Table Of ContentPhys. Rev. B (accepted)
Weak antilocalization in topological insulator Bi Te microflakes
2 3
Shao-Pin Chiu1 and Juhn-Jong Lin1,2,∗
1Institute of Physics, National Chiao Tung University, Hsinchu 30010, Taiwan
2Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan
(Dated: January 11, 2013)
Wehavestudiedthecarriertransportintwotopologicalinsulator(TI)Bi2Te3microflakesbetween
0.3 and 10 K and under applied backgate voltages (VBG). Logarithmic temperature dependent
resistancecorrectionsduetothetwo-dimensionalelectron-electroninteractioneffectinthepresence
3
of weak disorder were observed. The extracted Coulomb screening parameter is negative, which is
1
in accord with the situation of strong spin-orbit scattering as is inherited in the TI materials. In
0
particular, positive magnetoresistances (MRs) in thetwo-dimensional weak-antilocalization (WAL)
2
effectweremeasuredinlowmagneticfields,whichcanbesatisfactorilydescribedbyamultichannel-
n conduction model. Both at low temperatures of T < 1 K and under high positive VBG, signatures
a of the presence of two coherent conduction channels were observed, as indicated by an increase
J
by a factor of ≈ 2 in the prefactor which characterizes the WAL MR magnitude. Our results are
0 discussed in terms of the (likely) existence of the Dirac fermion surface states, in addition to the
1 bulkstates, in thethree-dimensional TI Bi2Te3 material.
] PACSnumbers: 73.20.Fz,72.15.Rn,73.25.+i
l
l
a
h
I. INTRODUCTION thevariousissues,suchasthecouplingmechanisms(e..g,
-
s inelasticelectronrelaxation)betweenthesurfaceandthe
e
bulk states, the electron/hole dephasing processes, the
m Topological insulators (TIs) belong to a new class of
Coulomb interaction effect in the limit of strong spin-
quantum matters, where unique gapless surface states
. orbit scattering, and the band bending effect in individ-
t
a with linear energy-momentum dispersion characteristics ualTI materials,is urgentlycalledforbefore a quantita-
m (i.e., Dirac fermion states) coexist with gapped bulk
tiveanalysisanddefinitiveconclusionaboutthecoherent
states.1–3 As a consequence of the inherit strong spin-
- surface transport could be unambiguously drawn.
d orbitcoupling,thespinorientationsofthehelicalsurface
n states are transversely locked to their translational crys- InthisworkwehavestudiedtheMRsintwoexfoliated
o tal momenta. In the absence of any magnetic scattering, Bi2Te3 microflakes between 0.3 and 10 K and under ap-
c
[ the time-reversal coherent backscattering would thus be pliedbackgatevoltagesVBG. We havefoundthe positive
suppressed, leading to the well-knownweak antilocaliza- MRs in the WAL effect in small perpendicular magnetic
1 tion (WAL) effect which manifests positive magnetore- fields B. Our results indicate an emergence of two co-
v sistances (MRs) in low perpendicular magnetic fields.4–7 herent conduction channels as either the temperature is
3
reduced to below 1 K or V is increased to be higher
2 Although an observation of the two-dimensional (2D) BG
thanafewtensofvolts. Thatis,theprefactorα[Eq.(2)]
0 WAL effect can be a signature of the existence of TI
2 surface states, the strong spin-orbit coupling in the bulk whichcharacterizestheWALMRmagnitudeincreasesby
. afactorof≈ 2from≈0.35to ≈0.7. These observations
1 conduction channel can also contribute to a WAL effect.
arediscussedintermsofthepossible(partial)decoupling
0 This situation often happens in the three-dimensional
of the surface states from the bulk states. Moreover, we
3 (3D) TIs, such as the p-type Bi Te and the n-type
2 3
1 haveobservedthe2Delectron-electroninteraction(EEI)
Bi Se , owing to the high levels of unintentional dop-
: 2 3 effect in the weakly disordered regime, which caused a
v ing which readily occurs during the sample growth as
logarithmictemperaturedependentresistanceriseatlow
i well as the device fabrication processes. Therefore, the
X temperatures. The extracted Coulombscreening param-
MR data of 3D TIs have often been analyzed in terms
r eter is negative, which faithfully reflects a situation of
a of a multichannel-conduction model which considers the
strong spin-orbit scattering as is inherited in the TI ma-
potential contributions from both the surface and the
terials.
