Quantum-limited shot noise and quantum interference in graphene based Corbino disk

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by  Grzegorz Rut

Quantum-limited shot noise and quantum interference in graphene based Corbino disk
Grzegorz Rut and Adam Rycerz
Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, PL–30059 Krako´w, Poland
(Dated: January 31, 2014)
This is a theoretical study of finite voltage effects on the conductance, the shot noise power,
andthethirdcharge-transfercumulantforgraphene-basedCorbinodiskinthepresenceofexternal
magneticfields. Periodicmagnetoconductanceoscillations,predictedinRefs.[1,2],becomeinvisible
forrelativelysmallsource-drainvoltages,asthecurrentdecaysrapidlywithmagneticfield. Quantum
interference still governs the behavior of higher charge-transfer cumulants.
4 PACSnumbers: 72.80.Vp,73.63.b,75.47.-m
1
0
2 I. INTRODUCTION with σ = (4/π)e2/h being the universal conductivity
0
of graphene. Analogous behavior is predicted for higher
n
a The Corbino geometry, in which electric current is charge-transfer cumulants [2, 16].
J passed through a disk-shaped sample area attached to Inthispaper,weextendtheanalysisbeyondthelinear-

0 two circular leads [see Fig. 1(a), the inset], was pro- responseregimebycalculatingtheconductance,theFano
3 posed over a century ago [3, 4] to measure the magne- factor quantifying the shot-noise power, and -factor
F R
toresistancewithoutgeneratingtheHallvoltage,making quantifying the third charge-transfer cumulant, in a sit-
]
l a significant step towards understanding the nature of uation when finite source-drain voltage is applied to
l
a charge transport in ordinary solids [5]. An interest in graphene-basedCorbinodiskintheshot-noiselimit. Our
h suchageometryhasreappearedduetothefabricationof resultsshowthatalbeittheconductanceoscillationsvan-
-
GaAs/AlGaAsheterostructures[6,7]. Also,afterthedis- ish rapidly with the voltage and magnetic field, and
s
F
e covery of high-temperature superconductivity, Corbino still oscillate periodically and their mean values ap-
m R
measurements have provided a valuable insight into the proach 0.76 and 0.55 (respectively) in the
. vortex dynamics, as the influence of sample edges was high-fielFd∞lim(cid:39)it. In the rRem∞ai(cid:39)ning parts of the paper, we
t
a eliminated [8]. first(inSectionII)recallbrieflytheformulaallowingone
m In the context of graphene, various transport proper- to determine the conductance and other charge-transfer
- ties of Corbino disks were recently studied experimen- characteristics of graphene-based Corbino disk at arbi-
d
tally [9–11] and theoretically [1, 2, 12]. Next to the case trary voltages and magnetic fields. Next, in Section III,
n
of a rectangular strip geometry [13, 14], the Corbino ge- our numerical results are presented in details. The con-
o
ometry provides another situation when transport prop- clussions are given in Section IV.
