Kondo temperature of a quantum dot
TABLE OF CONTENTS DEDICATION .................................. iii PREFACE ..................................... iv ACKNOWLEDGEMENTS .......................... v LIST OF TABLES ............................... viii LIST OF FIGURES .............................. ix SUMMARY .................................... xi I KONDO EFFECT IN A SINGLEELECTRON TRANSISTOR . 1 1.1 Conventional Kondo eect due to a magnetic impurity........2 1.1.1 Renormalization Group.....................4 1.1.2 Kondo singlet..........................6 1.2 Coulomb blockade in a quantum dot..................8 1.2.1 Model of a lateral quantum dot system............9 1.2.2 Charge quantization and Coulomb blockade oscillations...10 1.3 Kondo eect in a single electron transistor..............13 II KONDO TEMPERATURE OF A QUANTUM DOT. ....... 17 2.1 Anderson model:
E C .......................18 2.2 Realistic quantum dot:
E C ....................21 2.2.1 Kondo eect in the charge sector................21 2.2.2 Reduction to the Kondo model.................24 2.2.3 Discussion............................25 APPENDIXA —SCALINGFORTHE MULTICHANNEL KONDO MODEL .................................... 29 APPENDIX B —DISORDER IN A QUANTUM DOT ...... 36 REFERENCES .................................. 42 VITA ........................................ 48 vii
LIST OF TABLES B.1 A table of high energy cuto D of renormalization group in dierent regions over the gate voltage.......................39 viii
LIST OF FIGURES 1.1 Equivalent circuit for a quantum dot connected to two massive con ducting leads by tunnel junctions with conductances G L,R and capaci tances C L,R and capacitively coupled to the gate electrode V g ,adopted from Pustilnik and reprinted with the permission of John Wiley and Sons....................................8 1.2 (a) Average number of electrons in a dot N =
ˆ N
at T
E C as a func tion of the dimensionless gate voltage N 0 .The number of electrons N diers signiﬁcantly from integer values in the narrow mixedvalence re gions of the width
max {
,T } /E C about N 0 = N
0 = halfinteger. (b) At
T
E C the conductance is small in the wide Coulomb valley of the width of almost 1 in the dimensionless gate voltage due to the Coulomb blockade. Adopted from Pustilnik and reprinted with the permission of John Wiley and Sons..............................11 1.3 (a) The height of a Coulomb blockade peak vs. the temperature in the mixedvalence regions. (b) Conductance vs. the temperature in the middle of a Coulomb blockade valley. Adopted from Pustilnik and reprinted with the permission of John Wiley and Sons..............................13 1.4 Various secondorder (cotunneling) processes adopted from Averin et al. (a) Inelastic cotunneling:an electron tunnels from the left lead into one of the unoccupied singleparticle levels in the dot,whereas an elec tron occupying some other level tunnels out to the right lead,leaving an electronhole pair behind.The contribution to the conductance scales with temperature as T 2 . (b) Elastic cotunneling:unlike the inelastic case,occupation of elec trons in the dot is the same through the cotunneling process.This contribution to the conductance is T independent. (c) Spinﬂip cotunneling process:the origin of the Kondo eect in the dot. Adopted fromPustilnik ad reprinted with the permission of John Wiley and Sons..................................14 1.5 (a) The conductance vs. the temperature in the Coulomb blockade valley with an odd number of electrons in the dot. (b) Conductance vs. the gate voltage at T
0. Adopted from Pustilnik and reprinted with the permission of John Wiley and Sons..............................15 ix
2.1 Sketch of the dependence of the Kondo temperature T K on the dis tance in the dimensionless gate voltage to the charge degeneracy point ( N 0 ),see Eq.(2.38) The solid line corresponds to a quantum dot (
E C ) with the bare level width 0 .The dashed lines represent singlelevel Anderson impurity models (
E C ) with dierent level widths, and 0 .............................27 x
