Variable range hopping theory for the nodal gap in strongly underdoped cuprate Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6+\delta}$

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by  Wei-Qiang Chen

Variable range hopping theory for the nodal gap in strongly underdoped cuprate
Bi Sr La CuO
2 2−x x 6+δ
Wei-Qiang Chen,1 Jun-Qiu Zhang,1 T. M. Rice,2 and Fu-Chun Zhang3,4
1Department of Physics, South University of Science and Technology of China, Shenzhen, China
2Institut fu¨r Theoretische Physik, ETH Zu¨rich, CH-8093, Zu¨rich, Switzerland
3Department of Physics, Zhejiang University, Hangzhou, China
4Collaborative Innovation Center of Advanced Microstructures, Nanjing, China
Recentangleresolvedphotoemissionspectrascope(ARPES)experimentsonstronglyunderdoped
6 Bi2Sr2−xLaxCuO6+δ cuprates have reported an unusual gap in the nodal direction. Transport
1 experiments on these cuprates found variable range hopping behavior observed. These cuprates
0 have both electron and hole doping which has led to proposals that this cuprate is analogous to a
2 partiallycompensatedsemiconductors. ThenodalgapthencorrespondstotheEfros-Schklovskii(ES)
gap in such semiconductors. We calculate the doping dependence and temperature dependence of
n
a ES gap model and findsupport for an Efros-Schklovskii model.
a
J

0
The parent compound of high temperature cuprate samples whosenet carrierconcentration,P ,in the CuO
2 2
superconductor is a Mott insulator with antiferromag- plane is P < 0.10. Superconductivity was observed in

] netic(AFM) long range order1. With doping of holes samples at the upper end of the doping range. Low hole
l into the system, the AFM order is suppressed at first doping in the planar CuO can only be achieved in this
e 2
- and finally vanishes at hole concentration ∼ 3%−5%. way, when La donors and O acceptors are arranged in
r
t When the hole concentration exceeds a critical doping tightly bound neutral states leaving a small number of
s
x , the system becomes superconducting. The evolu- remainingholesinthemoreextendedstatesintheCuO
. c 2
t tion of the system with hole concentrations is one of the planes. The valuesofthenetcarrierdensitiesthatresult
a
m central issues in the field. Many experiments have been was estimated from a series of experiments, e.g resistiv-
performed in the very underdoped and low temperature ity, Hall coefficient and thermoelectric power. This pro-
-
d regime and various phenomena have been observed in- cedure led to estimates of the critical concentration for
n clude spin glass, spin fluctuations, charge perturbations superconductivityofP =0.1. Thespatialdistributionof
o etc1. One problem is that the impurity effect in this thesedifferentstatesisshownschematicallyinFig1. We
c
regime is strong because of the low carrier density and willconcentrateonthenonsuperconductingsampleswith
[
high impurity concentration (each impurity introduces net hole concentrations P < 0.1. We simulate the den-

1 one hole). This has led to debates about which phenom- sity of states with varying energy and temperature and
v ena are intrinsic or extrinsic due to the impurities. So it compare with the results of the ARPES experiments.
4
istimelytoinvestigatetheimpurityeffectsinthisregime.