bulk states.8–14 Due to the small surface-to-volume ra-
tios in real samples, a clear-cut separation of the possi- This paper is organized as follows. Section II con-
blesurfacecontributionfromtheoverallcarriertransport tains our experimental method for sample fabrication
has remained nontrivial. Even if a surface contribution and electrical-transport measurements. Section III con-
were separated, it still poses a very difficult task to as- tains our experimental results of resistance and MR as
sociate the contribution with either the top or the bot- functions of temperature, magnetic field, and backgate
tom surface states, or both. To reach a consensus on voltage. Comparisonandanalysesbasedonexistingthe-
this issue will definitely require more experiments em- oreticalconceptsandpredictionsaremade. Possiblelim-
ployingspecificallydesignedsamples(e.g.,withbothtop itationsonthededucedinformationbasedoncurrentthe-
andbottomgates)andwithimprovedmaterialqualities. oretical understanding are discussed. Our conclusion is
Equally important, a good theoretical understanding of given in Sec. IV.
2
II. EXPERIMENTAL METHOD
SinglecrystalsofBi Te werepreparedbymeltingand
2 3
annealing high-purity Bi Te and Te powders (99.999%
2 3
purity) in a sealed quartz ampoule which was continu-
ously stirred. The temperature was rapidly increased to
1000 ◦C for a few hours, and then slowly reduced to 500
◦C to allow the crystalline nucleation during a period of
five days. A subsequent annealing for another five days
wasfollowedtoallowtheampouletemperaturetoreduce
slowly from500 to 420 ◦C. The mixture was then cooled
toroomtemperature. Thex-raypowderdiffractionstudy FIG. 1. (Color online) (a) Resistivity as a function of tem-
demonstratedthegenuinecrystallineconditionofno. 166 perature for BT-15 and BT-24 devices. Inset: an optical mi-
space group as referred to the PDF card 820358. crograph of the BT-24 device. The scale bar is 10 µm. (b)
Sheetresistanceasafunctionofthelogarithmoftemperature
Bi Te microflakeswereplacedon300-nmSiO capped
2 3 2
of BT-15 and BT-24 at low temperatures. The straight lines
highly doped n-type Si substrates. Four-probe 20-nm
are least-squares fitsto Eq. (1).
thickAuelectrodesonthemicroflakeswerefabricatedby
photolithography. The inset of Fig. 1(a) shows an opti-
calmicrographofthe BT-24device. The thickness t and
T dependences[Fig.1(b)],stronglysuggestingadisorder
widthwofthemicroflakesweremeasuredwithanatomic
related phenomenon. The sheet resistance R(cid:3) of BT-24
forcemicroscope. Inthepresentexperiment,therelevant
increaseswithdecreasingT belowabout4K.Incontrast,
sample length L was taken to be the closest distance be-
the BT-15 device has a sheet resistance of R(cid:3)(10K) =
tween the two voltage probes in a four-probe configura-
17.1 Ω, which is nearly one order of magnitude smaller
tion. Thismannerofdefiningtheeffective samplelength
than that (124 Ω) of BT-24. As a consequence, R(cid:3) re-
could cause an error bar in the extracted sample resis-
mains roughly constant below 6 K in BT-15. The lnT
tivity (and thus, the sheet resistance R(cid:3) = ρ/t) by an increase in BT-24 between 0.4 and 4 K is due to the 2D
amountofatmost10%,seetheSupplementalMaterial.15
EEIeffectinthe weaklydisorderedregime. Similarloga-
MR measurements were performed on an Oxford He-
liox 3He cryostat with a base temperature of 250 mK rsietvhemrailcTteImsapmerpalteus.r8e,1d4,e1p7eTnhdeenEcEeIocfoRrr(cid:3)echtiaosnbteoetnhesereensidin-
and equipped with a 4-T superconducting magnet. The ual resistance in a quasi-2D conductor is given by18,19
temperature was monitored with calibrated RuO and
2
Cernox thermometers. The resistances were measured △R(cid:3)(T) e2 3 T
using aLinear ResearchLR-700acresistancebridge and R(cid:3)(T0) =−2π2~(cid:18)1− 4F(cid:19)R(cid:3)ln(cid:18)T0(cid:19) , (1)
a Stanford SIM-921 ac resistance bridge. An excitation
current of 100 nA was applied to avoidelectron heating. where △R(cid:3)(T) = R(cid:3)(T)−R(cid:3)(T0), 2π~ is the Planck
That is, the voltage drop along the sample length was
constant, F is an electron screening factor, and T is a
0
. k T/e with this amount of excitation current, where
B reference temperature (taken to be 4 K in this work).
