Theoretical Reproduction of the Lineshapes of Nonlinear Conductance Observed in a Quantum Point Contact

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by  Jongbae Hong

Theoretical Reproduction of the Lineshapes of Nonlinear Conductance Observed in a
Quantum Point Contact
Jongbae Hong∗
Max-Planck-Institut fu¨r Physik Komplexer Systeme, D-01187 Dresden, Germany
Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
(Dated: July 14, 2010)
The nonlinear conductance observed in a quantum point contact is theoretically reproduced for
theentirerangeofappliedbias. Thesingle-impurityAndersonmodelwithtworeservoirsatdifferent
0
chemicalpotentialsis studiedforasequentialchangeof thegatevoltage. Theimbalancein theleft
1
andrightKondocouplingstrengthisintroducedbythedisplacementoftheKondoimpuritybythe
0
electricfieldproducedintheconstrictionofquantumpointcontact. Werevealtheoriginoftheside
2
peaksand studythe behaviorof theheight and width of thezero-bias anomaly.

l
u
J
Understanding the nonlinear conductance in a system temperature is given by
3
with nanocontact is indispensable in the study of na-
1 dI e2
noelectronics or nanomagnetism. The Kondo effect is = Γtot ρ (eV), (1)
] often involved in studying its transport. A typical ex- dV ¯h Xσ iσ
l ample of nanocontact is quantum point contact (QPC)
l where e, σ, and ¯h denote electronic charge, spin, and
a or quantum wire [1–4] in which an interesting sequential
h the Planck constant divided by 2π, respectively. Here,
change of the dI/dV, where I and V denote the current
- ρiσ(eV) is the local density of states of the Kondo im-
s and the bias, respectively, lineshape is observed with an purity at bias V. The factor Γtot is given by the com-
e
m increase in the gate voltage VG. However, theoretical bination of featureless coupling functions, i.e., Γtot =
understanding for the observednonlinear conductance is ΓLΓR/(ΓL + ΓR) [7], where the superscripts L and R
t. insufficientandunsatisfactory. Thepurposeofthisstudy standfor leftandrightreservoirsofaQPC,respectively.
a istopresentthetheoreticaldI/dV lineshapesthatshowa
m We omit the term containing ∂ρiσ(ω)/∂V because its
similarsequentialchangeofthedI/dV withV totheone
G contribution to dI/dV is negligible. We show its neg-
- reported in Ref. [4]. Our theoretical lineshape remark-
d ligence in a different study.
ablyfitstheexperimentaldatafortheentirerangeofthe
n The spectral function of an up-spin electron, for ex-
o appliedbiasusedintheexperiments. Fromthisanalysis, ample, at the Kondo impurity is given by the real part
c one may understand the physics of electron transport in of the inverse of the matrix M [8], i.e., ρ (ω) =
[ aanmdessiodsecoppeiackssysatpepmeainrwinhtichhealizneersoh-abpiaesoafntohmeanlyon(ZlinBeAar) (1/π)Re[(M−r1)33], where r i↑
2
v conductance.  −iω′ −γLL UJL− γLR γJ− 
67 Inordertoexplaintheobservednonlinearconductance M = γULLL∗ −UiωL′∗ UJiLω+′ γUJ−R∗ γULRR∗ . (2)
.13 isnysatemQPwCit,hthaemterdainastpinogrtKmonecdhoainmispmuriintyamtuwsot-breeseinrvveosir- r −−−γγLJJ+R− −−−γγJLJ+R+ −UUJJRR+− −−−γiRωJR+′ −−γRiωJR−′ 
2 tigated when the system is under steady-state nonequi-
0 librium(SSN).Theissuesofthisstudyaretwo-fold: two- Here, ω′ ω ǫi U ni↓ , where ǫi, U, and ni↓ de-
≡ − − h i h i
0 note the energy level of the Kondo impurity, Coulomb
reservoir and SSN. The former duplicates all basis vec-
1 interaction, and the average number of down-spin elec-
torsdescribingthebackandforthmovementsoftheelec-
: trons occupying the level ǫ , respectively. All the matrix
v tron. For this reason, additional resonant tunneling lev- i
i els, which are the origin of the side peaks, occur. The elements, except the eight U-elements, have additional
X self-energyterms,β [iΣL(ω)+iΣR(ω)]=2β ∆,fora
latter shifts the Fermi level of the reservoir and allows mn 0 0 mn
r flatwideband,where∆ (ΓL+ΓR)/4andΓR VR 2,
a only unidirectionalresonanttunneling if the systemis in whereVRdenotesthehyb≡ridizationstrengthbet∝we|enkith|e
the groundstate. This unidirectionalresonanttunneling ki
Kondo impurity and the right reservoir [7]. The coeffi-
makesthestateofSSNdifferentfromthatofequilibrium.