5
Recently,YingyingPengetal. performedanARPESand In the ARPES experiments, one observes a peak-dip-
3
5 transportexperimentsonaseriesofBi2Sr2−xLaxCuO6+δ hump structure. In the case with hole doping 0.03 <
0 samplesinthisregime. Theyobservedthevariablerange P < 0.1, a gap is observed below the peak. The magni-
. hopping behavior in the transport experiments and an tude of the gapon the underlying Fermi surface is found
1
energygapalongnodaldirectionintheARPESspectra2. to be k-dependent, minimal at the nodal direction and
0
6 By analogy to a partially compensated semiconductor, maximal in antinodal region. The gap value is reduced
1 we attribute the energy gap to the soft gap opened in with increasing hole doping and vanishes at P ∼ 0.10
: the impurity band due to the Coulomb interaction be- wherethesystemundergoesaninsulator-superconductor
v
tween the localized electrons. This indicates that the transition. With increasing the temperature, the peak
i
X lowenergyphysicsinthismaterialisdominatedbyimpu- becomes weaker and weaker. Another very important
r rity effects. Further experimental investigations of these phenomena observed in transport experiments on sam-
a materials will help to separate the extrinsic phenomena ples in this doping range, namely variable range hop-
originating from impurity effects from the intrinsic phe- ping(VRH) behaviorinthe resistivity2,3. VRHwaspro-
nomena. posed by Mott and has been well studied in semicon-
ductors with localized impurity states forming impurity
In this paper, we examine the results of a recent se- bands.4Inapartiallycompensatedsemiconductors,both
ries of ARPES and transport experiments on a series of acceptors and donors are present which form the accep-
Bi Sr La CuO samples with low hole doping, ly- tor bands anddonor bands respectively. In a hole doped
2 2−x x 6+δ
ing just below the critical doping for the onset of super- case, the donor bands are completely empty, while the
conductivity. These cuprates are unusual in that large acceptor bands are partially filled. At low temperatures,
concentrations of La donors were added, which are com- the hopping of holes can only happen between acceptor
pensated by O acceptors. A series of samples were pre- states which lie veryclose to the chemicalpotential. Be-
pared with a large La donor concentration of x = 0.84 cause the localised acceptors are randomly distributed,
whichwereannealedinanOatmosphere. Theresultwas acceptors with a small energy difference are well sepa-
2
rated. So the characteristic hopping length of holes will To investigate the properties of the ES gap, we follow
increase with decreasing temperature, leading to VRH. earlier calculations5,7,8 and study the following classical
VRH makes the system an insulator though the accep- Hamiltonian
tor band is only partially filled, leading to a resistivity
V n n
which varies as ρ0exp[(T0/T)α] with temperature. The H =Xniφi+ 2 X ri j, (1)
exponent α is 1/3 in 2D if there is a finite DOS at the ij
i i6=j
Fermilevel4. ButEfrosandSchklovskii(ES)pointedout
laterthatbecauseofthe longrangeCoulombinteraction where i is the index of the centre of an impurity in the
between the localised charges, a soft gap will open at acceptor bands, φi is the energy of the correspondinglo-
chemicalpotentialatzerotemperature5. Thedirectcon- calisedstate, ni =0,1 is the occupationnumber of holes
sequence of the opening of ES gap is that the exponent in the state, V = e2/4πǫrǫ0a is the coupling strength of
becomesα=1/2atverylowtemperature. Withincreas- theCoulombrepulsionbetweentwoholes,ǫr isthedielec-
ing temperature,the ESgapis filledandthe exponentα tricconstant,rij isthedistancebetweentwoimpuritiesi
will be reduced to 1/3. and j. In our calculation, we assume that the impurities
form a square lattice for simplicity. This approximation
E will not affect the result because only the states around
the chemical potential are important, and those states
acceptor are spatially separated and have randomly distributed
energies as pointed out in ref.8. As mentioned above,
mostO acceptorsformtightly boundneutralstates with
Valence Band
La donors and only a small fraction of extra O2− show
tightly
up in the acceptor band, i.e. the effective concentration
bound
n of O acceptors per Cu-site is much smaller than the
states a
nominal one δ = (x+P)/2= 0.42+P/2. This leads to
the uncertainty to estimate the chargecarrier’sfilling ν
h
in the acceptor band, which is defined by
FIG. 1. A schematic figure of the band structure in lightly
doped Mott insulators, Bi2Sr2−xLaxCuO6+δ with both hole ν =P/n , (2)
h a
and electron dopants, similar to partially compensated semi-
conductors. The upper Hubbard band(not shown) acts as
with P the concentration of doped holes in a Cu-oxide
a conduction band, and the valence band is the lower Hub-
plane. Since n < δ, we have ν > P/δ. While the
bard(or Zhang-Rice) singlet band. La3+ impurities form a a h
precise relationbetween n andδ, hence the relationbe-
donor band below the bottom of the conduction band, and a
thecompensatingO2− ionsform anacceptorbandabovethe tween νh and P, depends on the specific material, it is
top of thevalenceband. Someholesand electrons form local reasonable to assume νh and P are monotonic: a larger
bound states due to strong attraction between nearby La3+ value of νh corresponds to a larger P in ARPES. In the
and extra O2− ions, with energies well below the top of the calculations below, we consider three different values of
valenceband. Thechemicalpotentiallieswithintheacceptor ν : 0.88, 0.92, and 0.96.