k is the Boltzmann constant, and e is the electronic
B From least-squares fits to Eq. (1), we obtain 1−3F/4 =
charge. A Keithley Model 2635A dc sourcemeter was
1.33 and F = −0.43, while the EEI theory in its sim-
utilizedtoprovidebackgatevoltages. Ourdeviceparam- plest form predicts 0 . F . 1.20 Recently, a negative
eters are listed in Table I. F value has also been extracted by Takagaki et al.8 in
Cu-doped Bi Se , but they did not connect their obser-
2 3
vation and discussion in terms of the strong spin-orbit
III. RESULTS AND DISCUSSION
coupling property of the TI materials. In the original
theory of Altshuler et al.,21 the triplet term in the EEI
A. Temperature dependence of sheet resistance effect in the diffusion channel is found to be suppressed
by strong spin-orbit coupling. However, the theory does
The temperature dependence of resistivity ρ of BT-15 not yield a negative F value. Empirically, Wu et al.22
and BT-24 reveals overall metallic behavior, as shown had previously demonstrated, by using a series of TiAl
in Fig. 1(a). A metallic feature of ρ(T) has often been alloysdopedwithdiluteheavyAuatoms,thatsufficiently
observedinBi Te andBi Se materialsbecausetheun- strongerspin-orbitcouplingcancauseamorenegativeF
2 3 2 3
intentional defects (free carriers) are readily generated value. Whether this is also the origin in the TI mate-
during the sample growth as well as the device fabrica- rials deserves further theoretical clarification.23 Here we
tion processes, which cause the Fermi level to shift into also would like to point out that, due to the finite width
the bulk valence or conduction band. In Bi Te , the of our voltage leads as compared with the relevant sam-
2 3
anti-structural defects in which Bi atoms are found on plelengthinafour-probeconfiguration,ourevaluationof
Te sites are responsible for the hole doping.16 At liquid- R(cid:3) couldbeoverestimatedbyanamountofatmost10%
heliumtemperatures,ourtwodevicesshowverydifferent (see the Supplemental Material15). Taking the largest
3
TABLE I. Parameters for two Bi2Te3 microflake devices. t is the thickness, w is the width, L is the device length (i.e., the
closestdistancebetweenthetwovoltageprobesinafour-probeconfiguration),µistheholemobility,pistheholeconcentration,
l is theelastic mean free path,and D is thediffusion constant. The values of µ, p,l and D are for 10 K.
Device t w L R(300K) ρ(300K) R(10K) ρ(10K) µ p l D
(nm) (µm) (µm) (Ω) (mΩcm) (Ω) (mΩcm) (cm2/Vs) (cm−3) (nm) (cm2/s)
BT-15 270 1.9 3.9 111 1.48 35.0 0.464 1690 8.0×1018 69 36
BT-24 65 2.8 1.9 424 4.06 84.0 0.805 1360 5.7×1018 50 23
possible 10% uncertainty in R(cid:3) [which appears on the and experimental studies.
right-handside of Eq. (1)] into account, our extractedF
value can be recalculated to be F ≃ −0.52±0.09.