cient β is given in [9]. We use ∆ to indicate the unit
mn
of energy.
We obtain the formula for dI/dV by differentiating
The matrix elements representedby γ are givenby [8]
the currentformula providedin Refs. [5] and[6] with re-
spect to the bias. We fix the chemical potential of the γ = i(V∗cL +V∗cR )c† [j−L,j+L]
Kondo impurity at the Fermi level of one of the metal- LL hX ki k↑ ki k↑ i↑ i↓ i↓ i
k
lic reservoirs. Then, the differential conductance at zero
[ (δj−L)2 (δj+L)2 ]−1/2,
× h i↓ ih i↓ i
γ = i(V∗cL +V∗cR )c† [j∓L,j∓R]
∗Electronicaddress: [email protected] J∓ hX ki k↑ ki k↑ i↑ i↓ i↓ i
k
2
[ (δj∓L)2 (δj∓R)2 ]−1/2,
× h i↓ ih i↓ i
and
γ = i(V∗cL +V∗cR)c† [j−L,j+R]
LR hX ki k↑ ki k↑ i↑ i↓ i↓ i
k
[ (δj−L)2 (δj+R)2 ]−1/2, FIG. 1: Motions of electrons indicated in γLR (+ sign) and
× h i↓ ih i↓ i γJ (−sign). GreensolddotindicatesaKondoimpurity. The
motiondenotedby(cid:13)2 inredperformsexchangeprocessalong
where the operator ck(i) denotes the fermion annihi- with (cid:13)2 in black or singlet hoppingwith (cid:13)2 in blue.
lation at the state-k(i) in the lead (impurity), j− =
i↓
i (V c† c V∗c† c ), j+ = (V c† c +
Pk ki i↓ k↓ − ki k↓ i↓ i↓ Pk ki i↓ k↓
V∗c† c ), and δ means the deviation from the average. impurity is formed spontaneously at the center or a dis-
ki k↓ i↓
placed position in the QPC constriction at a low V .
The element UL , on the other hand, is given by G
J∓ Whenabiasisapplied,anelectricfieldwillbecreatedin
the constriction and the magnetic impurity will be dis-
UL = iU h[ni↓,ji∓↓L](1−2ni↑)i+(1−2hni↓i)hji∓↓Li. placed to the direction opposite to the electric field, as
J∓ 2 (δj∓L)2 1/2 shown in Fig. 2 (a). As a result, one has γ > γ .
h i↓ i LL RR
As V increases, the magnetic impurity will move back
G
One may roughly understand the roles of matrix ele- to the center of the constriction because the number of
mentsasfollows: γ representsthedegreeofKondo electrons in the lead increases, as shown in Fig. 2 (b)
LL(RR)
coupling; γLR and γJ represent connecting mechanism; (γLL >γRR). When the magnetic impurity reaches near
The real part of U-elements determines the position of thece∼nteroftheconstriction,weassumethatγ =γ
LL RR
Coulomb peak, whereas the imaginary part gives rise to is established, as shown in Fig. 2 (c). Upon reaching the
the asymmetry in the dI/dV lineshape. A symmetric center, the magnetic impurity stays there even at a high
lineshape is obtained when hniσi=1/2. VG. However, both γLL and γRR increase continuously
The spectralfunctionρi↑(ω)givenbythe reducedma- as VG increases further.