SUMMARY The lowenergy properties of quantum dot systems are dominated by the Kondo eect.We study the dependence of the characteristic energy scale of the eect, the Kondo temperature T K ,on the gate voltage N 0 ,which controls the number of electrons in the strongly blockaded dot.We show that in order to obtain the correct functional form of T K ( N 0 ),it is crucial to take into account the presence of many energy levels in the dot.The dependence turns out to be very dierent from that in the conventional singlelevel Anderson impurity model.Unlike in the latter, T K ( N 0 ) cannot be characterized by a single parameter,such as the ratio of the tunneling induced width of the energy levels in the dot and the charging energy. xi
CHAPTER I KONDO EFFECT IN A SINGLEELECTRON TRANSISTOR Advances in nanoscale fabrications and manifestations allow one to establish systems unapproachable in the past;single electron transistors (SET) and single molecule de vices to name a few.Among a few interesting features of such devices,transport in nanostructures is of importance considering the weight of electronics and its applica tion.A quantum dot is a common implementation of single electron transistors,and an ideal candidate for the study of transport in nanostructures [1,2]. In quantum transport,a quantum dot (twodimensionally conﬁned region at the interface of two semiconducting layers,for instance,GaAs/AlGsAs in a lateral dot system) is capacitively connected via tunneling junctions to two massive conducting leads,the source and the drain [1,2].The dierential conductance dI/dV of such device displays dependence on the external parameters such as temperature T ,Zee man energy due to a magnetic ﬁeld B = gµ B H ,and the sourcedrain bias V .The dependence of the dierential conductance is well described by a formula in terms of the external parameters [3] dI dV
e 2 h
ln max { T,eV,B } T K
2 , (1.1) where T K is the characteristic energy scale in this transport.This anomalous be havior of the dierential conductance was reported in various systems such as lateral semiconductor quantum dots [4,5,6,7,8,9,10],vertical quantum dots [11,12], carbon nanotubes [13,14],and singlemolecule transistors [15,16,17],etc. The logarithmic behavior of the transport coecient over the external parameters has been known in condensed matter physics for a long time.When there is a magnetic 1
impurity in a host metal [18,19],the resistivity of the system exhibits nonmonotonic temperature dependence as well (the conventional Kondo eect) [20,21,22]. 1.1 Conventional Kondo eect due to a magnetic impurity The Kondo eect [23],the existence of the resistivity minimum at a certain nonzero temperature was discovered in the early 1930s [24].From the experimental data, the contribution to the resistivity of the impurity in a metal was empirically given by [23,25]
( T )
n i ln(
F /T ) , (1.2) with the impurity concentration n i and the Fermi energy
F .It is obvious that the impurity contribution Eq.(1.2) has a extremum to have a resistivity minimum,as the electronphonon scattering contribution to the resistivity monotonically decreases when decreasing T .It should be mentioned again that the resistivity minimum de velops only when the impurity atoms are magnetic [23].The proportionality
n i has been veriﬁed down to the lowest impurity concentrations experimentally allowed, which suggests that the phenomenon is due to a single magnetic impurity.Kondo suggested the simplest model considering the local exchange interaction J between the magnetic impurity and itinerant electrons at the impurity site from these obser vations [23], H K = H 0 + J ( s · S ) . (1.3) Here H 0 =
ks
k
ks
ks describes the noninteracting electron gas (
k are single particle energies of itinerant electrons), s = 1 2
kk
ss
ks ˆ
ss
k
s
is the spin density of itinerant electrons at the impurity site (with the Pauli spin matrices ˆ
= (ˆ
x , ˆ
y , ˆ
z )), and S is the spin1 / 2 operator for the magnetic impurity. It should be noted that the phenomenological Kondo model Eq.(1.3) (in other words,the s  d model) can be reduced from the microscopic Anderson impurity model 2