Figure 2(a) shows the MR data of the BT-15 and BT-
Figure 1(b) also plots the R(cid:3)(T) of BT-24 under an
24 devices at 0.3 K with the B field applied perpendicu-
applied backgate voltage of V = +60 V. The overall
BG
lar to the microflake plane. Note that the MR (ignoring
R(cid:3)(T) curve is slightly higher than the corresponding the dip around B = 0) reveals an approximate B2 de-
curve under zero backgate voltage, indicating hole dop-
pendence in the low magnetic field regime of |B| . 1
ing. The straight line is a least-squares fit to Eq. (1). It
T, see the inset of Fig. 2(a). This may be tentatively
indicates acoefficientof1−3F/4=1.23andF =−0.31.
ascribed to the classical MR and approximated by:27
(If taking a possible 10% overestimate in the R(cid:3) value [R(B)−R(0)]/R(0) ≃ (µB)2. We thus obtain the mo-
into account, our F value would read F =−0.39±0.08.)
bility µ ≈ 1690 (1360) cm2/Vs in BT-15 (BT-24). The
This result illustrates that the EEI effect persists in
hole concentration p = 1/(ρ|e|µ) is then calculated to
Bi Te when a large positive V was applied and the
2 3 BG be ≈ 8.0 × 1018 (≈ 5.7 × 1018) cm−3 in BT-15 (BT-
surfaceandbulkstatesbecame(partly)decoupled. (The
24).28 These values of µ and p are in line with those
possible decoupling under a V = +60 V is asserted
BG
values deduced for the Bi Te samples with compatible
through the measurements of the MR dips in the WAL 2 3
resistivities.12 Usingthefree-electronmodel,weestimate
effect, see below.) On the other hand, in Eq. (1), since
the thermal diffusion length L = D~/k T > 65 nm
the magnitude ofthe resistancecorrectiondue to the 2D T B
at T . 4 K in BT-24 (D being thepdiffusion constant).
EEI effect scales linearly with R(cid:3), a lnT increase in R(cid:3)
Therefore, the 2D EEI effect may occur at low T in this
thus must be minute in BT-15, as mentioned.
device, as we have seen above.
In the above discussion, we have ignored the WAL
correction to R(cid:3)(T). Theoretically, the WAL effect
is known to cause an opposite decrease of R(cid:3) with Furthermore, the overall feature of the MR curves of
reducing T, and is given by19,25 △R(cid:3)(T)/R(cid:3)(T0) = BT-15andBT-24inthe wide magneticfieldrange|B|≤
−αp˜(e2/2π2~)R(cid:3)ln(T/T0),whereαisdefinedinEq.(2), 4 T are nearly T independent between 0.3 and 10 K and
and p˜is the exponent of temperature in the electron de- canbe satisfactorilydescribedby thepowerlaw[R(B)−
phasing time τϕ ∝ T−p˜. In our case, this contribution R(0)] ∝ |B|γ, with γ = 1.53±0.01 (1.55±0.01) for BT-
doesnotseemtobecome importantuntil T isloweredto 15 (BT-24), see the solid curves drawn through the MR
subkelvin temperatures. Indeed, the seemingly satura- datainthemainpanelofFig.2(a). TheauthorsofRefs.
tion of R(cid:3) below ∼ 0.4 K in BT-24 under zero backgate 8 and 29 proposed that a magnitude of an intermediate
voltageis not due to the Joule heating (as was discussed power of 1 < γ < 2 would be suggestive of the presence
intheExperimentalMethod),itismostlikelyasignature of multiple types of carriers.
oftheonsetoftheWALeffect. Thisinterpretationissup-
ported by the fact that an onset of the downwarddevia-
tionfromthe lnT dependence occursataslightlyhigher In the inset of Fig. 2(a) we plot the MR curves mea-
T ≈ 0.7 K when a V = +60 V was applied, which suredinthelowfieldregimeof|B|≤0.4TandatT =0.3
BG
caused a slightly more negative α value [Fig. 4(b)], and K. A resistance dip in the MR curve of BT-24 is clearly
hence a slightly larger WAL contribution.4,19 We may seen, manifesting the WAL effect. On the other hand,
roughly estimate that the EEI term is ∼ 5 times greater the WAL effect in BT-15 is obscured by the relatively
than the WAL term, i.e., (1−3F/4)/|αp˜|≈5, where we largeuniversalconductance fluctuations (UCFs).30–32 In
have used our experimental values of α ∼ −0.5 and p˜≈ the rest of this paper, we shall thus focus our analysis of
0.5 (see Sec. III B). In short, the observation of increas- the WAL effect only on the BT-24 sample. It is, how-
ing R(cid:3) with decreasing T in Fig. 1(b) confirms the im- ever, worth noting in passing that the UCF signals do
portant role played by the 2D EEI effect in the presence allowus to extractthe electrondephasing length L (T).
