Charge transport in two dimensions limited by strong short-range scatterers: Going beyond parabolic dispersion and Born approximation

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by  Bretislav Sopik

Charge transport in two dimensions limited by strong short-range scatterers:
Going beyond parabolic dispersion and Born approximation.
Bˇretislav Sˇop´ık
Central European Institute of Technology, Masaryk University, Kamenice 735, 62500 Brno, Czech Republic
Janik Kailasvuori
International Institute of Physics, Universidade Federal do Rio Grande do Norte, 59078-400 Natal-RN, Brazil and
4 Max-Planck-Institut fu¨r Physik komplexer Systeme, No¨thnitzer Str. 38, 01187 Dresden, Germany
1
0 Maxim Trushin
2 University of Konstanz, Fachbereich Physik, M703 D-78457 Konstanz, Germany

r (Dated: April24, 2014)
p
A Weinvestigatetheconductivityof chargecarriers confinedtoatwo-dimensional system with the
non-parabolic dispersion kN with N being an arbitrary natural number. A delta-shaped scattering

3 potentialisassumedasthemajorsourceofdisorder. WeemploytheexactsolutionoftheLippmann-
2 Schwinger equation to derive an analytical Boltzmann conductivity formula valid for an arbitrary
scattering potential strength. The range of applicability of our analytical results is assessed by a

] numericalstudybasedonthefinitesizeKuboformula. WefindthatforanyN >1,theconductivity
l
l demonstrates a linear dependence on the carrier concentration in the limit of a strong scattering
a
potential strength. This finding agrees with the conductivity measurements performed recently on
h
chirallystackedmultilayergraphenewherethelowesttwobandsarenon-parabolicandtheadsorbed
-
s hydrocarbonsmight act as strong short-range scatterers.
e
m
. I. INTRODUCTION rangepotentialrepresentschargedimpuritiesandcanbe
t
a approximatedby a screenedCoulombpotential. The va-
m The band theory provides a simple effective-mass de- cancies or some adsorbed molecules act as short-range
- scriptionofchargecarriersnearthelocalminimum(max- scatterers which, in turn, can be approximated by a δ-
d imum)ofaconduction(valence)bandalmostinanysemi- shaped potential. To derive the conductivity formula for
n
conductor material where an energy gap separates the the weak scattering potential one usually makes use of
o
c bands. Indeed,theenergydispersioncanbeexpandedin the semiclassical theory based on the Boltzmann equa-
[ the momentum near the bottom (top) of the conduction tion with the golden-rule collision term.16 It is known
(valence) band. The linear term of the expansion is zero theFermigoldenruleisderivedwithinthe firstBornap-
2
andthequadraticonemimicsthefreeelectrondispersion proximation. However, it is not always safe to say that
v
8 withtheelectronmassreplacedbyaneffectivemass.4Al- thefirstBornapproximationisvalidaslongasthepoten-
7 ready several decades ago it was pointed out,5 however, tial strength is small. This is particularly important in
1 that the dispersion can differ from the parabolic one as the case of the short-range scatterers, as was mentioned
6 longasthecrystalsymmetrypermitsthecontactbetween inthe veryfirstchapterofthe famous bookby Peierls17.
.
1 thevalenceandconductionbandinthequasi-momentum In this paper, we focus on the scattering by a short-
0 space. Atthetime,theproblemwasdiscussedinconnec- range potential of carriers with non-parabolic disper-
4 tion with mercury telluride which is a three-dimensional sion. We investigate the applicability of the first Born
1 zero-gapsemiconductor.5 Recent advances in technology approximation (weak scattering potential) and the reso-
:
v have made it possible to fabricate a few peculiar high- nant scattering approximation (strong scattering poten-
i quality two- dimensionalgaplessconductors: singlelayer tial) and we also address breakdowns of these two com-
X
graphene6,7, bilayer graphene8, trilayer graphene9, and plementary approaches. The problem of the Born ap-
ar topological insulators10,11, such as Bi2Se3 and Bi2Te3. proximationbreakdownhasrecentlyariseninthefieldof
Themostofthesematerialsdemonstrateacarrierdisper- graphene, where the adsorbed hydrocarbons effectively
sion different from the parabolic one being standard for act as strong short-range scatterers18. The phenomenon
two-dimensionalelectrongasesconfinedinIII-Vsemicon- isknownasscatteringdueto“midgapstates”19 or“reso-
ductorheterostructures,12seeFig.1. Inparticular,thisis nantscattering”20. Since the problemhasbeenreviewed
the case of ABC-stacked N-layer-graphene1,13 or, equiv- by Peres in his colloquium paper21, we do not adduce
alently, thin flakes of rhombohedral graphite14,15, where the complete list of references here. We refer to the re-
the charge carriers described by a simplified model with centpublicationbyFerreiraetal.22 wheretheLippmann-
only nearest-neighbor interlayer hopping demonstrate a SchwingerequationtogetherwiththeT-matrixapproach
kN dispersion in both conduction and valence bands. havebeenutilizedtoshowthatthestrongshort-rangepo-
Electrontransportin semiconductorsis limited by the tential leads to a similar conductivity behavior in mono-
presence of localized impurities which can be described layer graphene with the linear dispersion and in bilayer
byeithershort-rangeorlong-rangepotentials. Thelong- with the parabolic bands. In what follows, we gener-
2
alize this setting for particles with kN-dispersion which Born approximation limit and the resonant scattering
is relevant, in particular, for the ABC-stacked N-layer- limit for different N is done in Table I.