trix of Eq. (2) has five peaks in general. The posi- According to the scenario discussed above, we deter-
tions of five peaks can be obtained from the zeros of mine the specific manner in which the matrix elements
the determinant of Mr in the atomic limit: a Kondo (UL,R and γ) and Γtot change with V . One can see
peak at ω′ = 0, two Coulomb peaks at ω′ U/2, J± G
≈ ± that the real part Re[UL,R] is composed of the corre-
and two additional resonant tunneling level peaks at J±
lation in numerator and the fluctuation in denomina-
ω′ = [(γ2 +γ2 )/2+(γ γ )2 +O(U−2)]1/2 for
± LL RR LR − J tor, i.e., Re[UL,R] = CL,R/FL,R. Since fluctuation is
a large U. One can also find the spectral weight of each J± U± U±
proportional to the number of particles N , where N
peak. It is interesting to note that the additional tun- c c
is the number of conduction electrons in a QPC, and
neling level can have a meaningful amount of spectral
N V [12], we obtain FL,R V1/2. We propose a
weight only when a bias is applied (γJ = 0). Therefore, c ∝ G U± ∝ G
we call the peak at the additional tun6 neling level the specific expression that shows the dependence of FL,R
U±
steady-state resonant tunneling level (SSRTL) peak. on V : FL,R =p√V q. The constants p and q are so
We havediscussedthe effectofbias,whichallowsonly deterGmineUd±that FL,GR−= 1 for V = 1 and FL,R = 6 for
unidirectional resonant tunneling at the ground state, U± G U±
V = 17, as shown in the first two columns of Table I.
G
above. InFig.1,wepresentthegraphicalillustrationsof
ThevaluesofV indicatethestepsinwhichV isvaried.
G G
γ and γ on the basis of the corresponding operator
LR J Inthe regionwhereγ >γ (Figs.2(a)and2(b)),
expressions,whicharethirdorderprocessesofhybridiza- LL RR
one must consider L R unbalance in the strength of
tion. Bothexchange(blackorred(cid:13)2)andsinglethopping Kondo coupling. Only−CL,R that includes j+L,R would
(blue or red 2) processes in equilibrium are depicted in U+ i↓
(cid:13) reflect the L R unbalance. We consider linear changes
Fig.1. Appliedbiasmakesthe secondpartofFig.1neg- in V for CL−,R up to V =9. For V >9, however, γ
ligible when we calculate the expectation value. Thus, G U+ G G LL
the condition γLR >γJ is obtained. Note that only sin- and γRR are balanced and CUL+ = CUR+ = 1. We choose
glet hopping proce∼ss can establish current. Therefore, VG = 9 as a border separating the L R balanced and
−
the unidirectional resonant tunneling (without black 2 unbalanced region. We also consider linear relationships
on the left part of Fig. 1) dominates the Kondo proce(cid:13)ss for γLL(RR) and Γtot. We set γLR = γJ for the unbal-
under bias. In this study, we do not calculate the values anced region, i.e., VG 9. However, in the balanced
≤
of γ. We will use them as free parameters within the region, VG > 9, we consider nonvanishing exchange pro-
condition γLR >γJ. cessinthesecondpartofFig.1andallowγLR toincrease
In order to ∼perform the calculation for the spectral further.
function, we must know what happens in the constric- We first fix the values of matrix elements of M for
r
tion of a QPC when V changes. Calculations by spin V = 1 as shown in the lowermost row of Table I that
G G
density functional theory [10, 11] show that a magnetic afford the lineshape similar to the one obtained experi-
3
FIG. 2: Location of a magnetic impurity (black dot) under
bias. The strength of Kondo coupling is represented by the
line thickness. (a) Low VG: γLL > γRR. Green dots denote
electronsintheleads. (b)VG beforebalance: γLL>∼γRR. (c)
VG after balance: γLL=γRR. FIG. 3: (a) Sequential change in the dI/dV lineshape when
VG changes from 1 to 17. The curve in red (VG = 9) is
Table I: Change in parameters with VG (γJ =0.5) theborderthatcorrespondstothe0.7conductanceanomaly.
