The role of spin-exchange on transport in quantum dots

PHYSICA blNbVIbR

Phvslca D 83 (1995) 280 285

The role of spin-exchange on transport in quantum dots Yt Wan a, Philip Phillips a, Qimlng Li b a Loom~s Laboratory of Ph)wt~s, Umver~zty of llhnot~ at Urbana-Champmgn, 1100 W Green St, Urbana, IL 61801-3080 USA b Ame~ Laboratory and Department or Physics, Iowa State Umver~tt3', Ames 1,4 50011, USA

Abstract

We analyze here a model for single-electron charging in semiconductor quantum dots that includes the standard Anderson on-site repulsion (U) as well as the spin-exchange (Jd) that is inherently present among the electrons occupying the vanous quantum levels of the dot A Schneffer-Wolffotype transformation is developed to analyze this model We show exphc~tly that for ferromagnetic couphng (Jj > 0), an s-d exchange for an S = 1 Kondo problem is recovered In contrast, for the antfferromagnetlc case, Jj < 0, we find that the Kondo effect is present only if there are an odd number of electrons on the dot Spin-exchange is then shown to produce a second penod m the conductance that is consistent with experimental measurements

In this lecture, we will focus on the role of spin-

trlcal leads connected to the dot Transport in quan-

exchange in on the transport properties of quantum

tum dots will be Coulomb limited if kBT < Ec and kBT > Ae, where Ae is the spacing between the single

dots, particularly the Kondo effect and the conductance To introduce this top~c, we will review the basic principles of Coulomb-limited transport in quantum dots When a gate voltage is applied to a nano-scale semiconductor inversion layer (or quantum dot), elec-

particle states of the dot Because of the central role played by on-site

trons will flow one at a time across this device pro-

with a Hubbard-like model In so far as a quantum dot can be reduced to a single site [7] with a charging energy U, the Anderson model [ 8] for the Interaction of a magnetic defect coupled to a non-interacting sea

vided that the applied voltage is an integral multiple of the capacitance charging energy of the quantum dot Experiments illustrating the p n n c l p l e of charge quantlZatlon by virtue of the charging energy have been performed recently on numerous semiconductor [ 15] as well as superconducting [6] nano-structures We focus here solely on the semiconductor devices It is now well-accepted [ 5 ] that In semiconductor quantum dots, the dominant contribution to the capacitance charging energy, Ec = e 2 / 2 C arises from the on-site Coulomb repulsion Here C IS the capacitance between the quantum dot, the tunnel junctions, and the elec-

Coulomb repulsions in the transport properties of quantum dots, it is natural to model a quantum dot

of conduction electrons is appropriate [7]

k,o-

o-

ta +~,g,~d(a~,~

(1)

It,or

=Hd+ZUd(a1~ad~+a~cra~

)

(21

kcr

= Hd + H1 1995 Elsevier Science B V S S D I 0167-2789(94)00264-9

t d,r+ad~ak~) + Und,ndl

(3)

281

Y Wan et al / Phystca D 83 (199.5) 280-285 where e j is the defect energy o f the magnetic ~mpurlty, ~ d the overlap integral between a band state wlth momentum k and the impurity, a~ creates an electron in the band states, atao. creates an electron with spin of on the impurity, and nd,~ = ad,~ad, lS the number operator for an electron of spin o- As a consequence of the on-site repulsion, the single particle states on the impurity have energies, ed and ed + U At high temperatures, the density o f states of this model has two Lorentzmn peaks centered at these two energy levels At low temperatures, however, the Anderson model displays a Kondo resonance [ 10,11 ] at the Fermi level Although the Kondo resonance is expected to occur for any value o f the defect energy within the range - - U < E d < 0, It IS most favourable at the defect energy corresponding to the greatest stability of the local moment at the d-impurity, namely, ed = - ½ U At this energy HA is particle-hole symmetric, and the Kondo resonance is pinned at eF = 0 The single-particle states e j and ed + U he symetrlcally, then, around the Fermi energy When the chemical potential o f the source lead coincides with the energy o f the Kondo resonance, a single electron should charge the dot In the symmetric limit, this state o f affairs should obtain at half-integer multiples o f the charging energy Thus far, no experimental hint of the Kondo resonance has been observed in quantum dots in zero bias voltage It is precisely the conditions under which the Kondo effect should be observed in quantum dots that we address here W h i l e it may be premature to draw any concluslon lrom the lack of experimental confirmation of the Kondo effect, it is certainly appropriate to Investigate the validity o f the Anderson model to a quantum dot It is in addressing this issue that we are (1) able to predict even-odd charging effects in semiconductor quantum dots as well as (2) a suppression of the Kondo effect when a quantum dot has an even number o f electrons The most obvious inadequacy o f the Anderson model in the context of quantum dots is the truncation of the multiple electronic levels on a quantum dot to a single state If multiple electronic levels are included on the dot, then other Coulomb interactions besides the on-site U become relevant A key quantity that comes Into play is the intrinsic Coulomb exchange energy, Jd. between two levels Consider for

