Transport and magnetotransport in 3D Weyl Semimetals

1.2 MB·English

by  Navneeth Ramakrishnan

Transport and magnetotransport in 3D Weyl Semimetals
Navneeth Ramakrishnan,1 Mirco Milletari,1,∗ and Shaffique Adam1,2,†
1Department of Physics and Center for Advanced 2D Materials,
National University of Singapore, 117551, Singapore
2Yale-NUS College, 16 College Avenue West, 138527, Singapore
(Dated: November 17, 2015)
We theoretically investigate the transport and magnetotransport properties of three-dimensional
Weyl semimetals. Using the RPA-Boltzmann transport scattering theory for electrons scattering
off randomly distributed charged impurities, together with an effective medium theory to average
over the resulting spatially inhomogeneous carrier density, we smoothly connect our results for the
minimum conductivity near the Weyl point with known results for the conductivity at high carrier
density. In the presence of a non-quantizing magnetic field, we predict that for both high and low
5
carrierdensities,Weylsemimetalsshowatransitionfromquadraticmagnetoresistance(MR)atlow
1
magnetic fields to linear MR at high magnetic fields, and that the magnitude of the MR (cid:38)10 for
0
realisticparameters. Ourresultsareinquantitativeagreementwithrecentunexpectedexperimental
2
observations on the mixed-chalcogenide compound TlBiSSe.
v
o
PACSnumbers: 71.23.–k,71.55.Ak,72.80.Ng,72.10.–d
N

6
I. INTRODUCTION In this work we use a random phase approximation
1
(RPA)methodtoevaluatetheeffectivescreenedpoten-

] Electronic band structures that have protected gap- tial of the Coulomb impurity that enters various disor-
ll less points – where the conductance and valence bands dered averaged quantities. We show that the RPA is
a are guaranteed to meet – have been of significant the- a much better approximation of the commonly used
h
oretical and experimental interest in recent years. The Thomas Fermi approximation due to the nature of the
-
s two dimensional manifestation of such band structures vacuum screening structure of Dirac materials. In the
e have been extensively studied in graphene [1], where homogeneous regime (far from the Weyl point in mo-
m
the gapless nature is protected by sublattice symme- mentumspace), weakimpurityscatteringisconsidered
t. try [2], and in 3D topological insulators [3], where the at the Born level to obtain the Drude conductivity. In
a crossing point is protected by topology [4, 5]. More re- the inhomogeneous regime (close to the Weyl point in
m
cently, attention has focused on the three dimensional momentum space), we find that it is important to con-
- analogues of these compounds, called Weyl semimet- sider structure in the disorder distribution when per-
d
n als [6–8]. Compounds such as Cd3As2 [9–11] TaAs [12] formingthedisorderaverage. WethenuseanEffective
o and TlBiSSe [13, 14] have been shown to have Weyl Medium Theory (EMT) to average over the inhomoge-
c pointsintheirbandstructure(seeRef.[15]forarecent neous carrier density distribution. This formalism has
[ review on the various candidate materials for semimet- been remarkably successful in providing a quantitative

3 als with 3D relativistic electronic dispersions). understanding of the transport properties close to the
v Theoretical efforts toward characterizing the elec- Diracpointingraphene[1],the2DcousinoftheseWeyl
5 tronicpropertiesofWeylsemimetalsareinthenascent semimetals. Our results allow us to make both quan-
1 stage and include the scattering properties of differ- titative and qualitative comparisons with experiment
8 ent impurity potentials [16], localization and delocal- and yield many qualitative insights into the behavior
3 ization [17, 18], thermoelectric properties [19], screen- of Weyl semimetals under various experimental condi-
0
ing[20,21]andtemperaturedependence[22],theinflu- tions.
.
1
ence of the chiral anomaly [23], diffusive transport [24]
0
and the effects of electron-electron interactions [25].
5
Inspired by unexpected observations in recent trans-
1
: port [11, 26] and scanning probe [27] experiments, we This paper is organized as follows: In Section II, we
v
study theoretically the transport and magnetotrans- discusstheDrudeconductivityusingboththeThomas-
i
X port properties of 3D Weyl semimetals in the presence Fermi and RPA screening approximations, far away
of randomly distributed Coulomb impurities. The ef- from the Weyl point and also discuss why the RPA
r
a fect of the charged impurities is twofold: they provide is required (unlike in the case of graphene). In Sec-
a momentum relaxation mechanism and they act as tion III, we discuss the effects of impurity correlations
dopantsforthelocalcarrierdensity. Atlowcarrierden- and induced charge carriers, that play a role near the
sity, the former mechanism introduces macroscopic in- Weyl point. Finally, we look at experimental results
homogeneities in the carrier density profile, giving rise from Ref. [26] and compare them with our theoretical
to positively or negatively charged puddles. modelsinSectionIV.Theexperimentaltransportdata
is found to be consistent with two possible theoretical
regimes and we note in Section. V that magnetotrans-
∗ [email protected] port provides a simple and experimentally accessible
† shaffi[email protected] mechanismtodistinguishbetweenthetwopossibilities.