h
band dueto theexcess of holes. First, we consider the density of states(DOS) at zero
temperature of a 10× 10 lattice with periodic bound-
IntheexperimentsonBi2Sr2−xLaxCuO6+δ,thetrans- ary conditions. The DOS are calculated and averaged
portdatawasfittedwiththeVRHformforbothα=1/3 over 106 impurity configurations. In the calculations,
and1/2. But the fitting to anexponent1/2is onlygood the bandwidth of the acceptor band is chosen to be
at low temperature, while the fitting to 1/3 is good at 200 meV, i.e. φ is generated randomly in the interval
i
muchlargertemperatureregion6. Ononehand,thisindi- [−100,100]meV. The Coulomb coupling strength V is
catesaVRHphysicsintheBi2Sr2−xLaxCuO6+δ system. chosen to 100meV which corresponds to ǫr ≈ 38. The
On the other hand, the exponent indicate a gap opens electron distribution with lowest energy for each config-
at low temperature which fills up at the temperature is uration is achieved with the procedure used in Ref. 7
raised. [Similarwiththe partiallycompensatedsemicon- and 8. At first, we generate an initial distribution of the
ductor, the Bi2Sr2−xLaxCuO6+δ material also has two holesrandomlyfor agivenimpurity configuration. Then
kinds of dopants, the La3+ and the extra O2− which in- we apply the so-called µ-sub procedure. We calculated
troducetheelectronsandholesrespectively. Soitisnatu- the single particle energy for each state with
raltomakeananalogybetweentheBi Sr La CuO
2 2−x x 6+δ
n
and a partially compensate semiconductors, where the Ei =φi+X j, (3)
La3+ and extra O2− play the role of donors and accep- rij
j6=i
tors respectively.]
As discussed above, the VRH behavior in wherer istheshortestdistancebetweentwositesiand
ij
Bi Sr La CuO material at low temperature j. If E of highest occupied site is larger than E of the
2 2−x x 6+δ i i
indicate the opening of an ES gap. This gap should lowestemptysite,wewillmovetheholefromthehighest
correspond to the gap observed in the experiments2,3. occupied site to lowest empty site. Then we recalculate
3
0.6
0.5
DOS(arb. unit) 000...234 OS(arb. units)
D
0.1
0
-0.1 -0.05 E-EF0(eV) 0.05 0.1(a) (b) -100 -50 0 50 100(a) (b)
E-E (meV)
F
FIG. 2. (a) Calculated electron density of states (DOS) for
various of filling factors of holes, νh in theimpurity acceptor FIG. 3. (a) Calculated DOS at impurity acceptor band fill-
band: νh =0.88(blue line), νh =0.92(green lines), and νh = ing factor νh =0.92 at various temperatures: kBT =0(black
0.94(red lines), respectively. (b) The ARPES data on energy line), 6meV(red line), 10meV(green line), and 20meV(blue
distributioncurve(EDC)forvariousholedopingsp,fromfig. line), respectively. (b) The ARPES result for energy distri-
3 in Ref. 2. The right panel in (b) shows only the occupied bution curve of at various temperatures, extract from fig. 4
electron states which are observed in ARPES. Note that the in Ref. 2. See the text for a discussion of the qualitative
position of theDOSpeak shown in (a) shifts towards zero as agreement between thetheory and experiment.