the moment a two-orbital model for a quantum dot For two degenerate levels and U ~ the &rect C o u l o m b exchange integral, the energy o f the two-body states predicted by this model are 2Ca + U - 1Jd, 2ed + U, and 2ed + U + 3Jd. The energy o f the 3-electron state is 3(ed + U) Consequently, the charging energy depends on Jd In fact, the general role o f spin-exchange is to introduce a spin-dependent charging energy that is determined by the p a n t y of the total number o f electrons on the dot It is worth pointing out that charging experiments on controlled-barrier atoms [ 2] in strong magnetic fields display systematic oscillation in the peak heights, widths as well as in the separations that are consistent with a second period in the conductance as a function o f the applied gate voltage Such systematic deviations have been attributed to a splitting o f the energy between the up and down Landau levels, rather than to spin exchange We propose here that such trends are also consistent with a spin-exchange model We explore then the simplest model of a quantum dot that includes the effects o f spin-exchange A natural way o f including spin exchange is simply to introduce another level Into the Anderson model The only qualitative change this level is going to provide is the spin-interaction with the d-level Consequently, we treat this level as a local spin, S Because spinexchange plays no role if S = 0, we will consider only the case in which the S-level is singly occupied, or equivalently, S: = ± 1 Hence, large N expansion techniques are inappropriate to solving this problem If we label the spin on the d-level of the impurity with Sd, we find that our Hamlltonian can be written as

H = HA + H j = Ho + H1,

(4)

H j =--JdSd

(5)

S,

S~= ~(ndT ~ -- ndJ,), S d =atdSadT

S~1 = atdTad.L, (6)

The first question we answer with this model ~s, does the Kondo effect still occur Before rigorous calculations are performed, a heuristic answer can be put forth immediately. Without loss of generality, the defect energy can be taken to be led] ~ U > > [Jd[, as Jd is

282

Y Wan et al

/ Phvstca D 83 (1995) 2 8 0 - 2 8 5

typically a fraction of U The tunnehng rate to the dot is determined by the matrix element Vka Because this quantity is an adjustable parameter determined by the width of the tunnel junction connecting the dot to the source lead, we can set [Jdl > IVl, rdl T h i s inequality is crucial m the analysis of what follows For example, in the absence of the Jd coupling, the form of the antlferromagnetxc interaction that gives rise to the Kondo effect scales as IJKI ~ IVk~dl2/U However, in the hmit that IJdl > IVk~dl,IJdl >> IJKI That is, the exchange coupling exceeds the Kondo couphng and could hence ultimately conspire to mask the Kondo effect Consider the case in which the d-level and the S-level are singly occupied. In the ferromagnetic case, Ja > O, the ground state of the dot is a triplet A Kondo effect should result in this case that is determined by the total spin on the dot However, in the antiferromagnetlc case, Jd < 0, the ground state on the dot is a slnglet As a consequence, there is no net spin to couple to the conduction electrons and the Kondo effect IS suppressed In the antiferromagnet~c case, there must be an odd number of electrons on the dot for the Kondo effect to be observed This is the essential physics of this model To prove the heuristic arguments given above, we diagonahze H in the subspace of all singlyoccupied states on the dot We first note that because [H0, H j] = 0, we can work entirely with the eigenstates of the dot Let us define a generalized elgenstate Iq) = iN, Stot, S(ot), where N refers to the number of electrons on the d-level of the dot Recall, we have set S z = +½ There are 8 elgenstates in the dot basis 10, ~,S~o t l : = ~21) with total energy E = 0 , the singlet 11,0,0) with energy e~ = e,t + ~3 Ja, the triplets ]1, 1, S(ot = 0, + l } , w,th energy Et = E d -- 1 J d and 1 z the doubly-occupied state, [2, 7, S~o t = ± 21) with energy 2ed + U Each of the Fermion operators as well as the bl-llnears such as Sd S can be expressed m terms of these 8 basis states Once this is done, matrix elements among these states can be determined straightforwardly For an arbitrary wavefunctlon [¢), we are interested in solving the Schrodinger equation