ϕ
of both weak disorder and strong spin-orbit coupling in Our analysis of the root-mean-square UCF magnitudes
Bi Te .26 To the best of our knowledge, the connection at various temperatures led to L values which are con-
2 3 ϕ
of negative F˜ values with strong spin-orbit coupling in sistent to within ∼ 40% with those corresponding values
TIs has not been pointed out in any previous theoretical deduced from the WAL method, see Fig. 3(d).
4
FIG.2. (Color online)(a)MRcurvesofBT-15andBT-24in
perpendicular B field at T = 0.3 K. The solid curve drawn
throughBT-15(BT-24)describesa|B|γ powerlaw,withγ =
1.53 (1.55). Theinset shows thelow magneticfield regimeof
|B|≤0.4T.ThesolidcurvedrawnthroughBT-24describesa
B2 powerlaw. (b)MRcurvesofBT-24inperpendicularand
parallelB fieldsatT =0.3K.Thesolidcurvedrawnthrough
the parallel MR data is the 2D WAL theoretical prediction
withanelectrondephasinglengthL ≃420nm(seeRefs. 34
ϕ
and 35). The inset shows the large magnetic field regime of FIG. 3. (Color online) BT-24 microflake in perpendicular B
|B|≤ 4 T. fields and under VBG = 0. (a) MR curves at several T val-
ues (from top down): 0.30, 0.50, 1.0, 2.0, 5.0, and 10 K. The
curves are vertically offset for clarity. (b) MR curves after
B. Weak-antilocalization magnetoresistance: subtractingawaythe10-Kbackgroundcurveandatfourrep-
Temperature dependence resentative T values, as indicated. The solid curves are the
theoretical predictions of Eq. (2). Note that universal con-
ductance fluctuations are visible. (c) Variation of extracted
InFig.2(b),weplottheMRcurvesofBT-24atT =0.3
parameter −α with temperature. As T decreases from 10 to
KandwiththeB fieldappliedeitherperpendiculartothe 0.3 K, −α increases by a factor of ≈ 2. (d) Variation of ex-
microflakeplaneorparalleltothemicroflakeplaneandin tracted electron dephasing length L with temperature. The
ϕ
the directionofthe currentflow. InbothB fieldorienta- straight dashed line is drawn proportional to T−0.24 and is a
tions, the MR dips aroundB = 0 areevident. Note that guide tothe eye.
in the parallel B field orientation, only the bulk states
can possibly contribute to the quantum-interference cor-
rection to the classical parabolic MR. Therefore, an ob- 10-K MR curve as the background curve and subtract
servedMRdipinthis caseunambiguouslyindicatesthat it away from those MR curves measured at lower T val-
the bulk states must lie in the strong spin-orbit scatter- ues. This procedure gives rise to those WAL MR curves
ing limit, giving rise to the WAL effect. This observa- shown in Fig. 3(b) (for clarity, only four representative
tiondoesnotsupporttherecenttheoreticalpredictionof curves are plotted). The 2D WAL MR in perpendicular
a weak-localization effect (which should manifest a MR B fields and in the limit of strong spin-orbit scattering
“peak”) in the Bi Te material.33 Also, we have carried (which is pertinent to TIs) can be written as5–7,38
2 3
outleast-squaresfitsoftheparallelMRcurvetotheper-
tinent2DWALtheoreticalprediction34,35[thesolidcurve △R(cid:3)(B) e2 1 Bϕ Bϕ
=−α Ψ + −ln , (2)
inFig.2(b)]andobtainedanelectrondephasinglengthof [R(cid:3)(0)]2 2π2~(cid:20) (cid:18)2 B (cid:19) (cid:18) B (cid:19)(cid:21)
L (0.3K)≈420nm. Thislengthscaleis largerthanthe
ϕ
