SOLVING THE KONDO PROBLEM FOR COMPLEX MESOSCOPIC SYSTEMS

Transcription

1 SOLVING THE KONDO POBLEM FO COMPLEX MESOSCOPIC SYSTEMS V. DINU and M. ÞOLEA National Intitute of Material Phyic, Bucharet-Magurele P.O. Box MG-7, omania eceived February 21, 2005 Firt we preent the calculation of the denity of tate for a ingle ite Quantum Dot with Coulomb interaction of trength U. We ue a decoupling cheme to obtain an explicit analytic approximate expreion for the dot Green function. Then we how how the denity of tate i changed when one more ite i coupled to the dot by a hopping parameter. 1. INTODUCTION The Kondo effect in metallic ytem with magnetic impuritie ha been well undertood for a long time [1]. The effect how up a a minimum in electrical reitivity in the low temperature region. In other word, the reitivity increae when the temperature decreae below a certain temperature (nown a the Kondo temperature). Due to the many body effect, a pea in the denity of tate (DOS) develop at the Fermi level, which lead to enhanced electron cattering, and therefore to increaed reitivity at low temperature. A related Kondo phenomenon in electronic tranport through quantum dot (QD ) trongly coupled to lead wa predicted theoretically [2] and alo oberved experimentally [3] in the pat decade. There i, however, a fundamental difference between the Kondo effect in QD and the Kondo effect in alloy with magnetic impuritie. In the latter cae there i a minimum in reitivity, wherea in the former one there i a minimum in conductance. However, the phyic reponible for thee minima i imilar. In both cae the effect i due to the formation of Kondo pea in the denity of tate at the Fermi level. Thi Kondo pea enhance cattering in the cae of impuritie but, on the other hand, it increae tranmiion through QD leading to almot perfect tranmiion at zero temperature. The Kondo pea in the denity of tate at the Fermi level i a conequence of many body effect. Exchange coupling of the electron located on the dot with conduction electron of the lead give rie to creening of the dot pin, which conequently lead to the Kondo reonance at ufficiently low temperature. Several theoretical technique have been developed in the pat decade to tudy om. Journ. Phy., Vol. 50, No. 3 4, P , Bucharet, 2005

2 448 V. Dinu, M. Þolea 2 thi intriguing phenomenon, including a Green function technique baed on the equation of motion or diagramatic method, caling approach, renormalization group technique, etc. From theoretical conideration, the Kondo pea i plit when an external magnetic field i applied to the ytem. In addition, the Kondo pea alo plit in a nonequilibrium ituation, when a ufficiently large bia voltage V i applied. In the latter cae the two pea are formed at the Fermi level of both lead attached to the dot, and are eparated in energy by ev. To calculate the denity of tate one need the retarded Green function, which i uually obtained by the equation of motion method in the framewor of ome approximation cheme. The mot commonly ued approximation i the one wored out by Meir et al [4]. Thi approximation decribe the ituation rather well for temperature cloe to the Kondo temperature T K a well a above T K. However, it i not correct for much maller temperature T T K. In thi regime the approximation developed by Lacroix [5] i more appropiate. 2. THE SINGLE-SITE ANDESON HAMILTONIAN We conider a ingle-level QD which i coupled to electron reervoir. The on-ite Coulomb interaction i taen in the Hubbard form. The whole ytem can then be decribed by the following Hamiltonian: H = HL + H + HD + HT, (1) where the term H α (with α = L, ) decribe the left and right lead in the noninteracting particle cae: Hα = ε αcα,cα,, (2) α where ε α i the ingle-electron energy in the lead α independent of the pin orientation. c α, ( c α, ) create (detroy) an electron with momentum and pin in one of the two lead. The term H D decribe the quantum dot and tae the form D 0 (3) H = E dd + Udddd, where E 0 denote the energy of the dicrete dot level and U i the Coulomb interaction between the two localized pin which we tae to forbid double occupancy (U ). d ( d ) create (detroy) a pin electron on the quantum dot.

3 3 Kondo problem for complex meocopic ytem 449 The lat term H T of the Hamiltonian decribe tunneling procee between the dot and lead, and i of the form HT = tαcα,d + H. c., (4) α where t α i a contant coupling between dot and lead. The ey objective of the paper i to calculate the denity of tate of the dot in a tationary tate and in a nonequilibrium ituation. To do thi we will apply the equation of motion method for the retarded Green function of the dot. The retarded Green function i defined a r G ( t, t ) = iθ( t t ) { d ( t), d ( t )}, (5) where { Aˆ, Bˆ} = AB ˆ ˆ + BA ˆˆ for arbitrary operator  and Bˆ. 3. DECOUPLING POCEDUE AND EQUATION OF MOTION FO THE GEEN FUNCTION To calculate the retarded Green function of the QD, we apply the equation of motion method. In the following the conduction electron denity of tate i taen to be contant, ρ ( ε ) = 1/ 2D when D< ε < D. The firt equation of motion for the energy dependent retarded Green function G ( ) d d = ; (the Fourier tranform of the time-dependent retarded Green function) i: which give: ( ) { } [ ] G = d ; d + d, H ; d, (6) ( E0 ) d ; d = 1+ t c ; d + U d nd ; d, (7) α α, Equation (7) and (8) imply:,α cα, d tα d d ( ε ) ; = ;, (8) ( E 0 i 0) d d 1 U dnd d + ; = + ;, (9) where =πρ ( tl + t) and n d = d d. In the previou equation we ued the approximation: D Λ ( ) 1 ( i0 + ) 1 d i Λ+ =ρ D ξ+ i0+ ξ ε πρ. (10)

