Full counting statistics of spin transfer through ultrasmall quantum dots
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Full counting statistics of spin transfer through ultrasmall quantum dots
T. L. Schmidt, A. Komnik, , and A. O. Gogolin Physikalisches Institut, Universit¨at Freiburg, D–79104 Freiburg, Germany Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, D–69120 Heidelberg, Germany Department of Mathematics, 180 Queen’s Gate, London SW7 2AZ, United Kingdom (Dated: November 5, 2018)We analyze the spin-resolved full counting statistics of electron transfer through an ultrasmallquantum dot coupled to metallic electrodes. Modelling the setup by the Anderson Hamiltonian, weexplicitly take into account the onsite Coulomb repulsion U . We calculate the cumulant generatingfunction for the probability to transfer a certain number of electrons with a preselected spin orien-tation during a fixed time interval. With the cumulant generating function at hand we are thenable to calculate the spin current correlations which are of outmost importance in the emergingfield of spintronics. We confirm the existing results for the charge statistics and report the discov-ery of the new type of correlation between the spin-up and -down polarized electrons flows, whichhas a potential to become a powerful new instrument for the investigation of the Kondo effect innanostructures. PACS numbers: 72.10.Fk, 72.25.Mk, 73.63.-b
Modern microelectronics is one of the most success-ful technologies ever conceived by humankind. However,in recent years a number of limitations, which can slowdown or even stop further progress began to come tothe fore. One possible way to overcome these difficultiesis to switch from charge current processing to spin cur-rent and spin configuration processing. Their advantagesare so enormous that recently a completely new scientificfield of spintronics has been established [1, 2].In the conventional microelectronics the properties ofthe basic circuitry elements are characterized by a num-ber of different quantities – by the nonlinear current-voltage relations, by the current noise spectra, by thecurrent correlations of third order (third cumulant) etc[3, 4]. However, there is one characteristic which (at leastin the low frequency range) contains information aboutcorrelations of all orders. This is the so-called full count-ing statistics (FCS), which answers all above questionsby providing the probability distribution P ( Q ) to trans-fer a certain amount of charge Q during a waiting timeinterval T [5, 6].While there is by now a vast amount of literature avail-able on the charge transfer statistics, its spin-resolvedrelative remains to a larger part unknown with some no-table exceptions [7, 8, 9, 10]. We would like to closethis gap and analyze the combined statistics of spin andcharge transfer through ultrasmall quantum dots withgenuine repulsive interactions. The distinctive featureof these devices are their extremely small lateral dimen-sions which allow for only very few energy levels to takepart in the transport processes. Typical realizations arenanoscale heterostructures on the semiconductor basis,or individual molecules coupled to metallic electrodes,which are even smaller [11, 12, 13, 14]. Both types ofsystems became available only during the last decadesand are promising candidates to become basic building blocks of future nanoelectronics and spintronics circuitry.The archetype system to describe such structures isthe single-impurity Anderson model [15, 16]. It consistsof a local fermionic level (also called dot level) whichis filled or emptied by d † σ , d σ creation and annihilationoperators and N different electronic continua modelingthe electrodes via fermionic fields ψ ασ ( x ), H = X σ = ± (∆ + σh ) d † σ d σ + N X α =1 X σ = ± H [ ψ ασ ] , (1)where ∆+ σh =: δ σ is responsible for the energy of the dotin the magnetic field h = gµ B B/
2, when it is occupied byan electron with spin orientation σ . The dot level energy∆ is an additional parameter which can in practice bechanged by varying the voltage on the background gateelectrode. The coupling of dot and leads is achieved bya local tunneling contribution H T = X α X σ γ α (cid:2) d † σ ψ ασ ( x = 0) + h.c. (cid:3) , (2)with energy independent tunneling amplitudes γ α . Inaddition to these terms one has to take into account theelectrostatic repulsion H U = U ( d †↑ d ↑ − n )( d †↓ d ↓ − n ) , (3)reflecting the energetic cost U of double dot occupationwith respect to the symmetric value n = 1 /