c
[ erties of a graphene nanodevice can be investigated ana-

lytically [1], by solving the scattering problem for Dirac
2
fermions, at arbitrary dopings and magnetic fields. In
v
7 particular,mode-matchinganalysisfortheeffectiveDirac II. CHARGE-TRANSFER CUMULANTS
4 equationgivestransmissionprobabilitiesforundopeddisk
2 in graphene monolayer [1, 2]
Intheshot-noiselimiteV k T,withV beingthe
7 eff B eff
(cid:29)
1. T(0) = 1 , (1) effective source-drain voltage [17], the charge Q passing
j cosh2[(j+Φ/Φ )ln(R /R)] a nanoscale device in a time interval ∆t is a random
0 0 o i
variable, with the characteristic function Λ(χ) given by
4
where j = 1, 3,... is the angular-momentum quan- the Levitov formula [18]
1 ±2 ±2
tum number labeling normal modes in the leads, Φ =
:
v π(R2 R2)B is the flux piercing the disk in the uniform
Xi magone−ticifield(B),Ri istheinnerradius,Ro istheouter lnΛ(χ)≡ ln(cid:104)exp(iχQ/e)(cid:105)=
ar rdaedriivuast,ioanndofΦE0q.=(1)2,(thh/eel)imlni(tRoof/hReia)v.ilyM-doorpeoevdegr,raipnhetnhee 4 ∆t µ0+(cid:90)eV2effd(cid:15) (cid:88)ln(cid:2)1+(cid:0)eiχ 1(cid:1)T ((cid:15))(cid:3), (3)
leads [13] is imposed. Summing Tj-s over the normal (σ,v) h − j
mcoonddeuscitnantchee,lienadtsh,eonlienefianrd-rsetshpaotnstehereLgainmdea,usehro-Bwu¨stptiekreir- µ0−eV2eff j
odic(approximatelysinusoidal)oscillationswiththeflux
where X denotes the expectation value of X, the fac-
Φ, with Φ0 being the oscillations period. Additionally, tor 4 (cid:104) (cid:105)accounts for spin and valley degeneracies, and
the disk conductance averaged over a single period re- (σ,v)
we have assumed V > 0 without loss of a generality.
stores the pseudodiffusive value [15] eff
The average charge Q , as well as any charge-transfer
(cid:104) (cid:105)
2πσ cumulant Qm (Q Q )m , may be obtained by
0
Gdiff = ln(R /R), (2) subsequent(cid:104)(cid:104)diffe(cid:105)r(cid:105)e≡nti(cid:104)ation−o(cid:104)f l(cid:105)nΛ((cid:105)χ) with respect to iχ
o i
2
at χ=0. In particular, the conductance where M(a,b,z) and U(a,b,z) are the confluent hyper-
geometric functions [19, 20].
(cid:12)
Q e ∂lnΛ(cid:12) It can be shown that in the zero-energy limit ((cid:15) 0)
G(Veff)= V(cid:104) ∆(cid:105)t = V ∆t ∂(iχ)(cid:12)(cid:12) Eq. (7) simplifies to Eq. (1). Similarly, in case th→e en-
eff eff χ=0
ergy is adjusted to a higher Landau level (LL), namely,
4(σ,v)e2 (cid:88)(cid:10)T (cid:11) , (4) (cid:15)˜2/(4β) = n = 1,2,..., the transmission probability for
≡ h j j |(cid:15)−µ0|(cid:54)eV2eff j-thnormalmode(inthehigh-fieldlimit)isT(n) =T(0)
j j 2n
[1]. In effect, periodic magnetoconductance oscillati−ons
where transmission probabilities T ((cid:15)) are averaged over
the energy interval (cid:15) µ (cid:54)eV j/2. Analogously, inthelinear-responseregimearefollowedbysimilaroscil-
| − 0| eff lations of and (for analytic Fourier decompositions,
F R
(V )= Q2 (cid:14) Q2 see Ref. [16]), with the mean values
eff Poisson
F (cid:104)(cid:104) (cid:105)(cid:105) (cid:104)(cid:104) (cid:105)(cid:105)
(cid:80) (cid:10) (cid:11)
T (1 T ) =1/3 and =1/15, (12)
= j (cid:80)j (cid:10) −(cid:11) j |(cid:15)−µ0|(cid:54)eV2eff (5) Fdiff Rdiff
T
j j |(cid:15)−µ0|(cid:54)eV2eff providedthediskisundopedorthedopingisadjustedto
anyhigherLL.Thesearethebasicfeaturesofanonstan-
and
dard quantum interference phenomenon, which may ap-
(V )= Q3 (cid:14) Q3 pear when charge transport in graphene (or other Dirac
R eff (cid:104)(cid:104) (cid:105)(cid:105) (cid:104)(cid:104) (cid:105)(cid:105)Poisson system) is primarily carried by evanescent modes [21].