graphene.1,13
First, we analytically solve the Lippmann-Schwinger
equation for particles with a kN-dispersion and subject II. SOLUTION OF THE
LIPPMANN-SCHWINGER EQUATION AND
to δ-shaped scattering potential. Using our solution, we
CALCULATION OF THE CONDUCTIVITY
calculatetheBoltzmannconductivityexpressionvalidfor
any potentialstrength. Second, we compute the conduc-
tivity numerically utilizing the finite-size Kubo formula Here, we utilize the effective low-energy two-
and comparethe results of these two approaches. Third, band Hamiltonian for carriers in N-layer ABC-stacked
we analyze the applicability of the first Born and reso- graphene1,13 asamodelsystemforparticleswiththekN
nantscattering approximationsfor different values of N. dispersion. In the simplest case of negligible interlayer
asymmetries and trigonal warping this Hamiltonian for
The main findings of this paper are: (i) The analyti-
a given valley can be parametrized as
cally derived formula for the conductivity in multilayer
graphene(N >2)reproducesnumericalresultsverywell
0 (k ik )N
in a broad range of conditions. (ii) We observe that the H0 =γ (k +ik )N x−0 y , (1)
firstBornapproximationbreaksdownforfillingscloseto (cid:18) x y (cid:19)
the neutrality point, and the transport in this region is
where k is the wave vector, and γ is a constant depend-
described within the resonant scattering limit. This re-
ing on the hopping between sublattices. (Note that γ
sult is confirmed by the experimental evidence. (iii) At
and its dimension depend on N. In particular,note that
largeenough filling the conductivity approachesthe first γ includes a factor ¯hN.) We focus on the conduction
Born approximation regime for any potential strength
band electrons which have the dispersionE =γkN, the
which is in contradiction with the monolayer graphene k
density of states
(N = 1), where under such conditions the Born approx-
imation breaks down. This discrepancy is due to dif- k2−N
ferent asymptotic of the density of states in multilayer D(k)= , (2)
2πγN
graphene. A comparison of the conductivity in the first
and the eigenstates of the form
(a) E ~ (k - k0)2 (b) E ~ (k - k0)3 φk(r)= √12L ei1Nθ exp(ik·r), (3)
(cid:18) (cid:19)
EF EF
with θ = atan(k /k ). We consider the finite doping
y x
regime where 1 l k applies, here k is the Fermi
≪ kF F F
wavevectorandl isthemeanfreepathofsuchelectron,
kF
and the Boltzmann approach is expected to be valid, in
k0 DOS k0 DOS
particular with the influence of the valence band being
(c) E ~ (k - k0) (d) E ~ (k - k0)N negligible. Thus, in contrast to our previous work, see
Ref. 23, it is the kN dispersion of carriers, rather than
EF EF the chiral structure of the effective Hamiltonian, that is
in the main focus of the present work. This approach
is therefore not limited to graphene but can be appli-
cable for the conductivity description of any other two-
k0 DOS k0 DOS dimensional conductor with such a peculiar dispersion.