with the help of the SchrieerWol transformation [26,27] H = H 0 + E d
n d + Un d
n d
+
k V kd c
k c k +H.c. , (1.4) where H 0 =
k
k n k (
) =
k  V kd  2
(
k ) and the Coulomb interaction U =
d ( r )
d ( r
) e 2  r
r

d ( r )
d ( r
) d r d r
(here,the Bloch wavefunction
). According to Kondo [23],the lowest order of perturbation theory in the exchange constant J is not sucient to show the logarithmic dependence in Eq.(1.2) due to the noncommutativity of the spin operators in the model,Eq.(1.3).Beyond the Born approximation,however,Kondo showed that the logarithmic temperature dependence appears in the third order in J [23],
n i ( J ) 2
1 +2 J ln( D 0 /T )
. (1.5) Here,
is the density of states of itinerant electrons (as a result, J
1 is a dimen sionless parameter) and D 0
F is the highenergy cuto in Eq.(1.3). Further study after Kondo showed that logarithmicallydivergent terms exist in all orders of perturbation theory,eventually leading to a geometric series [28]
( T ) / (0)
n =0 ( J ) n
ln( D 0 /T )
n
1
2 =
J 1
J ln( D 0 /T )
2 . (1.6) Rewriting this sum,bearing Eq.(1.1) in mind,reads
( T ) / (0)
ln( T/T K )
2 , (1.7) 3
with the Kondo temperature T K (characteristic energy scale in Eq.(1.1)) deﬁned by ln( D 0 /T K ) = ( J )
1 . (1.8) Eq.(1.8) gives the estimate of T K with logarithmic accuracy.A more accurate esti mate from Renormalization Group,see Appendix A,reads T K
D 0 ( J ) 1 / 2 exp(
1 /J ) . (1.9) It should be noted that the Kondo eect does not always bring out the increase of the resistance like Eq.(1.7) when lowering the temperature.In fact,the Kondo eect increases the probability for an electron to scatter o the impurity by forming a resonant ground state.Thus,the scattering probability increases when the energy of the scattered electron is close to the Fermi level due to the resonance.For a magnetic impurity in a bulk sample,the scattering probability o the impurity contributes to the resistivity increase as a result of nonspeciﬁc orientation of scattered electrons,as in the conventional Kondo eect. On the other hand,for a impurity in a tunneling barrier splitting two conductors, the increased scattering probability turns into the probability for an electron to tunnel through the barrier,and the dierential conductance thus increases when decreasing the external parameters in Eq.(1.1).These zerobias anomalies in dierential con ductance are also the signature of the Kondo eect,and are well understood in this context [29,30,31]. 1.1.1 Renormalization Group Eq.(1.7) is the leading term of the asymptotic expansion of the resistivity in powers of 1 / ln( T/T K ),and represents the impurity contribution to the resistivity in the leading logarithmic approximation.As explained above,it is a result of summing up the most diverging terms in all orders of perturbation theory.Moreover,it turns out that the Kondo temperature,Eq.(1.8) is a nonanalytic function of J .One 4
needs to come up with a better approach to resolve these issues.For a prescription, the Renormalization Group (RG),which is to be used in later chapters,provides the mathematical framework to study the Kondo eect in this thesis.RG [32] is based on the fact that the main contribution to observable quantities are from the electronic spectrum proportional to the temperature
T about the Fermi level.At temperatures of the order of T K ,when the Kondo eect governs the low temperature physics,the spectrum of relevance becomes much narrower (
T K ) compared to the bandwidth D 0 of the Hamiltonian (1.3). The exchange interaction in Eq.(1.3) induces transitions between the states near the Fermi level and the states near the band edges.Any such transition costs high energy (
D 0 ),and,therefore,can only occur virtually.Virtual transitions via the states near the band edges result in the secondorder correction
J 2 /D 0 to the exchange amplitude J for states in the proximity to the Fermi level.Consider a narrow strip of energies D
D 0 near the band edge.As the strip contains D electronic states,the total correction to the exchange amplitude due to virtual transitions is [32] J