2. Due tothis normalization, the particle-hole symmetric case cor-responds to ∆ = 0. The full system Hamiltonian is thesum of all three contributions H = H + H T + H U .The technology for the calculation of the FCS is bynow far advanced and allows for a number of differentapproaches. In the most widespread one, the quantityof interest is the so-called cumulant generating function(CGF) ln χ ( λ ↓ , λ ↑ ) = ln χ ( λ σ ) [6, 17, 18]. Its succes-sive differentiation with respect to the counting fields λ σ yields the respective irreducible momenta hh δQ nσ ii for theprobability distribution to transfer δQ σ charges with spinorientation σ through the system during the waiting time T , hh δQ nσ ii = ( − i ) n ∂ n ∂λ nσ ln χ ( λ σ ) (cid:12)(cid:12) λ =0 . (4)In analogy to the approaches taken in [19, 20], the firststep in the calculation of the CGF is to endow the tun-neling Hamiltonian (2) with counting fields λ σ . Outsideof the waiting time interval 0 < t < T the counting fieldsare zero. The tunneling Hamiltonian then transforms to H T → H λT = X α X σ γ α h e iλ ασ / d † σ ψ ασ + h.c. i . (5)In the noninteracting case ( U = 0, resonant level model[20, 21]) this quantity can easily be calculated by resum-mation of the perturbation series in γ α or by applyingthe Levitov-Lesovik formula [5]. Assuming the measure-ment time T to be large such that switching effects canbe neglected, one finds (setting e = ~ = 1)ln χ ( λ σ ) = T X σ Z dω π ln n X αβ T αβσ ( ω ) n α (1 − n β ) × (cid:2) e i ( λ ασ − λ βσ ) − (cid:3)o , (6)where n β ( ω ) denotes the Fermi distribution in lead β .The energy dependent transmission coefficients are givenby T αβσ ( ω ) = 4Γ α Γ β ( ω − δ σ ) + Γ , (7)where Γ β = πνγ β (with the density of states ν at theFermi level) is the hybridization of the dot with lead β and Γ = P β Γ β .While the non-interacting result (6) was derived for anarbitrary number N of fermionic leads, we shall restrictourselves henceforth to the symmetric two level case, i. e.,we assume N = 2 and Γ L = Γ R . The chemical potentialsof the leads are assumed to be at µ L,R where V = µ L − µ R denotes the applied voltage.It is quite inefficient to calculate χ ( λ σ ) using an ad-ditional expansion in U in the same way. As has beenrealized in Ref. [22] in a different context, as long as oneis only interested in small U , the CGF can be calculatedby a simple linked cluster like calculation. Thereby the full λ σ dependence can be shifted onto the unperturbedKeldysh Green’s functions D ( ω ) of the dot level. Usingthe notation of [23], these are given by D −− ( ω ) = h ( ω − δ σ ) + X i Γ β ( n β − / i / D ( ω ) ,D − + ( ω ) = hX i Γ β e iλ βσ n β i / D ( ω ) ,D + − ( ω ) = hX i Γ β e − iλ βσ ( n β − i / D ( ω ) ,D ++ ( ω ) = − [ D −− ( ω )] ∗ , (8)where we defined D ( ω ) = ( ω − δ σ ) + Γ (9)+ 4 X αβ Γ α Γ β n α (1 − n β ) h e i ( λ ασ − λ βσ ) − i . Note that the presence of the counting field causes a vi-olation of Keldysh’s sum rule. Using a linked clusterexpansion, the exact CGF ln χ ( λ σ ) can be expressed as acorrection to the noninteracting CGF (6). The quantitywe have to evaluate is then χ ( λ σ ) = χ ( λ σ ) (cid:10) T C exp (cid:2) − i Z C dt H U ( t ) (cid:3)(cid:11) . (10)The expectation value is to be taken with respect to thenoninteracting ground state and therefore contains the λ -dependent Green’s functions (8). It may appear thatsince the lowest order expansion in U only contains thedot occupation numbers n σ , one is not expecting anycounting field dependence to survive. However, this isno longer valid in the case of explicitly (quite artificially)time-dependent λ σ . In the limitmax { ∆ , h, V } / Γ ≪ , (11)the first order contribution is given byln χ (1) ( λ σ ) = − U T Vπ Γ X σ δ σ δ ¯ σ ( e − iλ σ − , (12)where ¯ σ = − σ . As it depends on δ σ , this term only con-tributes for finite magnetic field ( h = 0) and/or brokenelectron-hole symmetry (∆ = 0).The second order contribution is given by two differentdiagrams, see Fig. 1. One of these is again proportionalto the magnetic field and contains the average dot oc-cupation numbers while the other one is the double shelldiagram. The calculation procedure is rather lengthy butstraightforward and results in the following CGF expan-sion in the limit (11),ln χ (2) ( λ σ ) = T V χ o π Γ X σ δ σ ( e − iλ σ −
1) + T V ( χ e − π Γ X σ δ σ ( e − iλ σ − T V χ o π Γ (cid:26) e − iλ ↑ − iλ ↓ −