(b) Comparison of theoretical data (red line) with the ex-
VG FUL±,R CUL+ CUR+ γLL γRR γLR Γ perimental data (crosses) of the closest-spaced side peaks in
17 6 1 1 0.6 0.6 0.70 e1 Fig. 1 (b) of Ref. [4]. Weak asymmetry is given by setting
15 2.63 1 1 0.54 0.54 0.65 8.5/9 Im[UJL+] = 0.05. We set β11 = 0.252, β12 = β21 = 0.246,
β22 =0.256, and otherβmn =0.25.
13 1.92 1 1 0.48 0.48 0.60 8.0/9
11 1.59 1 1 0.42 0.42 0.55 7.5/9
9 1.39 1 1 0.36 0.36 0.50 7.0/9
7 1.25 1.4 0.8 0.42 0.30 0.50 6.5/9
5 1.15 1.8 0.6 0.48 0.24 0.50 6.0/9
3 1.08 2.2 0.4 0.54 0.18 0.50 5.5/9
1 1 2.6 0.2 0.6 0.12 0.50 5.0/9
mentallyforthe lowestV . Thesequentialchangeofthe
G
dI/dV lineshape with V is obtained by using the val-
G
uesinTableIthataredeterminedbytheV -dependence
G
discussedabove. Welistthevaluesonlyforoddnumbers
of V in Table I. Figure 3 (a) shows the characteristics
G
of the sequential change of the nonlinear conductance in FIG.4: Characteristics ofZBA.(a) ZBAheight (∆hZBA)vs.
a QPC, such as the change of ZBA and the behavior of Gmax. (b) Full width at half maximum (FWHM) of ZBA vs.
the side peak with an increase in VG. One can observe Gmax. The dots are obtained from Fig. 3 (a) and the crosses
two distinct groups of lineshapes, denoted by different arethedataofRef. [4]forVtop =+4V.Theunitsof∆hZBA
and G are 2e2/h. Inset: Conductance vs. gate voltage.
colors. As V increases, the SSRTL peaks first move in-
G
wards,andthenmoveoutwards. ThebehaviorofSSRTL
changesatV =9, atwhich pointthe two SSRTL peaks
G
are closest in bias. The values listed in Table I depend shown in Fig. 3 (b).
ontop-gatevoltageandsamplecharacteristics. Different The ZBA peaks shown in Fig. 3 (a) provide us with
sampleortop-gatevoltageaffordsdifferenttableandthe another interesting comparison with the experiment. In
corresponding set of dI/dV lineshapes. Fig. 4, we present the height and width of ZBA vs.
Aremarkablecomparisonwithexperimentisshownin G , the maximum value of the corresponding ZBA,
max
Fig.3(b). WechoosetheexperimentaldI/dV curvethat along with the experimental data reported in Ref. [4]
is the one with the closest-spacedside peaks ofFig 1 (b) for V = +4 V. Our theoretical data given in Fig. 3
top
of Ref. [4] for comparison. Our theoretical lineshape is (a) reproduce the ZBA characteristics quite well. One
obtained by setting γLL = γRR = 0.5, γJ = γLR = 0.65, may understand the peculiar behaviors shown in Fig. 4
Re[UL,R]=1.17,Im[UL ]=0.05,andΓtot =0.62. Since in terms of the competing relationshipbetweenthe ZBA
J± J+
the experimental curve is slightly asymmetric and the and the SSRTL peaks as V changes. In the region
G
peakisalittlebitoffsetfromV =0,weintroduceasmall V < 9, increasing V has two effects, i.e., recovering
G G
nonvanishing Im[UL ]. We discuss about asymmetry of the L R symmetry of Kondo coupling strength and
J+ −
the lineshape more below. Perfect fitting is obtained, as enhancing the Kondo peak by suppressing the fluctua-
4
tions (δj∓L,R)2 1/2. These two effects are conflicting in [1] S. M. Cronenwett, H. J. Lynch, D. Goldhaber-
h i↓ i
determiningthe ZBApeakbecausethe formercausesin- Gordon, L. P. Kouwenhoven, C. M. Marcus, K.
wardmovementoftheSSRTLpeakswithtakingspectral Hirose, N. S. Wingreen, V. Umansky, Phys. Rev.