J 2 D/D 0 . This correction is hence reﬂected on the exchange constant in the eective Hamil tonian
H ,which has the same form as the original Kondo hamiltonian (1.3),except that it is deﬁned for reduced energy bandwidth D 0
D , 
k  < D 0
D .The renormalized exchange constant reads J D 0
D = J D 0 + J 2 D 0 D D 0 , (1.10) with J D 0 in the original Hamiltonian. Reducing the bandwidth by inﬁnitesimal D can be considered as a continuous process during which the original Hamiltonian (1.3) with D = D 0 is transformed to the eective Hamiltonian with the bandwidth D
D 0 .From Eq.(1.10),one then 5
gets the dierential scaling equation of the exchange constant [32,33] dJ D d = J 2 D , = ln( D 0 /D ) . (1.11) This form of the scaling equation above is quite common in the Kondo eect up to the second order in J .The solution of the scaling equation with the initial condition J D 0 = J is J D = 1 ln( D/T K ) (1.12) with the scaling invariant T K = D 0 exp(
1 /J ) (the Kondo temperature).The reduction of the bandwidth in RG can be treated as a unitary transformation that decouples the high energy states near the band edges from the rest [34,35,36,37] (this point of view is discussed in the Appendix A).Any such transformation should also aect the operators of the observable quantities.However,the “current” operator is not aected during RG,and evaluation of the conductivity (and resistivity) can be carried out at any stage of RG,yielding the same result. The whole advantage of RG becomes apparent when it is pushed to its limit. The renormalization Eq.(1.11) works until the bandwidth D becomes of the order of the energy scale
T of real transitions.At this termination of RG,the thirdorder correction to the resistivity in Eq.(1.5) is ignorable,whereas the main (secondorder) contribution takes the form
( T ) / (0)
J D
T
2 =
ln( T/T K )
2 , (1.13) consistent with Eq.(1.7) of perturbation theory. 1.1.2 Kondo singlet We next replace the local spin density of itinerant electrons s in Eq.(1.3) with a single spin1/2 operator S
to capture the idea of the Kondo eect and the Kondo singlet in many body systems.The ground state of this toy model of two spins H
= J ( S
· S ) is a spin singlet for the antiferromagnetic case J > 0 (a triplet for the ferromagnetic 6
one J < 0).The excitation energy for a triplet is J .This energy J can be viewed as the binding energy of the singlet. Turning back to the Kondo Hamiltonian,one would expect the analogy of this spin singlet in the Kondo model.However, s (instead of a single spin S
) in Eq.(1.3) is a spin density of itinerant electrons at the impurity site.It is therefore dicult for the impurity to capture an itinerant electron and form a singlet as in the toy model.Nevertheless,RG suggests that even a weak bare exchange constant becomes eectively strong for the electrons close to the Fermi level,see Eq.(1.11),and therefore suces to forma singlet ground state [32,38,39,40] – the Kondo singlet.The binding energy for this Kondo singlet is not the exchange constant J but the exponentially small Kondo temperature T K in Eq.(1.8). It should be noted that the Kondo eect lifts the degeneracy of the ground state. It is the main reason for the logarithmic divergences in perturbation theory,too.By taking J = 0 as usual in perturbation theory,the ground state is doubly degenerate as a spinup and spindown state of the impurity.Then,perturbation theory in J is applicable when the temperature is greater than the binding energy for the Kondo singlet, i.e. , T
T K ,and the result is Eq.(1.7)
( T ) / (0)
ln( T/T K )
2 . In the opposite limit T
T K ,Fermi liquid theory (beyond the scope of this thesis) reads [41], 1
( T ) / (0)
( T/T K ) 2 ,T
T K . (1.14) To summarize,Eqs.(1.7) and (1.14) are valid in the weak ( T
T K ) and strong ( T
T K ) coupling limits,respectively.In addition,the Kondo eect is a crossover phenomenon,unlike a phase transition [32,38,39,40],and thus the resistivity
( T ) / (0) is a smooth function in the crossover region of T
T K . 7