2
II. TRANSPORT AT HIGH CARRIER
DENSITY
The Hamiltonian for a Weyl semimetal is given by
H =±(cid:126)ıv σ·∂ −ξ+V(r) (1)
F r
Nimp
(cid:88)
V(r)= U(r−R)
j
j=1
where σ is a vector of Pauli matrices, ξ is the chem-
ical potential, v is the Fermi velocity and ± ac-
F
counts for the two chiralities. The density of states
is ν(E) = g|E|2/2π2v3, where g is the the degeneracy
F
(here g =4dueto spinandthe presenceoftwocones).
InEq.(1),U(r−R)isthetotalscreenedpotentialseen FIG.1. Conductivityasafunctionofcarrierdensity.
j
by an electron at position r due to charged impurities RPA-Drude theory (Ref. [22], blue curve), self-consistent
at positions R . In this work, we consider Coulomb Thomas-Fermi (Ref. [28] , green curve) and the EMT ap-
j
impurities with momentum space screened potential proach (assuming uncorrelated impurities in both regimes)
discussed in the main text (red curve). The EMT predicts
4πe2 aminimumconductivityclosetotheWeylpointandrepro-
U(q)= , (2) ducestheDrudeoneσ∼n4/3/n farawayfromtheWeyl
(cid:15)(q)q2 imp
point. In order to compare with the results of Ref. [22],
whereeistheelectronicchargeand(cid:15)(q)isthedielectric we follow their parameters for this figure and use α = 1.2,
function. Herekandk(cid:48) aretheincomingandoutgoing g = 2 and nimp = 1024m−3 and Π˜V = 0. (Inset) Close-up
of the Weyl point.
momentaofthescatteredelectronandq=k−k(cid:48) isthe
transferredmomentum. NotethattheCoulombpoten-
tial suppresses large momentum scattering connecting
Aswediscussbelow, forWeylsemimetalstheThomas-
the two Weyls cones. For this reason, in the following
Fermi and RPA give qualitatively different results, and
we will work with one cone and consider the contribu-
in what follows, we use the more correct RPA approx-
tion of the second one in the degeneracy factor. For imation with q (q) = [e2D(E)Π˜(q/2k )]1/2 . Here,
a given concentration of impurities, n , the ensem- s κ F
ble averaged transport scattering time wimipthin the Born Π˜(x) is the ratio of the RPA polarization function and
approximation is given by the density of states, and is given by the sum of two
components: avacuumpartΠ˜ (x)andafinitedensity
V
τ(cid:126) =2πnimp(cid:90) (d23πk)(cid:48)3 U(|k−k(cid:48)|)21−c2os2θδ(Ek−Ek(cid:48)). part Π˜M(x) [20]
tr
Ttuorem[a2k2e, 2co8n]nweectfiiorsntwcoitnhsiedxeirsttihnegsriemsuplltesricnasteheofliteev(ra3al)-- Π˜M = 23(cid:20)1+ 41x(1−3x2)log(cid:12)(cid:12)(cid:12)(cid:12)11+−xx(cid:12)(cid:12)(cid:12)(cid:12)− x22 log(cid:12)(cid:12)(cid:12)(cid:12)1−x2x2(cid:12)(cid:12)(cid:12)(cid:12)(cid:21)
utiaotnin,gwEheqr.e(3t)heinscthateteTrhinogmpaso-tFeenrtmiail(cTaFn)baepptarokxeinmaas- Π˜V = 2x32 log(cid:12)(cid:12)(cid:12)(cid:12)xδ(cid:12)(cid:12)(cid:12)(cid:12). (5)
U(q) = 4πe2/(κ(q2 +q2)) with a transferred momen-
s
tum, q = |k−k(cid:48)|. Here κ is the dielectric constant of We remark that the Thomas Fermi calculation is
(cid:112)
the material and q = 4πe2ν(E )/κ is the inverse of merely the linearized result of the RPA which is valid
s F
the Thomas-Fermi screening length. Introducing the for q (cid:28)kF [30]. Since we consider transport both near
effective fine structure constant α = e2/(cid:126)v κ (where andawayfromtheWeylpoint,theTFapproximationis
F
α = 0.07 for Cd As and α = 0.68 for TlBiSSe), we insufficient. Moreover, the TF approximation neglects
3 2
have [22, 29] the vacuum term of the polarization function, which in
our case is the dominant term for q (cid:29) k . As seen in
F
e2v2τ e2 g k4 1 Figure 1, the large quantitative difference between the
σ = F ν(E )= F , (4)
TF 3 F h 12π2nimpα2H((cid:112)g2απ) two approximations implies that one must perform the
full numerical RPA calculation for accurate compari-
whereH(z)=(z2+1/2)log(1+z−2)−1. Inbothnon- sonwithexperiments. Finally,wewouldliketoremark
chiral two dimensional electron gases and graphene, thatunlikethecaseofoneandtwodimensions,forWeyl
the accidental coincidence at zero temperature of the fermions in d = 3 the vacuum polarization function is
polarization function for q ≤ 2k and the density of divergentandneedsanultravioletmomentumcutoff∆,
F
states implies that the Thomas-Fermi approximation whereinEq(5),δ =∆/2k . Inprinciple,thetransport
F
gives identical results to the RPA. However, we should coefficients calculated within the RPA approximation
emphasize that this is no longer true at finite temper- could depend on the choice of the cutoff [31], although
ature or in other materials like bilayer graphene [1]. in practice we have verified that such dependence is
3
weak for a realistic parameter range. Our results for
thehighdensitytransportintheabsenceofamagnetic
field are shown in Fig. 1. We find that far away from
the Weyl point, σ ∼ n4/3/n , where n is the carrier
imp
density. In Fig. 1, we show that this result is in agree-
mentwithcalculationsrecentlyreportedinRef.[22]for
the homogeneous regime.