~(qlHlp)(PIe) p

= E(ql¢)

(7)

in the subspace of singly occupied states, N = 1 This is the relevant phase space for considering the Kondo effect We have developed a perturbation scheme to do precisely this [9] Rather than focusing on this procedure here, we will &scuss an alternate route in the spirit of Schrleffer and Wolff [ 10] For the Anderson Hamlltonlan, Schrieffer and Wolff [ 10] have devised a slmdlarlty transformation that ehmlnates the hybridization to the conduction electrons The procedure involves finding a transformation, S, such that

H = e S H e -~ =H+

(8)

[S,H] + l[~, [S,H]] +

(9)

and H1 + [S, H0] = 0

(10)

If we can find an ,~ so that Eq (10) is sansfied, our Hamiltonian to lowest order in IVkl2/U << 1 will be

~,= n0 + ½[~,HI]

(l I)

After a considerable amount of algebra, it Is possible to show that the appropriate transformation for the spin-exchange model Introduced here Is

-2S+l

/~ o - , o -t

(S+ l

v, L\e-Ur_,+ s

]a ,

)

+ 2 ( s s~,) +

(,

E[+~

(1-n._,.)

nd -,r

J

~r~r

[(, l) ,) ]} E~_ ~

E~+~ .~_~,

E~_ ~ (1-nd_,~,) c~.'~, - h c

(12) In the above, E~,t ) = el, - es t and E s,t (+~ = e-k -- 2 e d -U + est The subscripts (s,t) refer to the singlet and triplet spaces, respectively In the slnglet subspace, S = 0 and only those virtual transitions involving the triplet state survive To construct the Hamdtonlan in the spin subspaces, we substitute S into Eq (11) In the triplet (S = 1) or ferromagnetic case (Ja > 0), the reduced HamiltonIan in the triplet subspace is

He~ffplet= He ÷ "t ÷ Z kU o-

W~U ( '~t~' '///k)

Y Wan et al I Phvst~a D 83 (1995) 280-285

- Z J~,t,(,0},¢rO,t)Stot

(13)

/,,U

where H~ is the Hamlltonian for the free conduction electrons, tr is the Pauh spin matrix, 0k is the twocomponent splnor

karl the antlferromagnetlc coupling constant is

J~k' = 2Vw~IV~d

el,, - 2ea - U + et

el< - et

(14) and

W~,

. (

l

1

= ~'VwdVI'd \ ek, - 2 e j - U + et

+

q E I~

,) Et

(15)

That J ~ , is negative can be seen immediately because the largest energy scale in the denominator is U which enters with a - sign The spin interaction obtained in this limit is identical to the usual Kondo coupling except in thls case the total spin on the dot enters For a triplet state Stot = l Consequently, ferromagnetic exchange gives rise to a S = 1 Kondo problem The S = 1 Kondo problem is an example of an undercompensated spin problem in which the conduction electrons only partially screen the spins on the dot The remaining unscreened spin couples ferromagnetlcally to the conduction band The only quahtative difference between the S = 1 Kondo problem and S = ½ is the behavior of the magnetic susceptlblhty As a result o f the undercompensation, the susceptibility does not vanish at T = 0 in the S = 1 problem Consider now the more experimentally-relevant antlferromagnetlc or slnglet ( S = 0) case l We recover in this hmlt an effective Hamlltonlan of the form HSmglet

eIf

_-.': Hc + e, +

ZW

~

k' ( ~ , ~ )

+~d~..JliwiJ,~21~;(¢WlO'Ok,)

(O~; O'O&~)

(16)

•~ I I{/I

/,2 ,t~ I In the N = 1 suhspace of our model the ground slate is a spin smgleI This follows from the general theorem of 112]