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G
to a local maximum of ∆h in the region V < 9 in
ZBA G
[2] L. DiCarlo,Y. Zhang,D. T. McClure, D. J. Reilly,
Fig. 4 (a). However, in the regime V > 9 in which the
G
C. M. Marcus, L. N. Pfeiffer, and K. W. West,
L Rsymmetryisretained,onlyoneroleofV ,enhanc-
− G Phys. Rev. Lett. 97, 036810(2006).
ingtheKondopeak,remains. Interestingly,however,the
unitary limit of conductance prevents the height of ZBA
[3] F. Sfigakis,C. J.B. Ford, M. Pepper, M. Kataoka,
from increasing further. As a result, another local maxi-
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increase in the width of ZBA from V = 9, as shown in
G [4] S.Sarkozy,F.Sfigakis,K.Das Gupta,I.Farrer,D.
Fig.4(b),isalsoresultedfromthisunitarylimit. Wealso
A. Ritchie, G. A. C. Jones, and M. Pepper, Phys.
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(b). A slight curve bending is shown near 0.7(2e2/h).
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inary parts of UL,R do not vanish. Since Im[UL,R] 2512 (1992).
J∓ J∓ ∝
(1/2 n )asshownabove,theasymmetryinthe line-
iσ
−h i [6] S. Hershfield, J. H. Davies, and J. W. Wilkins,
shape indicates the existence of spin polarization [13] in
Phys. Rev. B 46, 7046 (1992).
the magnetic impurity, i.e., n = n . Therefore,the
i↑ i↓
h i6 h i
Kondoeffectandspinpolarizationmaynotbeincompat-
[7] H. Haug and A.-P. Jauho, Quantum Kinetics in
ible. In order to identify the origin of spin polarization,
Transport and Optics of Semiconductors(Springer,
adeeperanalysisforthebehaviorofmagneticimpurities
Berlin, 1996).
createdinthe constrictionofaQPC.Recently,Songand
Ahn[14]proposedapossibilityofferromagneticcoupling [8] J. Hong, Preprint at
between the magnetic impurities. An extended Ander- <http://arxiv.org/cond-mat/1007.0615> (2010).
son model considering the ferromagnetic coupling may
be suitable for explaining both the Kondo and spin po- [9] The coefficients βmn are β11 = |ξi−L|2, β22 =
larization simultaneously. ξ+L 2, β = 1, β = ξ+R 2, β = ξ−R 2,
| i | 33 44 | i | 55 | i |
In conclusion, we obtained the dI/dV lineshapes ob- β = ξ+L∗ξ+R, β = ξ−L∗ξ+L, β = ξ−L∗ξ+R,
24 i i 12 i i 14 i i
served in a QPC in terms of the Anderson model with β = ξ−L∗ξ−R, β = ξ+L∗ξ−R, and β =
15 i i 25 i i 45
tworeservoirs. WeidentifiedthesidepeaksofthedI/dV ξ+R∗ξ−R, where ξ∓L, for example, is given by
as the SSRTL peaks. We showed that the sequential ξi∓L =iUL /[U (δni )2 1/2].
change ofthe dI/dV lineshape with V is separatedinto i J∓ h i↓ i
G
two parts (Figs. 3 (a)) and the border corresponds to [10] T. Rejec and Y. Meir, Nature 442, 900 (2006).
the point of 0.7 anomaly (Inset of Fig. 4 (b)) at which
the curve characteristics of ZBA height and width are [11] S. Ihnatsenka and I. V. Zozoulenko, Phys. Rev. B
apparently different (Figs. 4 (a), (b)). We also identi- 76, 045338(2007).
fied that the strengths of the L R Kondo coupling are
− [12] D. J. Reilly, Phys. Rev. B 72, 033309 (2005).
just balanced at the point of 0.7 anomaly (Table I and
Fig.1(c)). Thetheoreticalskillusedinthisstudycanbe [13] L. P. Rokhinson, L. N. Pfeiffer, and K. W. West,
extended to other mesoscopic systems with a mediating Phys. Rev. Lett. 96, 156602(2006).
Kondo impurity between two reservoirs under bias.
This work was supported by Korea Research Founda- [14] T. Song and K.-H. Ahn, unpublished.
tion Grant No. KRF-2008-314-C00140.