To illustrate the practical application of the effec-
FIG. 1. Panel (a) schematically shows a typical band struc- tive Hamiltonian we put it into a context of the ABC
tureforatwodimensionalelectrongasinIII–Vsemiconductor stacked trilayer graphene. Here, due to the split-off
heterostructures. The dispersion around the band minimum
bands, the maximum quasiparticle energy in the consid-
can beapproximated by that of free electron with certain ef-
ered two-bandmodel is limited by the value of the order
fective mass. If the crystal symmetry permits the contact
of 0.1eV.13 The explicit expression for γ in terms of the
between valence and conducting band, then the lowest order
interlayer hopping parameter t 0.4eV and the char-
term in the Taylor expansion around k = k can differ from ⊥
0 acteristic velocity v 108cm/s i≃s given by (¯hv )3/t2.1
theparabolicone. Inparticular(b)itcanbecubic,asitisin 0 ≃ 0 ⊥
thecase ofABC stackedgraphene1,(c)it can belinear, asit The maximum carrier concentration thus may not ex-
is for surface states in Bi Se 2 or for single layer graphene3, ceed 5 1012cm−2 which is a value comparable with
2 3 ≃ ×
and (d) of arbitrary natural power N which is the case of the oneobtainedfromtransportmeasurementsinmono-
ABC-stackedmultilayergraphene1. Theshortrangedisorder layer graphene. It is also worth to note that due to flat-
limited conductivity of the two-dimensional electron system ter bands for N > 2 the density of states (2) reaches
with thekN dispersion is in themain focus of this paper. higher values in the vicinity of the neutrality point, see
3
Fig.1. ThisresultsintoastrongerThomas-Fermiscreen- where the matrix σ is
ϕ
ingofthechargedimpuritieswhichmakesconsideringthe
0 e−iϕN
short-range disorder even more relevant. σ = , (10)
As the scattering potential model, we utilize the δ- ϕ eiϕN 0
(cid:18) (cid:19)
shaped potential, V(r)=V δ(r). The total Hamiltonian
0 with the angle ϕ defined by a projectionof a unit vector
withasingleimpurityreadsH =H0+V(r). Thismodel (r r′)/r r′ = (sinϕ,cosϕ)T. Following Ref. 22, we
allows a non-perturbative analyticalsolution and results app−roxim|at−e r| r′ r r r′/r, identify the outgoing
inanelegantconductivityformulavalidforanypotential wave vector a|s k− |=≃kr−/r,·and without loss of gener-
strength V0. To do that we follow the standard recipe ality take the incoiudtent wave vector k along the x-axis.
used by Ferreira et al., see Ref. 22, for the case of linear
The wave function of the scattered particle can then be
and parabolic bands with N =1, 2.
written as
eikr 1 1
A. The Lippmann-Schwinger equation ψk(r)=φk(r)+f(θ)√r √2L eiNθ , (11)
(cid:18) (cid:19)
where the scattering amplitude f(θ) reads
The Lippmann-Schwinger equation for the wave func-
tion ψk of a particle scatteredona single impurity reads 2i πD(k)V [1+e−iNθ]
f(θ)= 2 0 , (12)
ψk(r)=φk(r)+ d2r′G0(r r′)V(r′)ψk(r′), (4) −rπk1− π2D(k)V0[cot(π/N)−i]
−
Z with θ = 6 (k,k ) being the scattering angle. Note
out
where G (r r′)= r(E+i0 H )−1 r′ is the Green’s the qualitative difference between the scattering ampli-
0 0
− h | − | i
function of the problem which can be written down ex- tude(12)forN >2andtheonederivedinRef.22forthe
plicitly as case of bilayer graphene N = 2, where the denominator
does not contain the term with cot(π/N).
d2k′ eik′·(r−r′)
G (r r′)=(E+H ) . (5)
0 − 0 4π2 (E+i0)2 (γk′N)2
Z −
B. The Boltzmann dc conductivity
Since the scattering potential is V(r′) = V δ(r′), the in-
0
tegral in Eq. (4) becomes trivial. The amplitude of the
To calculate the conductivity out of Eq. (12) we
wavefunctionattheoriginψk(0)iseasytocalculatefrom
need the total scattering cross section Σ = dθ′(1
the equation cosθ′)f(θ′)2. Calculating the integral weTobtain −
| | R
ψk(0)=φk(0)+G0(0)V0ψk(0), (6) 2π2 D2(k)V2
Σ = 0 .
whereG0(0)canbefoundstraightforwardlyfromEq.(5), T k 1 πD(k)V cot(π/N) 2+ π2D2(k)V2
and for N >1 we obtain − 2 0 4 0
(13)
π (cid:2) (cid:3)
G (0)= D(k)[cot(π/N) i]. (7) The conductivity can be then written down in terms of
0 2 − eitherthemomentumrelaxationtimeτ−1 =n v Σ or
Thus, the Lippmann-Schwinger equation (4) becomes the mean free path lk = |vk|τk. (Here,kvk = ikh¯|γkN|kTN−2
rather simple and has the form is the particle velocity, and ni is the concentration of
scatterers.) In the latter case the conductivity is just
G (r)V
ψk(r)=φk(r)+1 πD(k)[c0ot(π0/N) i]V φk(0). (8) given by
− 2 − 0 e2l k
The remaining task is to find G (r). To do that we take σ = kF F, (14)
0 h 2
the integral in Eq. (5) in the polar coordinates k′,θ′ .