III. TRANSPORT AT LOW CARRIER
DENSITY
A. Correlations in Impurity Positions
In the regime near the Weyl point, we consider two
different effects. First, the conductivity may be mod-
ified from the homogenous case due to the presence
FIG.2. Comparison of n∗/n obtained using RPA
of correlations in the impurity positions. In order to imp
and Thomas-Fermi approximations. The error intro-
understand the origin of these correlations, one notes
ducedbyusingtheThomas-Fermiscreeningtocalculatethe
thattheeffective“size”oftheimpuritypotentialisap- effective minimumcarrierdensity, n∗, increaseswithanin-
proximately r0 (cid:39) λ (cid:39) n−1/3, where λ is the screening crease in the effective fine structure constant α. We note
length. This should be compared with the average dis- thatformaterialssuchasCd As ,thisisarelativelysmall
3 2
tance between the impurities, given by L (cid:39) n−1/3. At error but this is not the case for TlBiSSe. Here we use
imp
high carrier density, screening is more effective and the cutoff, δ=10
screening length is smaller, corresponding to r (cid:28) L
0
(or n (cid:29) n ). In this regime, impurities are well
imp
separated and can be essentially considered as point-
like, meaning that disorder is completely random. In
the inhomogeneous regime however, we find that the
electron density is comparable to the impurity den-
sity and therefore r ∼ L. In this case, disorder can- tributed impurities result in a spatially varying local
0
not be considered as completely structureless and cor- chemicalpotential,whichinducesapositiondependent
relations between impurity positions need to be in- carrier density. The induced carriers in turn screen the
cluded [32, 33]. Since the pair correlation function local potential and one eventually obtains a self con-
g(R ,R ) (cid:54)= g(R )g(R ), the self energy term propor- sistent relationship between the local induced carrier
i j i j
tionalton2 cannotsimplyberenormalizedaway. Fi- density and the disorder averaged impurity potential,
imp
nally, assuming that the impurities are homogeneously V(r,n(r)) = (cid:126)vF(6π2n(r)/g)1/3. Note that if we are
and isotropically distributed, the pair correlation func- far away from the Weyl point, the fluctuations in car-
tion reduces to the radial correlation function g(R). rier density are negligible compared to the total num-
The effect of correlations can be taken into account by ber of carriers but close to the Weyl point this is not
the modified correlator of the Gaussian random fields the case. The average induced carrier density corre-
(see Appendix A) sponds to the average value of the random Gaussian
(cid:90) field, V0 = (cid:104)V(r)(cid:105)d. The net effect of charge doping is
(cid:104)V(r)V(r(cid:48))(cid:105) =n d3qeıq(r−r(cid:48))U2(q)S(q) (6) the appearance of macroscopic regions of charge pud-
d imp
dles, each having an excess or a deficit of charge with
(cid:90)
S(q)=1+n d3R {g(R)−1}e−ıR·q, respect to the average value. Therefore, one performs
imp
an averaging over these spatial fluctuations in carrier
where(cid:104)...(cid:105) standsfordisorderaverageandwehavein- density using an effective medium theory (EMT). The
d
troducedthestructurefactorS(q). FollowingRef.[33], EMTisawellestablishedtechniquedevelopedbyBrug-
we take the radial distribution function as german[34]andlaterbyLandauer[35]inordertochar-
(cid:26) acterize the effects of macroscopic fluctuations on the
0 R<r
g(R)= 0 (7) global conductivity. In order to perform the average,
1 R>r
0 one considers each macroscopic region of definite local
When computing transport properties, the effect of conductivity to be embedded in a homogenous effec-
Eq. (6) is to replace U2(q)→U2(q)S(q) in Eq. (3). tivemedium,whoseconductivityisdeterminedinaself
consistent way over all the regions. Early EMT mod-
els assumed two types of regions with conductivities
B. Effective Medium Theory σ and σ occupying area fractions p and 1−p. This
A B
modelwaslatergeneralizedinthecaseof2Dmaterials,
The second effect that one must take into account in to continuous distributions of local, tensor conductivi-
the inhomogeneous regime is that non-uniformly dis- ties [36, 37]. Here we generalize the results of Ref. [37]
4
to the continuous three dimensional case and obtain
(cid:90) (σˆ(V)−σˆE)
DV P[V,V0,Vrms](cid:16)ˆI3+ 3σˆˆI3E (σˆ(V)−σˆE)(cid:17) =0
xx
(8)
(cid:18) σ σ (cid:19) (cid:18) σE σE (cid:19)
σˆ = xx xy , σˆE = xx xy . (9)
−σ σ −σE σE ,
xy xx xy xx
where DV is a functional measure, σˆ is the local con-
ductivity and σˆE is the effective medium conductivity
to be found self consistently. In obtaining Eq. (8) we
have assumed an isotropic material i.e. σE = σE =
xx yy
σE and that the local conductivity regions are spheri-
zz
cal in shape [37]. This assumption is valid in the case
of puddles that are small compared to the sample size.