4 450 V. Dinu, M. Þolea 4 A we ee in equation (9) the equation of motion method (E.O.M.) generate higher order Green function for which we apply again the E.O.M.: E 0 U dn d n t c n d α α, t c α α, dd d tα cα, d d d ( ) d ; = d + d ; + + ; ;. If we continue higher order Green function will be generated and the ytem of equation will never cloe. To cloe thi chain of equation we need to ue the approximation introduced by Meir et al. [4]: c α,d = 0, { d () t cqα, () t cα,() t, d(0)} = 0, (12) { c ( ) ( ) ( ) qα, t d t cα, t, d(0)} = 0, and { c ( t) c ( t) d ( t), d (0)} =δ f ( ε ) { d ( t), d (0)} qα, α, q, α q, where f L / ( ε ) i the Fermi function for the left/right lead. We ue the above approximation for the E.O.M. of the three Green function in the right ide of (11). The ytem of equation now cloe. A it i a little lengthy we give only the final reult (in the limit U ): 1 n d d; d, (13) 2 fα ( ε, ) E0 + 2i 0 tα ε where 2 α, D 2 α tα tα ε D i0+ α ξ α L, α f ( ε ) f ( ξ)dξ = ρ. In the previou equation the integral ha the following form: D fα( ξ) d ( ) ln 2 e 1 F i f πt ε α ξ= πα + + Ψ i, D ξ i0+ D 2 2πT (15) where eψ i the real part of the digamma function and ε Fα repreent the poition of the Fermi level in the lead α. For the lat term of (15) we ued the approximation of the digamma function: Ψ () z lnz for z 1, (16) which yield to: F (( F ) 2 2 T 2) 1/ 2 α ε α (11) (14) e 1 ε +π Ψ i ln. 2 2 T (17) π 2πT

5 5 Kondo problem for complex meocopic ytem 451 Fig. 1. (a) Denity of tate of a ingle Anderon impurity for the equilibrum cae ε = ε = 0. Other parameter are E 0 = 2, t L = t = 1 and temperature T = (b) The ame parameter a in (a) except for ε F = 0. 4 (nonequilibrum). We notice the plitting of Kondo pea. The two pea correpond to the L Fermi level. FL F 4. ONE MOE SITE COUPLED TO THE DOT We decribe the adding of one extra ite (referred by the and operator) to the dot by two more term in the Hamiltonian (one i diagonal and the other one i a hopping term). We do not conider Coulomb interaction on thi extra ite. The new term are: (18) E + t d + H. c. The calculation of the dot retarded Green function i performed a hown in the previou ection. The reult i: ( )( fα ( ε, ) d; d 1 nd E + 2i t α ε α (19) t2 (1 + f( E ))) 1 εf E The effect of the added econd ite on the dot i reflected in the lat term of the above formula and it conit in the introduction of a Fano linehape in the dot DOS. We ve written the formula for the cae of equilibrium ( εf = ε ) L F = ε F o that we could write = f( E ε F ). t2 The term i alo preent for U = 0 (noninteracting cae) while the E t2 term f( E ε F) come from econd-order correlation. It can be noticed E

6 452 V. Dinu, M. Þolea 6 that for = E the DOS equal zero that i the zero of the above mentioned Fano linehape. We have alo checed that the introduction of uch a virtual level doe not have any influence on the Kondo temperature. Fig. 2. Denity of tate of the QD connected to an extra ite of energy E = 1; t = 0.5. The other parameter are the ame a in Figure 1a. 5. CONCLUSIONS We have een that the decoupling procedure introduced by Meir et al [4] are enough for the obervation of the Kondo pea in the DOS. We alo howed the modification induced in the DOS by the adding of another ite on the dot. Such complication of the problem may be uefull for the analyi of tranport propertie in complex meocopic tructure. EFEENCES 1. J. Kondo, Prog. Theor. Phy. 32, 37 (1964). 2. T. K. Ng, and P. A. Lee, Phy. ev. Lett. 61, 1768 (1988). 3. D. Goldhaber-Gordon, H. Shtriman, D. Mahalu, D. Abuch-Magder, U. Meirav, and M. A. Katner, Nature (London) 391, 56 (1998). 4. Y. Meir, N. S. Wingreen, and P. A. Lee, Phy. ev. Lett. 66, 3048 (1991). 5. C. Lacroix, J. Phy. F 11, 2389 (1981).