1.2 Coulomb blockade in a quantum dot In recent years,interest in the Kondo eect grew again [42] due to advances in exper imental techniques as well as nanoscale fabrication.Progress in fabrication enables one to design artiﬁcial nanoscale magnetic impurities.Contemporary experimental techniques provide direct access to transport properties of such artiﬁcial impurities as well. The Coulomb blockade [20,21,43,44,45,46] is the key to understanding the nanoscale phenomena,especially quantum transport.The Coulomb blockade were ﬁrst reported in several pioneering experiments [47,48,49,50,51].In a single electron transistor (SET) setup [43,44,45],a quantum dot is connected to two conducting leads L and R via tunneling junctions and is capacitively coupled to the gate in Fig.1.1. L R dot C L C R C g V L V g V R G L G R Figure 1.1: Equivalent circuit for a quantum dot connected to two massive con ducting leads by tunnel junctions with conductances G L,R and capacitances C L,R and capacitively coupled to the gate electrode V g ,adopted from Pustilnik and reprinted with the permission of John Wiley and Sons. The electrostatic energy of a dot with charge Q is classically E ( Q ) = Q 2 2 C
Q C g C V g , (1.15) where C = C L + C R + C g is the total capacitance of the dot,and V g is the potential on the gate,see Fig.1.1.Plugging Q = eN into the energy,where N is the number 8
of excess electrons in the dot,one obtains E ( N ) = E C
N
N 0
2 +const , (1.16) where E C = e 2 / 2 C is the charging energy and N 0 = C g V g /e is the dimensionless gate voltage. 1.2.1 Model of a lateral quantum dot system The simplest Hamiltonian for the dot accounting for the electrostatic energy (1.16) is H d =
ns
n d
ns d ns + E C
ˆ N
N 0
2 . (1.17) Here,the ﬁrst termrepresents the singleparticle (noninteracting) part,and the second term is from Eq.(1.16) after replacing N with the corresponding number operator ˆ N =
ns d
ns d ns . The Hamiltonian (1.17) (known as the Constant Interaction Model) could be justiﬁed microscopically [20,21,46] for dots with no spatial symmetries,which are large compared with the eective Bohr radius a 0 =
2 /e 2 m (here m is the eective mass,and
is the dielectric constant).Both conditions are usually satisﬁed for lateral quantum dots formed by the electrostatic depletion of the twodimensional electron gas at the interface of semiconductor heterostructure such as GaAs/AlGaAs [4,43,44, 45].For a ballistic 2D dot of linear size L ,the capacitance C
L and the mean level spacing between the singleparticle energy levels can be estimated as
2 /mL 2 . Accordingly, E C /
L/a 0
1 . (1.18) For example,for GaAsbased semiconductor quantum dot systems [1,2,4,5,6,7,8, 9,10,43,44,45], a 0
10 nm,and a relatively small dot with L
100 nm contains about 10 electrons.The charging energy of such a dot is of the order E C
1 meV, while the mean singleparticle level spacing
is roughly 10 times smaller.Hence, both the charging eects and the eects associated with the quantization of the 9
singleparticle energy levels can be resolved in transport experiments performed in dilution refrigerators with base temperatures below 50 mK [43,44,45]. The electrostatic potential deﬁning lateral quantum dot systems varies smoothly on the scale of the de Broglie wavelength at the Fermi energy.Dotlead junctions thus act as electronic waveguides with a welldeﬁned number of propagating modes of an electronic wave.The Coulomb blockade emerges when the last propagating mode in each junctions is pinched o.This allows one to model the leads as reservoirs of onedimensional electrons [20,21,46,52,53], H leads =
ks
k c
ks c ks , = R,L (1.19) with the density of states
.Tunneling between the dot and the leads is H tunneling =
kns t
c
ks d ns +H . c ., (1.20) where we neglected the dependence of the tunneling amplitudes on n (see the discus sion in Chapter 2 and Appendix B) without loss of generality,so that each single particle energy level in the dot acquires the same levelwidth
= t 2
due to the tunneling to lead
. The conductance G
of the dotlead junction due to tunneling is G
= (4 e 2 /
)(