thickness (65 nm) of the BT-24 microflake, and hence where △R(cid:3)(B) = R(cid:3)(B) − R(cid:3)(0), Ψ(x) is the
our bulk channel must be 2D with regard to the WAL digammafunction,thecharacteristicmagneticfieldBϕ =
effect.36 Thus,in the caseofperpendicular B field orien- ~/(4eL2), and L = Dτ . α is a parameter whose
ϕ ϕ ϕ
tation,boththesurfaceandbulkstateswouldcontribute value reflects the numpber of conduction channels. For
to the 2DWAL effect andthe measuredMRcurveshave the 2D surface states in a 3D TI, α = −1/2 for a sin-
to be analyzed in terms of a multichannel-conduction gle coherenttopologicalsurface channel, and α=−1 for
model. two independent coherent transport channels with simi-
In Fig. 3(a), we show the MR curves of BT-24 mea- lar L ’s. In practice, for yet to be identified reasons,the
ϕ
sured in the low, perpendicular magnetic field regime of experimentallyextractedαvalues oftendiffer fromthese
|B| ≤ 0.4 T and at several T values between 0.3 and 10 two ideal values. For simplicity and also in order to re-
K, as indicated in the caption to Fig. 3. Note that the duce the number of adjustable parameters, in this work
curves are vertically offset for clarity.37 It can be seen wetakeL asaneffectivedephasinglengthaveragedover
ϕ
thattheMRdipsincreaseswithdecreasingT,suggesting the channels.39 The main objective of this paper is thus
themorepronouncedquantum-interferencecarriertrans- to showthat the value ofα is indeedtunable, whichto a
port at lower T. As T increases, the MR dip eventually good extent seems to signify the existence of conductive
diminishes around T ∼ 10 K. Therefore, we take this surface states.
5
Our measured MR data plotted in Fig. 3(b) can be applying the WAL theory in its current form5 to the TI
well described by the predictions of Eq. (2) (the solid Bi Te material. Whether any modifications might need
2 3
curves). The extracted α(T) and L (T) values are plot- to be incorporated into Eq. (2) to take into account any
ϕ
ted in Figs. 3(c) and 3(d), respectively. Our experimen- subtle effect(s) that might arise from the multichannel
tal value of −α monotonically increases from ≃ 0.35 to feature and the specific materials properties of Bi Te
2 3
≃0.64asT isreducedfrom10to0.3K.(Iftakingapos- have to await future theoretical investigations. For in-
sible 10% uncertainty in the R(cid:3) value into account, our stance, it would be of interest and importance to see if
extracted −α value would be slightly modified to vary the frequent experimental observations of an |α| value
from 0.39 to 0.70.) This variation of α with T is well smaller than 0.5 could be answered.
beyond our experimental uncertainties, and it is plau-
sible to think that such a change in the α value by a
factor of ∼ 2 does reflect a decoupling (doubling) of the
C. Weak-antilocalization magnetoresistance:
charge transport channels at sufficiently low T. Note Backgate-voltage dependence
that several recent studies of Bi Se have also reported
2 3
temperature dependent α values which lie between −0.3
We havefurther performedthe MRmeasurementsun-
and −0.6, but not between −0.5 and −1.9–14 For the
der different applied backgate voltages V and at T
Bi Te material, to our knowledge, the temperature be- BG
2 3 = 0.3 K. Figure 4(a) clearly shows that the size of the
havior of α has not yet been reported in the literature.