(cid:80) (cid:10) (cid:11)
T (1 T )(1 2T )
= j (cid:80)− (cid:10)j (cid:11) − j |(cid:15)−µ0|(cid:54)eV2eff , (6)
T
j j |(cid:15)−µ0|(cid:54)eV2eff III. RESULTS AND DISCUSSION
with Qm the value of m-th cumulant for the
Poisson
(cid:104)(cid:104) (cid:105)(cid:105) Several factors, not taken into account in the above
Poissonian limit (T ((cid:15)) 1), given by a generalized
j
Schottky formula Qm (cid:28) =em 1 Q . analysis, may make it difficult to confirm experimentally
Poisson −
(cid:104)(cid:104) (cid:105)(cid:105) (cid:104) (cid:105) the effects which are described in Refs. [1, 2]. These in-
Inthecaseofgraphene-basedCorbinodisk,theenergy-
cludetheinfluenceofdisorder,electron-phononcoupling,
dependent transmission probabilities are given by [1]
or electron-electron interactions; i.e., the factors which
16((cid:15)˜2/β)2j 1 (cid:20)Γ(γ )(cid:21)2 areabsent(orsuppressed)inseveralanalogsofgraphene
| − | j
Tj((cid:15))= (cid:15)˜2RR (X2+Y2) Γ(α ↑) , (7) [22–24], and which are beyond the scope of this paper.
i o j j j↑ Anotherpotentialobstacleisrelatedtothefactthatres-
where (cid:15)˜ = (cid:15)/((cid:126)vF) with vF 106m/s the energy- onances with distinct LLs shrink rapidly with increasing
independent Fermi velocity, β =(cid:39) eB/(2(cid:126)), Γ(z) is the field, making the linear-response regime hard to access.
Euler Gamma function, and Here we point out that theoretical discussion of charge
transport through graphene-based Corbino disk still can
1(cid:20) (cid:15)˜2 (cid:21) be carried out, in a rigorous manner, beyond the linear-
α = 2(j+m + j m +1) ,
js s s
4 | − | − β response regime.
Forthepurposeofnumericaldemonstration,wechoose
γ = j m +1, (8)
js s
| − | R /R = 5, and focus on the vicinity of the Dirac point
o i
with ms = ±12 for the lattice pseudospin s =↑,↓. The by setting µ0 = 0 [25]. The corresponding oscillation
remaining symbols in Eq. (7) are defined as magnitudes, in the linear-response limit, are [16]
Xj =wj−↑↑+zj,1zj,2wj−↓↓, ∆G(Veff →0)=0.11Gdiff, (13)
Yj =zj,2wj+ −zj,1wj+ , (9) ∆F(Veff →0)=0.27, ∆R(Veff →0)=0.14.
↑↓ ↓↑
where For any finite V and any flux Φ, the averages in Eqs.
eff
(4), (5), and (6) can be calculated numerically after sub-
wj±ss(cid:48) =ξj(1s)(Ri)ξj(2s(cid:48))(Ro)±ξj(1s)(Ro)ξj(2s(cid:48))(Ri), stituting Tj((cid:15)) given by Eq. (7). Our main results are
zj,1 =[2(j+sj)]−2sj, (10) presented in Figs. 1, 2, and 3.