The probability distribution in Eq. (8), P[V,V0,Vrms], FIG. 3. Comparison with the experimental data of
is the same one that has been used to evaluate the Ref. [26]. The experimentally determined mobility of Tl-
Drude transport time. Therefore, it is characterized BiSSe,µexp,decreasesasafunctionofthemeasuredcarrier
by the average and the variance of the disorder distri- densitynexp,whereastheDrudetheorywithconstantnimp
bution (see Appendix B ). Finally, we define the car- (blue curve) would predict the opposite trend. In the main
rier density associated with the variance V to be n∗ text,weproposetwopossiblescenarioscompatiblewiththis
rms
behavior. The red curve assumes that the experiments are
whichistheeffectiveminimumcarrierdensityinaWeyl
in the inhomogeneous regime with n (cid:28)n with corre-
semimetal. InFig.2,weshowthatonlyforα(cid:28)1does exp rms
latedchargedimpurities. Alternatively,thegreencurveas-
the n∗ calculatedwithin theRPAreduceto thatofthe
sumesthattheexperimentsareinthehomogeneousregime
Thomas-Fermi. In general, as in the case of transport, n (cid:29) n , with the charged impurities also responsible
exp rms
the full RPA polarizability function must be used. for doping. Here n =n ≈4 n , α=0.68 and δ=10.
exp 0 imp
C. Crossover Between Homogenous and Theimplicitassumptionmadehereisthattheimpurity
Inhomogeneous Regimes
concentrationdoesnotchangewiththesample. Indeed,
forfixedn ,Fig.1showsthatmobilityincreaseswith
imp
In Fig. 1, we show the conductivity at zero mag- increasing carrier density. This is represented in Fig. 3
neticfieldasafunctionofcarrierdensity. Wefindthat asthebluecurvewhichshowstheoppositetrendtothe
for large carrier density, our results reproduce those of experiment. Weproposetwopossiblescenariosthatal-
Ref.[22]butatzerocarrierdensity,andintheinhomo- low us to relax the constant n in a physically justi-
imp
geneous regime, our results differ from both the con- fiable way (and as we discuss below, it is not possible
ductivitycalculatedinRef.[28]andRef.[22]. Thisisa to determine from this data alone which of these two
consequenceofusingtheEMTwhichprovidesasmooth scenarios correspond to the experimental situation).
crossover between the two regimes instead of a hard In the first case, we consider the experiment to be in
floor as done in Ref. [28]. Note that Ref. [22] ignores the homogeneous regime with charged impurities also
the effect of inhomogeneity altogether. The transport actingasdopantsthatshiftthechemicalpotential[29].
theoryforWeylfermionsshowninFig.1representsthe The average induced carrier density is then given by
full crossover from the inhomogeneous regime (close to V =(cid:126)v (6π2n /g)1/3 =n U(q=0,n∗),fromwhich
0 F 0 imp
the Weyl) point to the homogeneous regime (far away we obtain n = 4 n in the density regime of inter-
0 imp
from the Weyl point). est. The green curve in Fig. 3 uses n =n =4 n
exp 0 imp
andshowsgoodagreementwiththeexperimentaldata.
Therefore, a plausible resolution of this experimental
IV. COMPARISON WITH EXPERIMENTS “mystery” is that the charged impurities that are re-
sponsible for scattering carriers are also responsible for
Next, we apply our theory to address recent experi- doping the samples.
mental findings on TlBiSSe (g =4,α=0.68). Ref. [26] Asecondpossibilityisthatthesamplesareinthein-
describetheirresultsofalargeincreaseinmobilitywith homogeneous regime where n (cid:28)n . In this regime,
0 rms
decreasing carrier density as surprising. In order to one equates the fluctuations in the impurity poten-
makeconnectionwiththeexperimentaldata,oneiden- tial V with the band energy to obtain an effective
rms
tifiestheexperimentalmobilityµ andcarrierdensity minimum carrier density n∗. The red curve in Fig. 3
exp
n from experimentally determined parameters [38] considers samples with disorder (n varying from
exp imp
6.1×1023m−3 to around 1.9×1027m−3). Using the
µ = lim σxy(B) , n = σxx(B =0). (10) EMT equations (8) and the definition of nexp (10), we
exp B→0Bσxx(B) exp µexpe see good agreement with the experimental data. Note
5
that here we must consider impurity position correla-
tions and take the correlation length r (cid:39) L ∼ n−1/3
0 imp
as discussed earlier. Both the homogeneous and inho-
mogeneous scenarios are generally consistent with the
scalinglaw,µ ∼n−2/3. Thisscalingisaconsequence
exp exp
of σ ∼ n4/3/n and n ∼ n . Note that this also
imp imp
presentsconvincingevidenceforusingCoulombimpuri-
tiesasopposedtoneutralorpoint-likeimpuritiessince
the conductivity scales differently with n for the lat-
ter[22]. Weremarkthatouranalysisiscompletelyfree
of fit parameters and the both the homogeneous and
inhomogeneous theories are in quantitative agreement
withthedata(theinhomogeneouscurveagreeswiththe
data to within a factor of 4). This also demonstrates
that transport measurements alone cannot distinguish
between the two regimes since both show comparable FIG.4. Magnetoresistance in TlBiSSe. Disordered3D
mobilities(sameorderofmagnitude). Weproposethat Weyl fermions can have large MR > 10 for realistic ex-
magnetoresistance is the appropriate measurement to perimental parameters. The figure shows that both in the
homogeneousandintheinhomogeneousregime,themagne-
distinguish between the two regimes.