{ } with l being the mean free path calculated for a given
The integral over θ′ results in the Bessel function of kF
the first kind, and the subsequent integration over k′ Fermi wave vector kF.
The mean free path can be written explicitly as
gives a combination of Bessel functions and Meijer G-
functions.24 To calculate the result of the action of H0 1 k 1 πD(k )V cot(π/N) 2+ π2D2(k )V2
on this expression in Eq. (5), H0 should be also trans- lkF = n 2πF2 − 2 F 0 D2(k )V2 4 F 0 ,
formed into the polar coordinates. We do not express i (cid:2) F 0(cid:3)
the general equation for G (r r′), since we employ its (15)
0
asymptotic formfor k r r′ −1 only,in whichcase the andrepresentsourmaintheoreticalresult. Letusdiscuss
| − |≫ its limiting regimes, beginning with the Born approxi-
Green’s function simplifies to
mation limit where D(k )V 1. In this limit we after
F 0
≪
G0(r−r′)=−sπk|r2−r′|eik|r−r′|+iπ4 π2D(k)(1+σϕ), expansion for smlBaolrlnD=(k1F)Vk0Fobtai1n , (16)
(9) kF n 2π2D2(k )V2
i F 0
4
the first correction, l = lBorn+∆lBorn, is of the order which leads to the mean free path
kF kF kF
of (D(k )V )−1,
F 0 1 k [1 D(k )V ln k R]2+ π2D2(k )V2
1 k cot(π/N) l = F − F 0 | F | 4 F 0 .
∆lBorn = F , (17) kF n π2 D2(k )V2
kF −n 2π D(k )V i F 0
i F 0 (23)
and the conductivity in the Born approximation reads Here we can also distinguish two limiting regimes, the
Born approximation regime, lBorn = 1 kF(D(k )V )−2,
σBorn = e2 n 1 . (18) and the resonant scatterkiFng rengiimπ2e, lFkreFs 0 =
h niπD2(kF)V02 1 kF ln2 k R + π2 . However, contrary to N > 2 for
niπ2 | F | 4
From (2) we see that for N > 2 the density of states N =1(cid:16)the density of(cid:17)states D(kF) is zero for kF =0 and
D(kF) → 0 for kF → ∞. This means that the Born diverges for kF → ∞ which means that for fixed V0 the
approximationcanbeapproachednotonlybydecreasing conductivity approaches Born approximation regime for
the potential V0 but also by increasing the filling n. On small filling kF 0 and the resonant scattering regime
→
the other hand the density of states diverges as kF →0, for kF →∞.
so at this point the Born approximation breaks down.
This is very different situation from N = 2 where the
density of states D is constant and kF independent. Let III. NUMERICAL STUDY OF THE
us also note that the first correction term (17) depends FINITE-SIZE KUBO FORMULA
on cot(π/N) which is zero for N = 2 but increases with
higher N and makes the Born approximation limit less
In this section we compare the dc conductivity ob-
accessible.
tained analytically in the previous section from the
The opposite limit of (15) is the regime where 1
Lippmann-Schwingerequationwithresultsevaluatednu-
≪
D(k )V which is also known as the regime of resonant
F 0 merically from the finite-size Kubo formula. We begin
scattering21. We approach this limit for very strong po-
with a discussionof details of the numericalmethod em-
tentials V and also for k 0 due to the divergence
0 F ployed.
→
of D(k ). The mean free path can be in this regime ex-
F
panded in powers of 1/(D(k )V ). We obtain
F 0
1 k A. Method
lres = F 1+cot2(π/N) , (19)
kF n 8
i
(cid:2) (cid:3) The finite-size Kubo formula is given by
which does not depend on D(k )V . The first correction
F 0
term, lkF =lkreFs∆+lr∆eslkr=eFs, is1ofkFthceoto(rπd/eNr o)f.(D(kF)V0)−(210,) σ{KRujb}o =−i¯hLe22 nX,n′ f(EEnn)−−fE(En′n′)hnE|vnx|−n′Eihnn′′+|vxiη|ni.
kF −n 2π D(k )V (24)
i F 0
Here L2 is a size of the system, f(E) = Θ(E E) is
F
The conductivity as a function of carrier concentration −
the Fermi distribution function at zero temperature and
turns out to be linear in n for any N
η =g /(D(k )L2) expresses broadening of levels due to
T F
e2 n π the possibility of the particle to escape the system, with
σres = 1+cot2(π/N) . (21) g beingthedimensionlessThoulessconductivity25. Vec-
h n 4 T
i
tors n andenergiesE areeigenstatesandeigenenergies
From this formula one can(cid:2) conclude that(cid:3)the resonant of an| eiffective mesosconpic Hamiltonian consisting of the
scatteringregimehasevenmoreuniversalcharacterthan kinetic term (1) and potential term which is represented
wasfoundbyFerreiraetal.22 Thedependenceofthecon- by N =L2n scattering centers describedby a δ-shaped
i i
ductivity on the carrier concentration is nearly linear at potential. The exact position of the scattering centers
V0 not only in the case of N = 1 or N = 2 but with respect to the underlying lattice is not addressed.