1) + X σ ( e − iλ σ − (cid:27) + T V ( χ e − π Γ X σ ( e − iλ σ − , (13) ↑ ↓ (a) ↑ ↓ ↑↓ (b) ↑↓↑↓ (c) FIG. 1: One first order (a) and two second order (b,c) dia-grams. where we introduced the equilibrium even/odd suscepti-bilities (correlations of n ↑ with n ↑ and n ↓ , respectively),which are known to possess the following expansions forsmall U [24, 25, 26], χ e = 1 + (cid:18) − π (cid:19) U π Γ , χ o = − Uπ Γ . (14) Following the reasoning along the lines of Ref. [20], wemay speculate that (13) is the exact result to all ordersof U and Γ as soon as one inserts the exact values for χ e,o which have been obtained by, e.g., Bethe ansatz cal-culations [27, 28]. Now we are in a position to establishcontact to known results and to discuss new effects. Thusfar, similar results have been obtained only for the chargetransport statistics of the same system for the much morerestrictive particle-hole symmetric parameter constella-tion [20]. The complete spin resolved statistics for largetransmission Γ is given by the following CGF,ln χ ( λ σ ) = iG T V ( λ ↑ + λ ↓ ) + T V π Γ X σ ( χ e δ σ + χ o δ ¯ σ ) ( e − iλ σ − χ o T V π Γ ( e − iλ ↑ − iλ ↓ −
1) + ( χ e + χ o ) T V π Γ X σ ( e − iλ σ − , (15)where G = 1 / (2 π ) is the conductance quantum per spinorientation. In order to go over to the pure charge CGFone has to set λ ↑ = λ ↓ = λ . In this case, the resultof Ref. [20] is perfectly reproduced for h = ∆ = 0.Moreover, it had been speculated that while the termscontaining a single λ correspond to single electron tun-neling events, the term with the doubled counting fieldis brought about by a coherent electron pair tunnelingin a singlet state. Eq. (15) represents the proof of thisconjecture since the term giving rise to 2 λ part indeedstems from the contribution originally containing the sum λ ↑ + λ ↓ .Yet another justification of the validity of (15) for arbi-trary U is brought about by comparing the above resultto the spin-resolved statistics of charge transfer througha Kondo impurity in the unitary limit presented in [22].Similar to the parameter mapping for the conventionalcurrent statistics one identifies the two limits of small and large U (Kondo regime) by φ/T K = χ o / Γ and α/T K = χ e / Γ , (16)where T K is the Kondo temperature and φ and α areFermi liquid parameters of the Kondo fixed point [29].The similarity of these two results can be traced back tothe similarity of the corresponding Hamilton operators,which not only both contain a resonant level part butalso possess analogous interaction terms.Next, we would like to discuss the linear response (lin-ear in V ) contribution. It can easily be verified that un-der the conditions (11) one obtains the following CGF,ln χ ( λ σ ) lin = (17) G T V X σ ln (cid:26) ( χ c ∆ + σχ s h ) + Γ ( e iλ σ − (cid:27) . This result perfectly coincides with the conjecture of the binomial theorem formulated in [20]. It predicts that thelinear response charge transfer statistics of any interact-ing region coupled to noninteracting continua is binomialand governed by the value of the transmission coefficientat the Fermi edge. In fact, the road to the constructionof the spin-resolved CGF from Eq. (26) of [20] (whichcontains only the charge transfer generating function) isvery natural and intuitive: the logarithms with differentsigns in front of the magnetic field term should containcounting fields for different spin projections.The spin current statistics can easily be recovered fromthe above results after transition to the charge currentand spin current counting fields λ , µ via λ ↑ , ↓ = λ ± µ .One feature of (15) is the fact that the odd cumulants ofspin currents are only non-zero in finite field and for theparticle-hole asymmetric case ∆ = 0. This is the precisecondition for the spin flow existence in a conventionalnoninteracting resonant level system as well. In the linearresponse regime the corresponding odd order cumulantsare then given by hh ( δQ ↑ − δQ ↓ ) n +1 ii = − T V ∆ hπ Γ ( χ e − χ o ) . (18)The even order cumulants are non-universal but n -independent as well, so that the ratio of even/odd orders(it can be seen as a generalization of the Fano factor) isgiven by hh ( δQ ↑ − δQ ↓ ) n iihh ( δQ ↑ − δQ ↓ ) n +1 ii = (19)= − ∆ ( χ e + χ o ) + h ( χ e − χ o ) h ( χ e − χ o ) . Going beyond the linear response regime, we find thatthe most fundamental feature emerging from (15) is theexistence of the invariant cross-cumulant , hh δQ n ↑ δQ m ↓ ii = ( − i ) n + m ∂ n + m ∂λ n ↑ ∂λ m ↓ ln χ ( λ σ )= ( − n + m χ o T V π Γ , (20)for n, m ≥