/ ) . The tunneling Hamiltonian Eq.(1.20) is valid for an almost closed dot, i.e. ,when G
e 2 /h ,and hence the total width 0 = L + R ,the mean level spacing
,and the charging energy E C establish a welldeﬁned hierarchy in a later quantum dot
0
E C . (1.21) 1.2.2 Charge quantization and Coulomb blockade oscillations Let’s consider an isolated dot ( t
0) where the number of electrons N in the dot is a good quantum number at low temperatures.The electrostatic energy to add an 10
electron to the dot is E N +1
E N = 2 E C
N
0
N 0
,N
0 = N +1 / 2 = halfinteger . From this,one sees that the N and N + 1 electron states are degenerate at each N 0 = N
0 = halfinteger.At low enough temperatures T
E C ,the average number of electrons N ( N 0 ) =
ˆ N
over the dimensionless gate N 0 should be staircaselike as in Fig.1.2(a) with the stepwidth
T/E C .These regions in the vicinity of the degeneracy within are called the mixedvalence regions.In other words,the charge quantization is expected over the entire region of the gate voltage except for the mixedvalence regions.The charge quantization remains intact at low temperatures, even when the tunneling is introduced.At very low temperatures T
,however,the stepwidth
/E c comes with the renormalized levelwidth
0 (see Chapter 2) instead of the temperature T ,and hence the width of the mixedvalence regions is renormalized as well.
Figure 1.2: (a) Average number of electrons in a dot N =
ˆ N
at T
E C as a function of the dimensionless gate voltage N 0 .The number of electrons N diers signiﬁcantly from integer values in the narrow mixedvalence regions of the width
max {
,T } /E C about N 0 = N
0 = halfinteger. (b) At
T
E C the conductance is small in the wide Coulomb valley of the width of almost 1 in the dimensionless gate voltage due to the Coulomb blockade. Adopted from Pustilnik and reprinted with the permission of John Wiley and Sons. The charge quantization in the dot translates into the conductance G through the dot.At high temperature T
E C ,the Coulomb interaction in Eq.(1.17) (and hence the gate voltage dependence) has no eect since the thermal excitation is big enough to wipe out the staircaselike charge quantization.The conductance in this high T 11
limit is small, G
e 2 /h ,and the classical resistance addition formula gives 1 G
= 1 G L + 1 G R . (1.22) Things change dramatically at low temperatures.The conductance starts to de pend on the gate voltage N 0 at all T
E C .When the gate voltage is outside the mixedvalence regions,in Fig.1.2(a),adding or removing an electron costs approx imately the charging energy E C in Eq.(1.17).From the energy conservation for a real transition,the probability to have an electron with energy E C is proportional to exp(
E C /T ) outside the mixedvalence regions.That is,the conductance is ex ponentially suppressed at T
E c (Coulomb blockade valley,hereafter CB valley). On the other hand,the energy cost in the mixedvalence regions is much smaller due to the degeneracy,and the conductance is relatively large in these regions (Coulomb blockade peak,hereafter CB peak) In the low T limit, i.e. ,at T
E C ,the conductance G ( N 0 ) displays a quasi periodic behavior of narrow CB peaks of the width
1 accompanied by wide CB valleys in Fig.1.2(b).In terms of the dimensionless gate voltage N 0 ,the spacing between two neighboring CB peaks
1 in Fig.1.2(b). Thorough study on the Coulomb blockade was done in the light of the orthodox theory by Shekhter [54,55,56,57,58].The orthodox theory is based on the rate equation formalism at T
(Numerical approach to solve the rate equations of a quantum dot is shown in Bonet [59]).The orthodox theory assumes that the inelastic electron relaxation rate in a dot is large compared with the electron escape rate /
. In this approximation,the tunneling via each junction through the dot can be treated as an independent process. According to the orthodox theory,the Coulomb blockade peaks saturate to half of their hightemperature conductance G
when decreasing T .However,the dis creteness of the energy levels becomes more relevant at T