→ ∞
for any other N > 2. Similar to the Born approxima- We have
tion limit, the resonant scattering regime becomes less
accessible for larger N. Ni 1 0
Note, that all the formulas given above are valid for H({Rj})=H0+ V0 0 1 δ(r−Rj), (25)
N 2 only, and that the case of N =1 must be consid- j=1 (cid:18) (cid:19)
X
ere≥d separately. As shown by Ferreira et al.,22 in order
to calculate the integraloverk′ in Eq. (5) it is necessary where locations Rj are randomly distributed in the
{ }
continuum of the sample. For every distribution of the
to introduce a momentum cut-off corresponding to the
scattering centers R we diagonalize the Hamiltonian
smallest length scale of the system, R. At N = 1 the j
{ }
(25), using a large momentum-space cut-off k , so that
Green’s function thus reads Λ
k < k , and evaluate the conductivity (24). This con-
F Λ
G (0)=D(k)(ln kR iπ/2), (22) ductivity is then averaged over random distributions of
0
| |−
5
N = 3, Ni = 80 N = 4, Ni = 80 N = 4, ci = 80
9 (a) ci = 4 (d) ci = 80 (g) Ni = 80
]
h
2/ 6
e
σ [ Kubo
3 Theory
Born
Resonant
0
9 (b) ci = 16 (e) ci = 200 (h) Ni = 240
]
h
2/ 6
e
[
σ
3
0
9 (c) ci = 40 (f) ci = 600 (i) Ni = 320
]
h
2/ 6
e
[
σ
3
0
0 2 4 6 8 0 2 4 6 8 0 1 2 3
n/ni n/ni n/ni
FIG.2. Theplotsshowthedependenceoftheconductivityontheconcentrationofcarriersrelativetothedensityofimpurities
n/n . Themainobservationisthatatstrongpotentialtheconductivitybecomesalinearfunctionofn/n foranyN >2which
i i
generalizes the conclusion made by Ferreira et al.22 for N = 1,2. Red squares represent the Kubo conductivity σKubo, blue
solid line is the theoretical conductivity σ calculated from (14) and (15) using the exact solution of the Lippmann-Schwinger
equation, green dash-dotted line is the Born approximation limit σBorn and yellow dashed line is the resonant scattering limit
σres. The strengthof thepotentialV isexpressed bythedimensionless parameter c using (26). (a), (b),(c) Conductivityfor
0 i
N = 3, fixed number of impurities N = 80 and potential strengths c = 4, 16 and 40. (d), (e), (f) Conductivity for N = 4,
i i
N =80 and potential strengths c =80, 200 and 600. (g), (h), (i) Conductivity for N =4, c =80 and increasing numberof
i i i
impurities N =80, 240, 320. Plots (h),(i) were calculated with k =202π.