toresistance is quadratic at low fields (see inset) and linear
athighfieldsinagreementwithexperimentalobservations.
Notice that the MR in the inhomogeneous regime is much
V. MAGNETORESISTANCE largerthanthatofthehomogeneousregime,suggestingthat
increasing disorder is an easy way to enhance the MR.
TocalculatethemagnetotransportofWeylsemimet-
als,weassumethatthechargeandtheaxialcurrentare
For Weyl semimetals in general, our theory predicts
weaklycoupled,sinceCoulombimpuritiessuppressthe
that the MR should be quadratic at low magnetic
scatteringbetweenthetwoWeylnodes[39]. Indeed,the
fields and linear at high magnetic fields. In Fig. 4
results ofRef. [26] show no negative magnetoresistance
we show our results for n = 3.8 × 1023m−3 and
oranyeffectofaninplanemagneticfield,bothofwhich 0
n = 9.5 × 1022m−3 (homogenous regime), and
aresignaturesofthechiralanomaly[40,41]. Moreover, imp
for n → 0, n = 2.3 × 1024m−3 (inhomogeneous
wenotethatintheabsenceofparallelelectricandmag- 0 imp
regime). These values were chosen so that they
neticfields,theeffectsofthechiralanomalyareabsent.
correspond to similar µ and n and therefore they
Thus,theoriginoftransversemagnetoresistanceinthis exp exp
cannot be distinguished from transport measurements
system is likely due to be disorder-induced which can
alone. The results in Fig. 4 demonstrate convincingly
be treated in a semiclassical regime. We then assume
that within the semi-classical theory presented here,
that one can define a conductivity matrix with
the magnetoresistance in the inhomogeneous regime
1 µB is much larger than that of the homogeneous regime.
σxx =σB(r)1+µ2B2 , σxy =σB(r)1+µ2B2, (11) We also note that in the inhomogeneous regime MR
is comparable to that seen by Ref. [26] for matching
parameters, suggesting that those samples were in the
where σ (r)=σ(n(r)) is the RPA-Boltzmann conduc-
B
inhomogeneous regime. Within this model, having
tivity discussed earlier and the magnetic field is taken
MR > 10 is easily achievable in the inhomogeneous
along the z axis. The above set of equations con-
regimeformoderatevaluesofB,butitisseveralorders
stitute the input for the EMT model. The solution
of magnitude weaker in the homogeneous regime. This
of the EMT equations contains both the magnetore-
suggests a clear way to experimentally distinguish
sistance caused by having two types of carriers (elec-
between these two regimes. Moreover, an easy way
trons and holes) [42] and the disorder-induced mag-
to increase MR for technological applications is to
netoresistance discussed in the context of silver chalco-
make the sample dirtier, a counterintuitive, yet easily
genides[37,43]andothertwodimensionalsystems[44],
achievable, experimental goal.
where MR≡(ρ (B)−ρ (B =0))/ρ (B =0).
xx xx xx
A simple physical picture for this disorder induced
MR was proposed in Ref. [44] and this still holds in
Weyl semimetals. Essentially, disorder results in a sit-
ACKNOWLEDGEMENTS
uation where the charge carriers move with varying
drift velocities in various regions. The global Hall field
thuscannotcancelthevelocitydependentLorentzforce This work was supported by the National Research
(as would be the case in a homogenous system) and Foundation of Singapore under its Fellowship program
theelectrontrajectoriesbecomelongerasthemagnetic (NRF-NRFF2012-01). WewouldliketothankB.Feld-
fieldisincreased. Thisistheoriginofdisorder induced manandS.DasSarmaforsuggestingthisproblem,and
magnetoresistance. A. Yazdani and Y. Ando for correspondence regarding
6
corresponding to the diagrams of Fig. 5. Performing
the impurity average, one obtains three diagrams: two
Σ = + = + +
proportional to N and one to N2 . For completely
imp imp
random disorder, the term proportional to N2 is a
imp
constant that can be renormalized away. However,
FIG. 5. Disorder averaged self energy. Amputated di-
as we are now going to show, if there is any residual
agrams defining the self energy at the Gaussian level. The
structure in the disorder, the N2 term cannot be
brackets represent disorder averaging, the “x” accounts for imp
renormalized away.
theimpurityconcentration,thedotfortheimpuritypoten-
tial insertion and the solid line is the clean propagator.