First, in Figs. 1(a)–(c), we have depicted the values
z =2(β/(cid:15)˜2)sj+1/2,
j,2 taken by G(V ), (V ), and (V ), when the flux
eff eff eff
F R
is varied in separate intervals, each of which having Φ
with s 1sgn(j). The functions ξ(1)(r) and ξ(2)(r) in 0
j ≡ 2 js js width, namely
the first line of Eq. (10) are given by
(m 1)Φ Φ m Φ , m =1,2,.... (14)
Φ 0 Φ 0 Φ
ξj(νs)(r)=((cid:15)˜r)|j−m(cid:26)s|eMxp((α−β,rγ2/2,)βr2), if ν=1, The two−shaded≤area≤s are for mΦ = 3 and mΦ = 7;
js js (11) distinctsolidline(ateachpanel)depictsthecorrespond-
× U(αjs,γjs,βr2), it ν=2, ing charge-transfer characteristic at Φ = 0. It is clear
3
(a) (a)
1.5
4.0 Ri Ro � = 0 eVe↵Ri/(~vF)=0
h] 3 ]
2e/ �/� 0  2/h 1.0
4 e
[ 2  4 2
G 2.0 6�/�07 G[ 0.5 0.25 0.05 0.01 ·10�3
1.25
0 0
5
(b) 6�/�07 (b) =1.2
0.8 Ri
↵
0.6 eVe ~vF 0.25 5 03�
0 1
F 2�/ F 0. 0.01 2·
�
0.4 0 0
3
0.4
�=0
0 0.2
(c) (c)
0.8
0.6
6�/�07 = 1.25
R i
R 0.4 eVe↵~v F 0.25 5
R 0 3
0.4 0. 1 0�
0 1
0. 2·
0.2
2
0
�/
�
0
0 �=0 3 0
0 2 4 6 8 0 2 4 6 8
eVe↵Ri/(~vF) �/�0
FIG. 1: Variation ranges for the finite-voltage conductance FIG. 2: Magnetic flux effect on the finite-voltage conduc-
(a), Fano factor (b), and R-factor (c) in cases the magnetic tance (a), Fano factor (b), and R-factor (c). The effective
flux Φ piercing the disk area R <r <R [see inset in panel source-drain voltage V is specified for each curve.
i o eff
(a)]isvariedinthelimitsgivenbyEq.(14)withm =3and
Φ
m =7. The values for Φ=0 are also shown.
Φ
data presented in Fig. 2, where the conductance and
other charge-transfer characteristics are plotted directly
fmroamgneFtiigc.fi1eldthpartovGid(eVdefft)haist estVreoffn(cid:46)gly(cid:126)vsFu/pRprie.sFedorbhyightheer aGs(Vfueffn)ctdioencsayosf rΦe,laftoivreslyelefcatsetdwviatlhueΦs ofofrVaeffn.y AVelfftho=ug0h,
Veff the ballistic transport regime is entered, leading to suchthatmagnetoconductanceoscillationsarevisibl(cid:54)efor
GeV(eVffeff(cid:29)) ∝(cid:126)vFV/effR,i Flim(Vite.ff)M(cid:46)ost0.r2e,maanrdkaRbl(yV,efffo)r 0(cid:39)<0eiVneffth(cid:46)e sehVoeffw(cid:28)per(cid:126)ivoFdi/cRoisocinlllayti[osenesFatigh.i2g(ha)fi]e,lFds(Vfoerff)arabnidtraRry(VVeffeff)
(cid:126)vF/Ri andthehighestdiscussedfluxinterval(mΦ =7), [see Figs. 2(b) and 2(c)]. In order to describe these os-
(Veff) and (Veff) take the values from narrow ranges cillations in a quantitative manner, we have calculated
F R
around 0.7 and 0.5 [see Figs. 1(b) and 1(c)], numerically the average values of (V ) and (V ),
coincidinFg(cid:39)with recentRfin(cid:39)dings for transport near LLs in as well as the corresponding oscillaFtioneffmagnituRdes,efffor
graphene bilayer [26]. severalconsecutivefluxintervalsdefinedbyEq.(14),and
These observations are further supported with the depicted them as functions of the interval number (m )
Φ
4
A striking feature of the results presented in Table I
(a)
is the total lack of effects of both the radii ratio R /R
o i
and the source-drain voltage V on limiting values of
eff
0.6 and . (In contrast, ∆ and ∆ strongly
Fde∞pends oRn∞R /R.) This factFal∞lows us tRo∞expect the
o i
R quantum-limited shot noise, characterized by
0.4
| 1/3
0.76 and 0.55, (16)
F
F∞ (cid:39) R∞ (cid:39)