i Λ L
N = 1, Ni = 40, ci = 1.2, R/L = 0.01 the scattering centers, σKubo = hσ{KRujb}oiav, until a suffi-
Kubo cient precision is achieved. To improve the averaging we
9 Theory impose a smallrandomshift δk ( π, π) ( π, π) on
Born wavevectorgridforeverydistrib∈uti−onL RL ×of−scLattLering
2/h] 6 Res centers. Results in this work have been{ caj}lculated using
e
σ [ gT =12,inagreementwiththediscussioninRef.25. The
3 momentumspacecut-offkΛ wassetto162Lπ ifnotstated
differently. To express the strength of the potential V
0
we use the following parametrization
0
0 1 2 3 4 5 6 7 8
V =γ 2π NL2N c , (26)
n/ni 0 L 2π i
with a dimensionless pa(cid:0)ram(cid:1)eter c . This parametriza-
FIG. 3. The figure shows conductivity dependence on n/n i
i tion provides that both the kinetic and the potential
for N = 1, N = 40, c = 1.2 and R/L = 0.01. It reveals
comparison ofithe Kubio conductivity σKubo represented by term of the hamiltonian (25) scale like (1/L)N and so
red squares with the theory σ from (23) and also its Born theKubo formula(24)isindependentonthe lengthLat
approximation limit σBorn and resonant scattering limit σres. zerotemperature. Thatis because eachofthe twoterms
WlineedσiBstoirnngouvieshrlatphsewliintehszienroacsoanmdeucmtiavnitnyeraxaiss.inTFheigr.e2s.onTahnet h1n/|Lv2x|inn′if/r(oEntno−f Ethn′e)ssucmalmesatliikoen.∼NLotewhthicahtfLacmtoursstobuet
scatteringlimitσres wasalreadyderivedbyFerreiraetal.,see takeninto accountexplicitly atfinite temperatures,as it
Ref. 22. has been done for the description of thermally activated
electron transport in gapped bilayer graphene.26 For
more details about the numerical method, see Refs. 23
6
and 25. comparison with results from Fig. 2. We determine the
smallest length scale as R = 1/k which gives R/L =
Λ
0.01. The theoretical conductivity overshoots the Kubo
B. Results conductivity for all n/ni, however, both conductivities
obeyalinearscalingforlargen/n . The linearscalingof
i
theconductivityinthediffusiveregimewasreportedalso
The main results are depicted in Fig. 2 which shows
by Kl os et al., see Ref. 28, that have done the numerical
conductivitydependenceonthefillingrelativetotheden-
calculations within the Landauer approach.
sity of impurities n/n for several values of the power N
i
ThecaseofN =2hasbeenconsideredinRef.29with
of the dispersion, number of impurities N and potential
i
the application to the pseudo-spin coherent conductiv-
strengthc . RedsquaresrepresenttheKuboconductivity
i
σKubo, the blue solid line is the theoretical conductivity ity of bilayer graphene. The first Born approximation
(16) has been utilized there in order to fit the numeri-
σ calculated from (14) and (15), the green dash-dotted
line is the Born approximation limit σBorn and the yel- cal Kubo conductivity curves. This approximation once
lowdashedline is the resonantscatteringlimitσres. The established at N = 2 remains valid for any kF because
theoretical curves are plotted using k =√4πn. the correction (DV0)−1 does not depend on kF for the
F
parabolic bands. This lucky circumstance made it pos-
The most important observation in Fig. 2 (a)–(i) is
sible to consider the pseudo-spin coherent terms in the
that the theoretical conductivity σ obtained from the
Boltzmann equation within the golden-rule approxima-
Lippmann-Schwinger equation gives a good agreement
with the Kubo formula conductivity σKubo in a broad tion and reach a good agreement between the numerical
andanalyticalmodelsevenatlowercarrierdensities.29 It
range of conditions. This is because we did not rely on
is clear from Eq. (16) now that this approach could not
the golden-rulerelaxationtime employedinthe previous
papers23,27,28. Since the first correction term of lBorn as workforN >2equallywellasitdidforN =2: Thecor-
well as lres is negative, the value of σ is alwayskFbelow rection depends on kF and the first Born approximation
kF breaks down at low enough carrier concentrations.
σBorn or σres. At the vicinity of the neutrality point the
Kubo conductivity drops to a finite minimal conductiv-
ity which is given by interband scattering events. Since
IV. CONCLUSIONS
our theoretical approach considers intraband scattering
exclusively, such contribution is not present in the theo-
retical conductivity and thus it goes to zero. Inthispaperwehavestudiedthedctransportofquasi-
Plots (a), (b), (c) reveal the conductivity for N = 3, particles with kN dispersion in the presence of δ-shaped
with fixed number of scattering centers N = 80 and in- scattering centers. This model can be applicable to mul-
i
creasing potential strength c = 4, 16 and 40. In (a) we tilayer(N 2)graphenecontaminatedbyhydrocarbons.
i ≥
seethatbothσandσKubofollowσBorn startingfromvery Special attention was payed to two complementary lim-
small values of n/n . As the potential strength c grows iting regimes — the first Born approximation limit and
i i
in (b) and (c), the region where the resonant scattering the resonantscattering limit — with respect to the limi-
limit is valid and σ is linear in n/n , enlarges. Similar tations of these approaches. The results are summarized
i
situation which confirms that this trend is universal for inTableI whichshowsσBorn, σres andtheir firstrelative
allN >2,occursinplots(d),(e),(f)forN =4,N =80 corrections for N =1, 2 and N >2. We conclude that
i
and potential strengths c =80, 200 and 600. In (d) the
i
BoththefirstBornapproximationandtheresonant
conductivity σKubo reaches the Born approximation be- •
scatteringregimeoverestimatetheconductivityfor
haviorforlargen/n . Withhighervaluesofc in(e)and
i i
N >2, in contrast to the case of N =2, when the
(f)bothσandσKubogetgraduallyclosertothelinearbe-
conductivity turns out to be underestimated.