1. Not only is this quantity non-zero in inter-acting systems only, it is also independent of magneticfield strength and (up to the sign) of its orders n, m . Itexists in the strong coupling Kondo case as well and isfound using (16) for the parameter translation betweenweak and strong coupling. Despite the formally identicalmathematical shapes, the origin of this phenomenon iscompletely different for small U and in the Kondo regime.While in the weak coupling case it signifies the begin-ning of the spin singlet formation, in the strong couplinglimit it starts to appear as soon as it becomes possible tobreak up (though virtually) the Kondo spin singlet. Theamplitude of these correlations grows as one approachesthe strong coupling fixed point. In principle, in addi-tion to the conventional linear conductance (which ap-proaches the unitary limit of almost perfect conductance) the cross-cumulant can also be regarded as a measure ofhow deep in the Kondo regime the system in question isbeing.To conclude, we have analyzed the non-equilibriumspin resolved FCS of the Anderson impurity model bycalculating the CGF of the probability distribution totransfer a fixed amount of charge with preselected spinorientation during very long waiting time interval. Ourresults perfectly agree with existing predictions for thestatistics of charge transfer. The CGF indeed supportsthe interpretation that the electron transport in such asystem is mediated not only by single charge tunnelingbut by correlated transfer of electron pairs in a singletstate. Moreover, the emerging expressions confirm thepreviously conjectured statistics in finite field beyond theparticle-hole symmetric situation. In the linear responseregime, it turns out to be binomial and to factorize in dif-ferent spin channels. Finally, we have discovered a newtype of correlation between the spin-up and spin-downcurrents: a cross–cumulant. It is universal and field in-dependent. In our view, it has a potential to become oneof the quantities to measure and control the ‘quality’ ofthe Kondo effect in nanostructures.We would like to thank Hermann Grabert and DarioBercioux for many valuable discussions. This work wassupported by DFG under grant No. KO 2235/2 (Ger-many). [1] D. Awschalom, D. Loss, and N. Samarth, eds., Semicon-ductor Spintronics and Quantum Computation (Springer,2002).[2] A. R. Rocha, V. M. Garc´ıa-Su´arez, S. W. Bailey, C. J.Lambert, J. Ferrer, and S. Sanvito, Nature Materials ,335 (2005).[3] Ya. Blanter and M. B¨uttiker, Phys. Rep. , 1 (2000).[4] Ya. M. Blanter, in CFN Summer School 2005 on Nano-Electronics , edited by Ch. Roethig, G. Schoen, andM. Vojta (Springer, 2006), Springer Lecture Notes.[5] L. S. Levitov and G. B. Lesovik, JETP Lett. , 230(1993).[6] L. Levitov, H. Lee, and G. Lesovik, J. Math. Phys. ,4845 (1996).[7] A. Di Lorenzo and Yu. Nazarov, Phys. Rev. Lett. ,046601 (2004).[8] M. Kindermann, Phys. Rev. B , 165332 (2005).[9] T. L. Schmidt, A. O. Gogolin, and A. Komnik, Phys.Rev. B , 235105 (2007).[10] K.-I. Imura, Y. Utsumi, and T. Martin, Phys. Rev. B ,205341 (2007).[11] S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwen-hoven, Science , 540 (1998).[12] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu,D. Abusch-Magder, U. Meirav, and M. A. Kastner, Na-ture , 156 (1998).[13] J. Schmid, J. Weis, K. Eberl, and K. von Klitzing, Phys-ica B , 182 (1998).[14] J. Reichert, R. Ochs, D. Beckmann, H. B. Weber, M. Mayor, and H. von Loehneyesen, Phys. Rev. Lett. , 176804 (2002).[15] P. W. Anderson, Phys. Rev. , 41 (1961).[16] A. C. Hewson, The Kondo problem to heavy fermions (Cambridge University Press, 1993).[17] Yu. Nazarov and M. Kindermann, Eur. Phys. J. B ,413 (2003).[18] K. Sch¨onhammer, Phys. Rev. B , 205329 (2007).[19] L. S. Levitov and M. Reznikov, Phys. Rev. B , 115305(2004).[20] A. O. Gogolin and A. Komnik, Phys. Rev. B , 195301(2006).[21] M. J. M. de Jong, Phys. Rev. B , 8144 (1996).[22] A. O. Gogolin and A. Komnik, Phys. Rev. Lett. , 016602 (2006).[23] E. M. Lifshits and L. P. Pitaevskii, Physical Kinetics (Pergamon press, Oxford, 1981).[24] K. Yamada, Prog. Theor. Phys. , 970 (1975).[25] K. Yamada, Prog. Theor. Phys. , 316 (1975).[26] K. Yosida and K. Yamada, Prog. Theor. Phys. , 1286(1975).[27] N. Kawakami and A. Okiji, J. Phys. Soc. Jap. , 1145(1982).[28] P. B. Vigman and A. M. Tsvelik, JETP Lett. , 100(1982).[29] P. Nozi`eres, J. Low Temp. Phys.17