For completeness, we start considering the standard
single impurity diagram; in momentum space it reads:
their experimental data.
Nimp
(cid:88)(cid:104)U(k−k(cid:48))e−ıRi(k−k(cid:48))(cid:105)=n U(0)δ(k−k(cid:48)),
imp
Appendix A: Correlated Impurities i=1
(A4)
wheren =N /Ld isthedensityofimpuritiesand
Hereweconsidertheeffectofspatialcorrelationsbe- imp imp
U(0) is the impurity potential evaluated at zero trans-
tween impurities in the inhomogeneous regime. This
ferred momentum q. Note that in principle this term
problem has been considered before in Refs. [33, 45],
is singular and needs to be regularized, for example
butitsrelationtodiagrammaticperturbationtheoryis
by screening. Here we will assume that the impurity
now made explicit. Disordered averaged Green’s func-
potential is screened and therefore consider U(0) as a
tions are defined in terms of their self energy. At the
finite quantity. Consider now the second order term
Gaussian level, the disorder averaged self energy is de-
fidneredavienrtaegrimngsiosfgtehneedraiallgyradmefisndeedpiinctteedrminsFoifga.5w.eiDghistoerd- N(cid:88)imp 1 (cid:88)(cid:104)U(k−k(cid:48))e−ıRi(k−k(cid:48))G (k(cid:48),E)
sum over impurities position of a disorder dependent Ld 0
i,j=1 k(cid:48)
quantity O(R ) [46]
i × U(k(cid:48)−k(cid:48)(cid:48))e−ıRj(k(cid:48)−k(cid:48)(cid:48))(cid:105), (A5)
(cid:90) Nimp
(cid:104)O(r)(cid:105)= (cid:89) dRig(Ri)O(r,Ri) (A1) whereG0(k(cid:48),E)isthefreeelectronpropagator. Taking
the disorder average, there are two contributions: for
i=1
i=j one finds
(cid:90) Nimp Nimp
(cid:89) (cid:89)
(cid:104)O(r)O(r(cid:48))(cid:105)= dR dR g(R ,R ) (A2)
i j i j Nimp
i=1 j=1 (cid:88)(cid:104)e−ıRi(k−k(cid:48))e−ıRj(k(cid:48)−k(cid:48)(cid:48))(cid:105)=n δ(k−k(cid:48)(cid:48)),
imp
×O(r,R )O(r(cid:48),R )
i j i=1
... (A6)
corresponding to the rainbow diagram shown in Fig. 5.
Here(cid:104)...(cid:105)standsfordisorderaverage,Ri istheposition For i (cid:54)= j, we need to make some assumptions on the
of the impurities in a d-dimensional volume Ld, Nimp pair correlation function. We assume that it still de-
is the total number of impurities in the system and the pends only on the coordinate difference R −R [32].
i j
indicesi,j,l...labeldifferentimpurities. Theimportant As a consequence, momentum is conserved on average,
objectsintheabovedefinitionsarethecorrelationfunc- i.e. k = k(cid:48)(cid:48). We also assume that the pair correlation
tionsg(Ri,Rj,...,Rz), describingcorrelationsbetween functiondoesnotdependontheanglebetweenRi and
one impurity, two impurities and so on [32]. As usual, R . Within these assumptions, Eq. (A5) reads
j
thehierarchyofcorrelationfunctionscannotbeworked
out explicitly and one has to perform some physically 1 (cid:88)G (k(cid:48),E)U2(|k−k(cid:48)|)n (A7)
motivated ansatz in order to truncate the hierarchy. Ld 0 imp
k(cid:48)
For completely random disorder, all correlation func- (cid:26) (cid:90) (cid:27)
tions factorize as product of single particle correlations × 1+n ddRg(R)e−ıR(k−k(cid:48)) ,
imp
g(R ) [46]. These are simply equal to the probability
i
of finding an impurity at site i, i.e. 1/Ld. The pair
where the first term in the curly bracket corresponds
correlation function g(R ,R ) gives the probability of
i j to the Born term (single particle scattering) and the
findinganimpurityatsite R givenoneatsiteR . Let
i j second takes into account the effect of two particles
us consider the total disorder potential V(r) defined in
scattering. Notethatthesecondtermissingularatk=
Eq. (1) k(cid:48) [32]; to take care of this singularity, one subtracts it
Nimp astheFouriertransformofunityandwecandefinethe
(cid:88)
V(r)= U(r−R ). (A3) regularised structure factor as
i
i=1 (cid:90)
S(|k−k(cid:48)|)=1+n ddR[g(R)−1]e−ıR(k−k(cid:48)).