283

where W~k, and J~k' are identical to their triplet counterparts with e~ replaced by et As is evident the spin coupling only involves the conduction electrons and is O( (Vta) 4) Further, the overall sign o f this interaction is negative or ferromagnetic Consequently, there is no antlferromagnetlc exchange interaction that can produce a Kondo effect to fourth order in the couphng to the band electrons The physical origin of the absence of the Kondo effect here is the stablhty of the slnglet on an energy scale Ja As a result, the Kondo coupling constant must be cut off at this energy scale Consequently, it cannot diverge and give rise to a bound state at the dot This result Is consistent with the heuristic arguments of Ng and Lee [ 13] on the role of spin exchange in a quantum dot and a mean-field limit o f the 2-impurity Anderson model in the presence o f spin-exchange [ 14] Jones, Kothar and Mllhs [ 14] found that in the N --, oc limit o f this model, a phase transition occured which suppressed the Kondo effect if the bare exchange interaction exceeded a critical value The critical condition is similar to the one used here, namely IJa] > ]VkdI2/U There still remains one chance for the Kondo effect to be observed when Ja < 0 II the number of electrons on the dot is odd, or equivalently we restrict ourselves to the N = 0 subspace, the standard S = ½ Kondo problem is recovered If experiments are going to detect the Kondo effect, the total number of electrons on the dot must be carefully controlled We now calculate the conductance at finite temperature in the presence of spin-exchange To facilitate this we need the average occupancy on the dot (ha,,) This quantity is obtained by Integrating the imaginary part of the d-electron Green function, G,t<,( w ) = ((aa,,, afa,,)), weighted with the F e r m l - D l r a c distribution function Standard equations of motion methods [ 15,16] can be used to formulate an accurate expression for Ga<,(w) Each level o f iteration generates a new hierarchy of Green functions for which new equations of motion must be derived To illustrate, the Helsenberg equations o f motion for G a , , ( w ) generate two new Green functions, ((nd-o-ao-, atd,,)) and ((S Saado., atao.)) Equations of motion for this set of Green functions as well as for the new Green functions that appear at this level were derived and solved self-

284

Y Wan et al

Physzca D 83 (1995) 280-285

06

05

k T = O01U - k T = O05U

04

03

02

01

0 -04

i -02

i 02

i

04 06 C h e m i c a l potential

i

O8

i 12

Fig 1 Conductance (measured m units of e2/h) as a function of the chemical potenual (measured m umts of U) as computed from Eq (17) using the third-level decouphng of the equatmns of motion for the Green function for Jd = --0 IU and .5 = 0 001U Each set of two peaks corresponds to a smglet and triplet pare the smglet being lower m energy m the antfferromagnetlc case consistently for the impurity density of states by invoking the Hartree-Fock closure The density of states obtained at the 3rd level of iteration is sufficient to describe the Kondo effect As we have already described the T = 0 phase, we focus on the experimentallyaccessible high-temperature limit The conductance -2e 2 A / F-

~-

fFD(tO) (1 -- fFD(tO))

kiT

x I m Gdcr( tO + O)dto

(17)

was calculated using the standard Landauer formula [7,17] In Eq ( 1 6 ) , A = ( - 1 / 7 " r ) ~ ~ k l V k l 2 ~ ( t o e t ) To Illustrate the role of spin-exchange, we report here the infinite U hmlt of G d ( w ) at the second level equations o f motion We find that the second level Green function

3-4(1 Gd(w)

=

+ ½Is s./

o~ -- es -- X O ( 1 - (rid-a))

+¼

(i -

-

o . - e , - Zo (i -

s.>

(is)

contains a contribution for the slnglet and triplet states with differing spectral weights In Eq. (18), the selfenergy IS X 0 = ~-~k 1v k ] 2 ( O ) - - e k ) - 1 ~ - - l A T h i s expression clearly illustrates that the singlet (first term)

and the triplet (second term) spectral weights differ We expect that the conductance into the slnglet and triplet levels should reflect the asymmetry m the spectral weights The conductance calculated from Eq (17) (with the 3rd level Green function) is shown in Fig 1 as a function of the chemical potential. Illustrated clearly is the asymmetry in the conductance peaks centered at es (the first peak) and at 6t (2nd peak) In Fig 1, J,l = - I U Hence, the slnglet is the ground state In the absence of Jd, the slnglet and triplet peaks would coalesce into a single peak as in the standard Anderson model The higher peaks m the conductance occur at energies 2ca - et + U and 2ed -- es+u, respectively The upper peaks appear reverted because for Ja < 0, e~ < 'ft In the ferromagnetic regime, the triplet peak dominates and It ~s the nelghbourlng sInglet states that lead to the asymmetry m the peak heights in the conductance as illustrated m Fig 2 We conclude then that spin exchange in zero magnetic field leads to peak height alternation m the conductance that is identical in form to the experimental [2] trends seen in the presence of a magnetic field UlUmately, the ferromagnetic and antlferromagnetlc cases can be distinguished by a low temperature study of the Kondo phase Experiments by Westervelt and

Y Wan et al 0 35

J

285

Phystca D 83 (1995) 2 8 0 - 2 8 5

J

J

03

kT= kT=

i

i

O01U - O05U . . . .