νh increases, qualitatively consistent with the ARPES data,
showing the peak position in EDC shifts towards zero as p
henceνh increases. highertemperaturetolowertemperaturewithastandard
Monte Carlo algorithm. At each temperature, we drop
thefirst104holeconfigurationstoremovethememoryof
E and do the check again and again until all E of oc-
i i higher temperature and keep the single particle energies
cupied sites are less than any Ei of the empty sites. for the next 105 hole configurations. By averaging over
Inthenextstage,wechecktheenergyforasinglehole,
all the impurity configurations,we get the DOS at finite
which hops with
temperature. The result for ν = 0.92 at temperatures
h
k T =0(blackline),6meV(redline),10meV(greenline)
1 B
Eji =Ej −Ei− , (4) and 20meV(blue line) are depicted in fig. 3(a). With
r
ij increasing temperature, the amplitude of the peak de-
creaseswhiletheDOSatthechemicalpotentialincreases
where i is an occupied site and j is an empty site. The
leading to the closing of the ES gap. Note the position
hop will be performed if E < 0. After each hop, we
ji
of the peaks shift only slightly to lower energy with in-
redo the µ-sub process and check all possible single hole
creasing temperature. These results are consistent with
hopuntilallE arepositive. Thenwetakethe resulting
ji
the experiment shown in fig. 3(b) extracted from fig. 4
hole distribution as one candidate ground state. After
in Ref. 2.
5,000iterationsofthewholeprocessforagivenimpurity
configuration,wechoosetheonewithlowestenergyfrom Anotherfeature ofthe ARPESexperiments is the mo-
all the candidates and choose it as the ground state for mentum dependence of the ES gap which is minimal at
that impurity configuration. thenodalpointandincreasesawayfromthenodalpoint.
With the single particle energy of the ground states, Thismomentumdependencemayoriginatefromthespa-
we can get the DOS by averaging over all the impurity tialdistributionoftheimpuritieswhichcouldbeaffected
configurations, the result is shown in fig. 2(a), where by various effects like clustering etc. at such high impu-
the DOS at ν = 0.88(blue line), 0.92(green line) and rity densities. But in our model, we only consider the
h
0.96(redline)aredepicted. TocomparewiththeARPES random distributed impurities because of the lack of in-
experiment,theDOSispresentedinelectronnotation. It formation on the actual spatial distribution.
is obvious that a soft gap opened at the chemical poten- We have carried out simulations of the electron den-
tial for all of the three cases at low temperatures. The sity of states for lightly doped cuprates with strong ran-
finite DOS at chemical potential at larger hole doping dom potentials and compensated electron and hole dop-
case may due to the finite size effect. The most impor- ings and compared them to the results of ARPES ex-
tant feature is that the peak position is shifted towards periments on nonsuperconducting Bi Sr La CuO
2 2−x x 6+δ
zeroastheholedopingpincreases,whichisqualitatively samples whose transport properties show variable range
consistentwiththeexperimentalresultshowninfig.2(b), hopping at low temperatures. The ARPES experiments
which is extracted from fig. 3 in Ref. 2. measure only states occupied by electron with energies
Then we study the DOS at finite temperature with strictlybelowthechemicalpotentialatlowtemperatures,
the Monte Carlo technique used in ref.8. At first, we extending to slightly above as T is raised. Our simula-
generate 104 random configurations of φ. For each con- tions show good agreement with the key features in the
figuration, we calculate the single particle energies from densityandtemperaturedependenceoftheelectronden-
4
sity of states and supportthe interpretationput forward Chen is partly supported by NSFC 11374135, and FC
by X. J. Zhou and collaborators of their experimental Zhang thanks NSFC 11274269, and National Basic Re-
results. search Programof China (No. 2014CB921203).
ACKNOWLEDGMENTS
We thank X. J. Zhou for interesting discussions and
allowing us to reuse the figures in their paper. WQ
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