In the weak scattering limit (Born limit), only the first imp
two moments of the distribution of V(r) are relevant, (A8)
7
This has been obtained in Refs. [33, 45] and the struc- interested in evaluating the disordered averaged gener-
ture factor can be measured e.g. in neutron diffraction ating functional
experiments. Next,weconsiderthecasewheretheterm
proportionalton2 isimportant. Forthechargecarri- (cid:90)
imp (cid:104)logZ[V](cid:105) = DV P[V] logZ[V], (B1)
ers, the “size” of the charged impurity is roughly given d
by the effective Bohr radius a of its lowest impurity
0
level; this can be orders of magnitude larger [47] than whereDV isafunctionalmeasure. Herewearenotin-
the underlying lattice constant [32]. If one compares terested in the actual calculation of this quantity, but
the average spacing L between the impurities and a0, onlyinfindingP[V]. Generally,logZ[V]canbesubsti-
onecomesat theconclusion thatcorrelation effectsare tutedwith aneffective functionalof V asin thecase of
important if L < a0. Note that the ratio L/a0 is rem- the Effective Medium Theory (EMT) employed in the
iniscent of the parameter rs used to quantify correla- main text. Let us consider the total disorder potential
tionsinanelectrongas. Weconcludebyconnectingthe of Eq. (A3), where the potential U is due to the N
imp
above analysis to the relaxation time τ. By definition impurity potentials and V(r) is the resulting effective
1/τ(k)=ImΣ(k), where potential. In order to find P[V], one needs to interpret
the above definition as a constraint. In the functional
ImΣ(k)= nimp (cid:88) U2(|k−k(cid:48)|)S(|k−k(cid:48)|)ImG (k(cid:48),E) formalismthisisaccomplishedbymeansofafunctional
Ld 0
k(cid:48) Dirac delta function averaged over impurity positions
(A9)
It follows that the variance of the random Gaussian Nimp
(cid:89) (cid:88)
field V(x) can be written as P[V]= (cid:104)δ[V(r)− U(r−R )](cid:105) (B2)
i
(cid:90) r i=1
(cid:104)V(r)V(r(cid:48))(cid:105)d =nimp d3qeıq(r−r(cid:48))U2(q)S(q) =(cid:90) Dξ(cid:10)eı(cid:82)d3rξ(r)[V(r)−Ni(cid:80)i=m1p U(r−Ri)](cid:11),
(A10)
that is our Eq. (6) of the main text.
wherewehaveusedthefunctionalrepresentationofthe
Dirac delta function and ξ(r) is a Lagrange multiplier
Appendix B: On the Gaussian approximation field. The first step in the evaluation of P[V] consists
inassessingthecorrelatednatureoftheimpurities. Ac-
Here we consider the form of the disorder probabil- cording to Ref. [46], and as explained in the main text,
ity distribution used in Eq. (8) of the main text. As if the distance between the impurities is much larger
explained in the main text, within the Drude trans- thanthescreeninglength,thenthepositionsoftheim-
port theory, this function is the same as the one used purities are uncorrelated. On the other hand, if the
to evaluate the transport time. Here we provide an screening length of the Coulomb potential is compa-
explicit proof of the validity of the Gaussian approx- rable to the average distance between the impurities,
imation. This discussion is mostly based on Ref. [48] then there may be correlations in the positions of the
and is based on the functional approach to disordered impurities themselves (see Appendix A). Here we con-
systems. This method is completely equivalent to the sider for simplicity the case of completely uncorrelated
“sum over impurities” approach used in diagrammatic disorder and comment on the effect of correlations at
perturbationtheory. Inthefunctionalapproach, oneis the end. In the thermodynamic limit one obtains [48]
P[V]=(cid:90) Dξei(cid:82)d3rξ(r)V(r)exp(cid:26)−n (cid:90) d3R(cid:16)1−e−ı(cid:82)d3rξ(r)U(r−R)(cid:17)(cid:27)=(cid:90) Dξei(cid:82)d3rξ(r)V(r)eΦ(ξ), (B3)
imp
where in the second step we have identified the cumu- For screened Coulomb disorder, and for the considered
lant function of the stochastic process Φ(ξ) and the values of the electromagnetic coupling α, the Born cri-
related characteristic function χ(ξ) = eΦ(ξ). We now terion is satisfied for carrier density n ≥ n∗, see Sec-
follow Ref. [49] (where more details can be found) and tion IIIB. Finally, integrating out the Lagrange mul-
assess the Gaussian nature of the characteristic func- tiplier field, one finds the Gaussian distribution of the
tion using the central limit theorem. One needs to ex- random fields used in the main text. When n (cid:39) n∗,
pand the cumulant function and show that the magni- the screening length of the Coulomb potential is com-
tudeofthesecondcumulantisthedominantone. This parable to the average distance between the impurities
isbasicallythewellknownBorncriterion,c.f. Ref.[46]. and one needs to use the modified propagator of the
random fields, Eq. (A10).
8
[1] S.DasSarma,S.Adam,E.Hwang, andE.Rossi,Rev. [25] W.Witczak-Krempa,M.Knap, andD.Abanin,Phys.
Mod. Phys. 83, 407 (2011). Rev. Lett. 113, 136402 (2014).
[2] A. Castro Neto, F. Guinea, N. Peres, K. S. Novoselov, [26] M. Novak, S. Sasaki, K. Segawa, and Y. Ando, Phys.
and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). Rev. B 91, 041203 (2015).
[3] M.Z.HasanandC.L.Kane,Rev.Mod.Phys.82,3045 [27] S. Jeon, B. Zhou, A. Gyenis, B. Feldman, I. Kimchi,
(2010). A. Potter, Q. Gibson, R. Cava, A. Vishwanath, and
[4] C.L.KaneandE.J.Mele,Phys.Rev.Lett.95,226801 A. Yazdani, Nat. Mater. 13, 851 (2014).