025

02

015

O~

005

0 04-

~ -02

'J~ 0

, 02

04 06 Chem,cal potent=al

08

.~,

, 1

12

l ~g 2 Same as Fig 1 but for the ferromagnetic couphng, Jd = 0 1U Each set of two peaks corresponds to a smglet and triplet pmr, the triplet bemg lower in energy m the ferromagnenc case c o - w o r k e r s [ 18] are consistent with the predictions

References

o f e v e n - o d d a s y m m e t r y The m a g n i t u d e o f the weak peaks in the c o n d u c t a n c e that arise from the principal peaks is consistent with predictions from antlferromagentlc e x c h a n g e as illustrated in Fig

i Further ex-

periments are needed, particularly those in the K o n d o regime, to confirm the s p i n - e x c h a n g e m o d e l presented here

Acknowledgements We thank Patrick Lee, Ylgal Meir, and Robert Westervelt for insightful remarks This w o r k is supported in part by the N S F and the donors o f the P e t r o l e u m Research F u n d o f the A m e r i c a n C h e m i c a l Society, and the D i r e c t o r o f Energy Research, Office o f Basic E e r n g y Sciences A m e s Laboratory is operated for the U S D O E by I o w a State University under C o n tract N o

W - 7 4 0 5 - E N G 82 W h i l e this w o r k was in

preparation, a paper appeared ( L Wang, J K Zhang, and A R

Bishop, Phys

Rev Lett

73 ( 1 9 9 4 ) 585)

which reported numerical calculations on an extensive m o l e c u l a r orbital m o d e l for a q u a n t u m dot E v e n - o d d a s y m m e t r y was predicted in the overall conductance However, the w e a k peaks in the vicinity o f the central coductance peaks were not predicted

[1] K Likharev, IBM J Res Dw 32 (1988) 144, D V Averm and K K Llkharev, m Mesoscoplc Phenomena m Sohds, B L Altshuler, PA Lee and R Webb, eds (Elsevier, Amsterdam, 1991) [2] EB Foxman, PL McEuen, U Meirav, NS Wmgreen, Y Mere PA Belk, NR Belk, MA Kastner and SJ Wind, Phys Rev B 47 (1993) 10020, for an overview, see M A Kastner, Physics Today 46 (1993) 24 [ 3 ] R C Ashoorl, H L Stormer, J S Wemer, L N Pfelffer, K W Baldwin and KW West, Phys Rev Lett 71 (1993) 613 [4] H Chang, R Grundbacher, T Kawanura, J -P Leburton and 1 Adeslda, J Appl Phys, submitted [5] H van Houten and CWJ Beenakker, Phys Rev Lett 63 (1989) 1893 [6] M Tuommen, J M Hergenrother, TS Ttghe and M Tmkham, Phys Rev Lett 69 (1992) 1997 [7] Y Me]r, N S Wmgreen and PA Lee, Phys Rev Lett 66 (1991) 3048, 70 (1993) 2601 [8l PW Anderson, Phys Rev 124 (1961) 41 [9] Y Wan, P Phdhps and Q LI, Phys Rev Lett (1994), submitted [10] JR Schneffer and PA Wolff; Phys Rev 149 (1966) 491 I l l ] AM Tsvehck and PB Wlegmann, Adv Phys 32 (1983) 453 [12] E L,eb and D Maths, Phys Rev 125 (1962) 164 [13] TK Ngand PA Lee, Phys Rev Lett 61 (1988) 1768 [ 14] B Jones, B G Kothar and A Mlllls, Phys Rev B 39 (1989) 3415 [ 15] J A Appelbaum and D R Penn, Phys Rev 188 (1969) 874 [16] C Lacrolx, J Phys F 11 (1981) 2389 [17] R Landauer, Phdos Mag 21 (1970) 863 [ 181 R Westervelt, unpubhshed