to appear generically in graphene-based nanosystems at
0.2
high magnetic fields and for finite source-drain voltages,
F1
1/15 similarly as pseudodiffusive shot-noise (with = 1/3
0.6 Fdiff
and =1/15) appears generically at the Dirac point
0 R1 in thRedliiffnear-response limit.
R
0.4
(b) |
0.3 F
IV. CONCLUSIONS
0.2
R
(c)
� 0.2 0 We have investigated the finite-voltage effects on the
| 0 0.2 0.4 0.6 magnetoconductance, as well as the magnetic-field de-
F
1/m pendence of the shot-noise power and the third charge-
� �
transfer cumulant, for the Corbino disk in ballistic
0.1 graphene. Periodic magnetoconductance oscillations,
earlierdiscussedtheoreticallyinthelinear-responselimit
[1, 2], are found to decay rapidly with increasing field at
0 finite voltages. To the contrary, the and -factors,
F R
1 3 5 7 9 11 quantifying the higher charge-transfer cumulants, show
m� periodic oscillations for arbitrary high fields, for both
the linear-response limit and the finite-voltage case. Al-
FIG. 3: Average values X (a) and oscillation magnitudes though such oscillations must be regarded as signa-
∆X =max(X)−min(X)(b),withX =F(squares)andX = tures of a nonstandard quantum interference phenom-
R (circles), calculated for several consecutive flux intervals ena, specific for graphene-based disks near zero doping
defined by Eq. (14). Open (or closed) symbols at each panel
(and having counterparts for higher Landau levels), the
correspond to eVeffRi/((cid:126)vF)=0.25 (or 0.5). Lines in panels parameter-independent mean values of 0.76 and
(a) and (b) depict the linear-response values given by Eqs. 0.55 suggest the existence of aFg∞en(cid:39)eric, finite-
(12) and (13). Panel (c) illustrates the scaling of F and R Rvo∞ltag(cid:39)eandhigh-fieldanalogofafamiliarpseudodiffusive
with 1/m →0 (see the main text for details).
Φ
charge transport regime in ballistic graphene.
Wehope ourfindings willmotivatesome experimental
attempts to understand the peculiar nature of quantum
TABLEI: Limitingvaluesofperiod-averagedF,Randoscil-
lation magnitudes ∆F, ∆R obtained by least-squares fitting transport via evanescent waves in graphene, which man-
of the parameters in Eq. (15). Numbers in parentheses are ifests itself not only in the well-elaborated multimode
standard deviations for the last digit (see also Ref. [27]). case of wide rectangular samples [13–15], but also when
averylimitednumberofnormalmodescontributetothe
R /R F ∆F R ∆R
o i ∞ ∞ ∞ ∞ systemconductanceandothercharge-transfercharacter-
2.5 0.761(1) 0.0014(1) 0.552(3) 0.0064(2) istics, as in the case of Corbino disks with large radii
5.0 0.763(1) 0.061(1) 0.555(2) 0.017(1) ratios R /R 1. Albeit the discussion is, in principle,
o i
(cid:29)
10 0.771(5) 0.191(2) 0.56(1) 0.170(2) limited to the system with a perfect circular symmetry
and the uniform magnetic field, special features of the
results, in particular the fact that mean values of the
F
in Figs. 3(a), 3(b). Next, the scaling with 1/m 0 is and -factorsareinsensitivetotheradiiratioandtothe
Φ → R
performed by least-squares fitting of the approximating voltage, allow us to believe that quantum-limited shot
formula noise as well as the signatures of quantum interference
should appear in more general situations as well.
(cid:18) (cid:19)2
1
Y [m ] Y +A , (15)
Φ Y
(cid:39) ∞ mΦ
Acknowledgments
for Y = , , ∆ , and ∆ . The examples of [m ]
Φ
F R F R F
and [m ] are presented in Fig. 3(a); the values of Y TheworkwassupportedbytheNationalScienceCen-
Φ
for dRifferent ratios R /R are listed in Table I [27]. ∞ treofPoland(NCN)viaGrantNo.N–N202–031440,and
o i
5
partly by Foundation for Polish Science (FNP) under formed using the PL-Grid infrastructure.
the program TEAM. The computations were partly per-
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by the electrochemical potential µ −eV /2 (or µ + detected for other V -s in the 1/m →0 limit.
0 eff 0 eff Φ
eV /2); the actual source-drain voltage may differ from
eff
V due to charge-screening effects.
eff