regular. In both the cases, voltage is scaled by the Kondo temperature.
The Kondo effect draws growing interest in physics of transport through semiconducting quantum dots [1]. With the recent advance of nanotechnology in semiconductor physics, it has become possible to fabricate tunnel junctions of low dimension connected in series. In this system, charging effects become important and lead to, for example, so called the Coulomb blockade for a certain regime. It is expected that the system approaches in near future the utmost limit where an electron travels from a 'source' to a 'drain' using a single quantum state localized in a dot being influenced by the strong onsite Coulomb interaction (see Fig. 1). Such a system can be modeled by the Anderson Hamiltonian for a single magnetic impurity and provides a synthesized Kondo system. This system possibly exhibits the Kondo effect which is peculiar to a one dimensional system far from equilibrium. Since the couplings between the quantum dot and two electrodes are weak, the tunneling Hamiltonian, that is, the s  d mixing Hamiltonian of Anderson, can be treated as perturbation [2]. The onsite Coulomb interaction, on the other hand, is so strong that its effect should be considered completely [3]. The lowest order term of a current in the tunneling Hamiltonian corresponds to sequential flows from the source to the dot and, then, from the dot to the drain [3]. When electrochemical potentials of both the electrodes are arranged as shown in Fig. 1, a total of only one electron is stationarily localized in the quantum dot and consequently the sequential current vanishes. In this case, electrons can flow through the dot using the virtual processes over the localized state and whole the processes can be described by the Kondo Hamiltonian [5]. Main subjects of the transport through the quantum dots have been the I  V characteristics and it would be
* Corresponding author. Email:
[email protected]; fax: + 81455631761.
fruitful to study the Kondo effect in the nonlinear transport. For that purpose, the established theories of the Kondo effect in magnetic alloys can be invoked. In addition, the Kondo effect in current fluctuation is worth studying, since the fluctuations of any physical quantities are comparable to their mean values in such a small system. By collecting the most divergent terms in the perturbation expansion, we obtain expressions for the current I through the quantum dot and zerofrequency current noise spectrum (I2)~o=o (this is essentially timeaveraged current fluctuation) as functions of applied bias voltage V as follows:
3~r e [ eV ~ 2 ( I ) = ~eVtlog~~~K ) ,
(la)
6'rre 2 [ eV ~ 2 (12),o=0 = T e V ~ l O g ~ K )
(lb)
at zerotemperature limit. In (1), T K is the Kondo temperature of the system.
(a)
(b)
Fig. 1. (a) A schematic design of the device and (b) energy level structure in a quantum dot and two electrodes. The single energy level Ed is arranged by changing the gate voltage so that Ed + U / 2 is located between chemical potentials of both electrodes.
03048853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 03048853(94)005818
F. Yamaguchiet al. /Journal of Magnetism and MagneticMaterials140144 (1995) 1207I208
1208
Our calculation begins with the s  d Hamiltonian, H = H o + Hex ,
Ho =
(2)
E r/
of currentcurrent correlation function,
%ka~k~ank~,
J
atn'k'o'am~o(S"°')o"o',
E
==f dt{
e i'°,
(7)
Especially, we pay attention to the zerofrequency (i.e, oJ = 0). This can be calculated in the same manner as the current and it becomes at zerotemperature limit,
{S,D},k,o
Hex =   
(12),o
2 N ,oko,rfk'o'
which describes our system. The electrodes are labeled by q. ( ' S ' and ' D ' denote source and drain respectively.) We treat //ex as perturbation which is switched on adiabatically at t = zo. The first step to calculate the current is to take statistical average with respect to the full Hamiltonian H of a current operator defined by I ( , ) =  e / V s ( t ) = eND(t ),
(3)
in terms of the Keldysh Green function [13]. Here Nn is the number of electron in each electrode. The next step is perturbation expansion in Hex as follows:
<'(0> =
f'
dtl
<[Hex(tl)' 1(1)]>0
(i) 2
+ ~ ft
dqf' _~
{I2)o,=0 =
 ~~
x
(4)
where all operators are in the interaction picture of Ho, and average (  .  )0 is taken by unperturbed states• As a result, the current has the following expression:
j)2
3"r;e (
<1> = T
.spDf
x E p , f d,'
1  2 f n)(Ee ', E
,q
fD(ff)]
+..
}
.
(s)
Here, fn and P,7 are the fermi distribution function and the density of states of the electrode 77, respectively, and integrals of the energy parameters • and E' are performed within the band width 2 D in the electrodes. At zerotemperature limit, (5) becomes 3'rre (
~
=T
x
(
1+
j )2
PsPB(eV/D)
2J(Ps+PD)(l°g(eV/D)l)+ N
"")"
(6) Current noise spectrum is defined by the Fourier transform
OD(eV/D)
2 J ( Ps A PD) N
1+
X (log(eV/D) 
1) +  .  ).
(8)
Collecting the most divergent terms in (6) and (8), we find the expressions in (1) for current and zerofrequency current noise at zerotemperature limit. It should be noted that current noise is equal to the classical shot noise 2 e ( I ) . The perturbation theory employed above is valid for the regime that eV > kT K. If eV < kTK, we should take it into account that the ground state is the spin singlet bound state which is not reached by perturbation theory [4]. In this region, invoking the Fermi liquid theory of Nozi~res [6], we can derive
(I2)~o0
X ([ Hex(t2) , [ H e x ( t l ) , l ( t ) ] ] )o + ...,
Ps
(I)=
h eV

(
~
,
(9)
which does not lead to singularity in the gradient of the current. In summary, we have studied the zerobias anomalies in IV characteristics in a quantum dot on semiconductor devices, using the Keldysh Green function technique. Perhaps the most important discovery in our results is the nonlinear current as function of applied bias voltage. The present study is based on the s  d Hamiltonian on the assumption that a small voltage difference is applied. To extend our theory for the case where sufficiently large voltage difference is applied between source and drain, we should employ the Anderson model, but it might need much more troublesome procedures. References [1] S• Hershfield, J.H. Davies and J.W. Wilkins, Phys. Rev. Lett. 67 (1991) 3720; Phys. Rev. B 46 (1992) 7046. [2] Y. Meir, N•S. Wingreen and P.A. Lee, Phys. Rev. Lett. 70 (1993) 2601. [3] F. Yamaguchi and K. Kawamura, J. Phys. Soc. Jpn. 63 (1994) 1258. [4] K. Yosida, Prog. Theor. Phys. 36 (1966) 875. [5] J. Kondo, Prog. Theor. Phys. 28 (1962) 846; P.W. Anderson, Phys. Rev. 124 (1961) 41. [6] P. Nozi~res, J. Low Temp. Phys. 17 (1974) 31.