(2005). [28] B. Skinner, Phys. Rev. B 90, 060202 (2014).
[5] C.L.KaneandE.J.Mele,Phys.Rev.Lett.95,146802 [29] A. A. Burkov, M. D. Hook, and L. Balents, Physical
(2005). Review B 84, 235126 (2011).
[6] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. [30] N.AshcroftandN.Mermin,SolidStatePhysics (Saun-
Savrasov, Phys. Rev. B 83, 205101 (2011). ders College, Philadelphia, 1976).
[7] Z. Wang, Y. Sun, X. Q. Chen, C. Franchini, G. Xu, [31] Inthiswork,theresultsofFig.2andFig.3areobtained
H.Weng,X.Dai, andZ.Fang,Phys.Rev.B85,195320 using δ=10 following Ref. 20.
(2012). [32] J. M. Ziman, Models of Disorder: The Theoretical
[8] A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, Physics of Homogeneously Disordered Systems (CUP
127205 (2011). Archive, 1979).
[9] S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolot- [33] Q. Li, E. H. Hwang, and E. Rossi, Solid State Com-
nyy,B.Bu¨chner, andR.J.Cava,Phys.Rev.Lett.113, mun. 152, 1390 (2012).
027603 (2014). [34] V.D.Bruggeman,Annalenderphysik416,636(1935).
[10] M. Ali, Q. Gibson, S. Jeon, B. Zhou, A. Yazdani, and [35] R.Landauer,inElectrical Transport and Optical Prop-
R. Cava, Inorg. Chem. 53, 4062 (2014). erties of Inhomogeneous Media, Vol. 40 (AIP Publish-
[11] T. Liang, Q. Gibson, M. Ali, M. Liu, R. Cava, and ing, 1978) pp. 2–45.
N. Ong, Nat. Mater. (2014). [36] E. Rossi, S. Adam, and S. Das Sarma, Phys. Rev. B
[12] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, 79, 245423 (2009).
G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, [37] V. Guttal and D. Stroud, Phys. Rev. B 71, 201304
C.-C. Lee, et al., Science 349, 613 (2015). (2005).
[13] S. Xu, Y. Xia, L. Wray, S. Jia, F. Meier, J. Dil, J. Os- [38] Y. Ando (private communication).
terwalder,B.Slomski,A.Bansil,H.Lin,R.Cava, and [39] A.A.Burkov,arXivpreprintarXiv:1505.01849 (2015).
M. Z. Hasan, Science 332, 560 (2011). [40] D. Son and B. Spivak, Physical Review B 88, 104412
[14] B. Singh, A. Sharma, H. Lin, M. Z. Hasan, R. Prasad, (2013).
and A. Bansil, Phys. Rev. B 86, 115208 (2012). [41] P.Goswami,J.Pixley, andS.D.Sarma,arXivpreprint
[15] Q. D. Gibson, L. M. Schoop, L. Muechler, L. S. Xie, arXiv:1503.02069 (2015).
M. Hirschberger, N. P. Ong, R. Car, and R. J. Cava, [42] E.H.Hwang,S.Adam, andS.DasSarma,Phys.Rev.
Physical Review B 91, 205128 (2015). B 76, 195421 (2007).
[16] Y.OminatoandM.Koshino,Phys.Rev.B89,054202 [43] M. M. Parish and P. B. Littlewood, Nature 426, 162
(2014). (2003).
[17] B. Sbierski, G. Pohl, E. J. Bergholtz, and P. W. [44] J. Ping, I. Yudhistira, N. Ramakrishnan, S. Cho,
Brouwer, Phys. Rev. Lett. 113, 026602 (2014). S. Adam, and M. S. Fuhrer, Phys. Rev. Lett. 113,
[18] H.-Z.LuandS.-Q.Shen,PhysicalReviewB92,035203 047206 (2014).
(2015). [45] T.KawamuraandS.D.Sarma,SolidStateCommuni-
[19] R. Lundgren, P. Laurell, and G. A. Fiete, Phys. Rev. cations 100, 411 (1996).
B 90, 165115 (2014). [46] J. Rammer, Quantum Transport Theory (Perseus
[20] L.MinandS.C.Zhang,Int.J.Mod.Phys.B27(2013). Books, 1998).
[21] B. Roy and S. Das Sarma, Phys. Rev. B 90, 241112 [47] An impurity concentration of 10−4 −10−2 impurities
(2014). per unit volume should be considered quite dense ac-
[22] S. Das Sarma, E. H. Hwang, and H. Min, Physical cording to this principle.
Review B 91, 035201 (2015). [48] V. M. Galitski, S. Adam, and S. Das Sarma, Physical
[23] A. A. Burkov, Phys. Rev. Lett. 113, 247203 (2014). Review B 76, 245405 (2007).
[24] R. R. Biswas and S. Ryu, Phys. Rev. B 89, 014205 [49] R. Kubo, M. Toda, and N. Hashitsume, Statistical
(2014). Physics (Springer-Verlag, 1991).