havioroftheresonantscatteringlimitσres foralln/n in
i
theplot. TheregionoftheBornapproximationbehavior
In contrast to the case of N = 2, the conductivity
isshiftedtolargern/ninotshownhere. Lastcolumn(g), • correctiontoσBorn isnotquadraticbutlinearinV
0
(h), (i) shows the example with N =4, constant c =80
i for N > 2. This makes the first Born approxima-
and increasing number of impurities N = 80, 240, 320.
i tionregimelessaccessible. Thesameistrueforthe
From (15) we see that l depends on the number of im-
kF resonant scattering regime with respect to 1/V .
0
puritiesonlyviafactor1/n whichscalesthedependence
i
of σ on n/ni. This is confirmed by numerical calcula- The conductivity corrections are concentrationde-
tions. In (g) for Ni = 80 the conductivity σ and σKubo • pendent for N > 2. At large enough filling the
are very close to σres within the studied range of n/ni. conductivity approaches the Born approximation
With increasing of Ni in (h) and (i) the region, where σ regime for any potential strength.
starts to follow the trend of Born approximation limit,
gradually shifts to lower n/n . We conclude that σKubo In the limit of very strong scattering potential,
i
•
obeys the same scaling by 1/n as theoretical σ. V , i.e., in the resonant scattering regime,
i 0
→ ∞
Fig.3revealsadependenceoftheconductivityonn/n the dependence of the conductivity on the carrier
i
for N = 1, N = 40, c = 1.2 and represents a useful concentration is always linear for any N >2. This
i i
7
N First Born approximation (V 0) Resonant scattering regime (V )
0 0
→ →∞
σBorn [e2/h] δBorn [σBorn] σres [e2/h] δres [σres]
N =1 n2iγV202 −2Vγ0pnπln(R√4πn) nni (cid:2)π2 + π2ln2(R√4πn)(cid:3) −V2γ0pπn π42l+n(lnR2√(R4π√n4)πn)
N =2 16πγ2n + V02 π n +64γ2
niV02 64γ2 4ni V02
N >2 Nni2Vγ022(4πn)N−1 −2NVγ0(c4oπt(nπ)/NN2−)1 4πnni (cid:2)1+cot2(π/N)(cid:3) −8NγV(04[π1n+)cN2ot−2(1πc/oNt()π]/N)
TABLE I. Conductivity in the two limiting cases of the potential strength V — thefirst Born approximation, σBorn, and the
0
resonant scattering regime, σres — and its first relative corrections δBorn and δres, with σ˜α =σα(1+δα). The rows for N =1
and N =2 are taken from Ref. 22, the third row follows from Eqs. (16) and (19). The dependence of conductivity on carrier
concentration nisqualitatively differentforN =1, N =2,andN >2in theBorn approximation, butdemonstratesuniversal
linear response in theresonant scattering regime.
generalizestheconclusionmadebyFerreiraet al.22 thattheelectrontransportofthesamplewasinaregime
for N =1,2. of strong resonant scattering described by Eq. (21) with
N = 3. We understand this as yet another evidence of
These outcomes have been confirmed by the numerical
significantimpact ofthe scatteringonshort-rangeimpu-
conductivity calculation using the finite-size Kubo for- rities on the electron transport in graphene.21
mula, see Figures 2 and 3.
Let us compare our theoretical results with experi-
mental data for trilayer graphene, N = 3. Although
ACKNOWLEDGMENTS
related experimental studies are already present in the
literature,9,30–33 we found the comparisondifficult. This
is because there is often lack of information about the M. T. thanks Wolfgang Belzig and Aires Ferreira for
stacking(ABAorABC)ofthegraphenesampleonwhich discussions and acknowledges financial support by the
themeasurementwasperformed. Forthisreasonwecom- DFG through SPP 1285. This work was also sup-
pare our results with work of Zhang et al.31 only, since ported by the program ”Employment of Newly Grad-
in this case we are sure the trilayer graphene with chi- uated Doctors of Science for Scientific Excellence”
ral ABC stacking was utilized, because in order to fit (CZ.1.07/2.3.00/30.0009)co-financedfromEuropeanSo-
the data authors used Hamiltonian (1) identical to the cial Fund and the state budget of the Czech Republic.
one discussed in this work. We observe that the conduc- Theaccesstocomputingandstoragefacilitiesofthe Na-
tivity in Figure 1 (e) of Ref. 31 is linear in n, similar to tional Grid Infrastructure MetaCentrum providedunder
theconductivityinmonolayerandbilayer.9Thissuggests the programLM2010005 is also highly appreciated.
1 H. Min and A. H. MacDonald, 10 Y.Xia,D.Qian,D.Hsieh,L.Wray,A.Pal,H.Lin,A.Ban-
Phys. Rev.B 77, 155416 (2008). sil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan,
2 H.Zhang,C.-X.Liu,X.-L.Qi,X.Dai,Z.Fang, andS.-C. NaturePhysics 5, 398 (2009).
Zhang, Nature Physics5, 438 (2009). 11 Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K.
3 P. R.Wallace, Phys.Rev.71, 622 (1947). Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang,
4 C.Kittel,Introduction to SolidState Physics (JohnWiley S. C. Zhang, I. R. Fisher, Z. Hussain, and Z.-X. Shen,
& Sons, INC., 1968). Science 325, 178 (2009).
5 N. N. Berchenko and M. V. Pashkovskii, 12 S. Datta, Electronic Transport in Mesoscopic Systems
Soviet Physics Uspekhi19, 462 (1976). (Cambridge University Press, 1995).
6 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, 13 F. Zhang, B. Sahu, H. Min, and A. H. MacDonald,
M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and Phys. Rev.B 82, 035409 (2010).
A. A.Firsov, Nature438, 197 (2005). 14 R.R.Haering,CanadianJournalofPhysics36,352(1958).
7 Y.Zhang,Y.-W.Tan,H.L.Stormer, andP.Kim,Nature 15 J. W. McClure, Carbon 7, 425 (1969).
438, 201 (2005). 16 N. W. Ashcroft and N. D. Mermin, Solid State Physics
8 K.S.Novoselov,E.McCann,S.V.Morozov,V.I.Fal/’ko, (Holt, Riwehart Winston, New York,1976).
M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and 17 R. Peierls, Surprises in Theoretical Physics (Princeton
A. K.Geim, Nat. Phys.2, 177 (2006). University Press, Princeton, NJ, 1979).
9 C.M.F.,R.S.,Y.M.,O.J.B.,M.A.F., andT.S.,Nat. 18 J. P. Robinson, H. Schomerus, L. Oroszl´any, and V. I.
Nano. 4, 383 (2009). Fal’ko, Phys.Rev.Lett. 101, 196803 (2008).
8
19 T. Stauber, N. M. R. Peres, and F. Guinea, 27 H. Xu, T. Heinzel, and I. V. Zozoulenko,
Phys. Rev.B 76, 205423 (2007). Phys. Rev.B 84, 115409 (2011).
20 S. Yuan, H. De Raedt, and M. I. Katsnelson, 28 J. W. K los and I. V. Zozoulenko,
Phys. Rev.B 82, 115448 (2010). Phys. Rev.B 82, 081414 (2010).
21 N. M. R. Peres, Rev.Mod. Phys.82, 2673 (2010). 29 M.Trushin,J.Kailasvuori,J.Schliemann, andA.H.Mac-
22 A. Ferreira, J. Viana-Gomes, J. Nilsson, E. R. Muc- Donald, Phys. Rev.B 82, 155308 (2010).
ciolo, N. M. R. Peres, and A. H. Castro Neto, 30 T. Taychatanapat, K. Watanabe, T. Taniguchi, and
Phys. Rev.B 83, 165402 (2011). P. Jarillo-Herrero, Nat. Phys.7, 621 (2011).
23 J. Kailasvuori, B. opk, and M. Trushin, 31 L.Zhang,Y.Zhang,J.Camacho, M.Khodas, andI.Zal-
Journal of Statistical Mechanics: Theory and Experiment 2013, Piz1n2y0a0k6,N(2a0t1.3P).hys. 7, 953 (2011).
24 A.P.Prudnikov,O.I.Marichev, andY.A.Brychkov,In- 32 C. H. Lui, Z. Li, K. F. Mak, E. Cappelluti, and T. F.
tegrals and Series, Vol. 3: More Special Functions. (Gor- Heinz, Nat. Phys.7, 944 (2011).
don and Breach, Newark, NJ, 1990). 33 W. Bao, L. Jing, J. Velasco, Y. Lee, G. Liu, D. Tran,
25 K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 98, B. Standley, M. Aykol, S. B. Cronin, D. Smirnov,
76602 (2007). M. Koshino, E. McCann, M. Bockrath, and C. N. Lau,
26 M. Trushin,EPL (Europhysics Letters) 98, 47007 (2012). Nat. Phys.7, 948 (2011).