MR dip increases monotonically as V progressively
He et al. have recently reported an α = −0.39 at T BG
increases from −40 to +60 V. From least-squares fits to
= 2 K,12 while our value is α(2K) ≃ −0.54. In our
Eg. (2), we have obtained the values of α and L un-
case, an −α(0.3K) value smaller than unity might im- ϕ
der various V values. Figure 4(b) reveals that, as
ply that the decoupling is incomplete (partly due to the BG
V increases from −40 to +60 V, −α monotonically
high carrier density in BT-24). The exact reason why BG
increasesfrom ≃ 0.36 to ≃ 0.72,while L monotonically
ultralow temperature facilitates decoupling should await ϕ
decreases from ≃ 450 to ≃ 250 nm. (If taking a possi-
further theoretical investigations. Tentatively, if there
ble 10% overestimate in the R(cid:3) value into account, our
should exist an inelastic relaxation process between the
−αvaluewouldbeslightlymodifiedtovaryfrom0.40to
surface carriers and the bulk low-lying excitations (e.g.,
0.79, while the extracted L values are barely affected.)
surface-carrier–bulk-phononscattering24), the scattering ϕ
These amounts of variationin α and L are remarkable,
strengthwoulddecreasewithreducingT. Then,ahigher ϕ
since a change of V from −40 V to +60 V only causes
degreeofdecoupling ofthe surfaceandbulk states could BG
a ≃ 2% reduction in the hole concentration p in this
likely take place atlower T.40 For comparison,we would
device. [Such a minor change in p can be ascribed to
like to stress that in the conventional metals and alloys
the facts (1) that our applied V partly dropped in the
with strong spin-orbit scattering, such as Au-doped Cu BG
highly doped n-type Si substrate, (2) that the relatively
(Ref. 41) and AuPd (Refs. 38 and 42) thin films, the
thick SiO layer reduced the ability to electrostatically
MR dips in 2D WAL effect have been firmly observed, 2
modulate the interface carrier density, and (3) that our
where one always finds a constant α = −1/2 in the wide
sample possessed a relatively high p level. Recall that,
T range from subkelvin temperatures up to above 20 K.
in Fig. 1(b), the R(cid:3) value of BT-24 increased only by a
Our extracted Lϕ values of BT-24 are plotted in tiny amountof . 0.2%as VBG was variedfrom0 to +60
Fig. 3(d). As T decreasesfrom10 to 0.3K, L increases V.] For comparison,we would like to point out that sev-
ϕ
from ≃ 200 to ≃ 440 nm. The straight dashed line is eral previous experiments using the n-type Bi2Se3 have
drawn proportional to L ∝ T−0.24 and is a guide to also found that, as the electron density (n) is lowered
ϕ
the eye. This slope corresponds to an effective exponent by a sufficiently large negative VBG, the value of −α in-
of temperature p˜ ≃ 0.48 in τ ∝ T−p˜, which is consid- creases (i.e., α becomes more negative) while the value
ϕ
erably lower than that (p˜ = 1) would be expected for of Lϕ decreases.9–11,13 Since the p/n carriersin the bulk
the quasielasticNyquistelectron-electronscatteringpro- channel are (partly) depleted away under a large posi-
cess in 2D.43,44 This suggests the existence of additional tive/negativeVBG,adecouplingofthetwochannelsthus
weakly T dependent magnetic or nonmagnetic electron couldpossiblytakeplace. Thisinturnisreflectedbythe
dephasing processes which are noticeable in this device increase in the −α value by a factor of ≈ 2.
over our measurement temperature range.41,45 Our ex- Becauseourextracted−αvalueisalwayssmallerthan
tracted L values are larger than the thickness of the unity, we have presumed that there exists only one con-
ϕ
microflake, justifying the 2D WAL characteristics in our ductivesurfacechannelinBT-24. Thispresumptionisin
samples. Furthermore, our value of L (0.3K) inferred accordwitharecentreportbyTakagakiet al.8 whohave
ϕ
from the perpendicular MR, Eq. (2), is in good agree- pointedoutthattheseeminglyabsenceoftheWALeffect
ment with that (≃ 420 nm) deduced from the parallel foroneofthesurfaceswasacommonfindinginanumber
MR, see Fig. 2(b) and Ref. 35. This close agreement ofstudies of3DTIs.9–12,14,17 Inourcase,the topsurface
in the extracted L (B ) and L (B ) values strongly in- layermight have been deterioratedowing to its constant
ϕ ⊥ ϕ k
dicates the self-consistency of our experimental method exposure to the air atmosphere as well as its undergoing
and data analysis as well as the reasonable validity of the device fabrication process. Furthermore, our BT-24
6
other explanations, for example, based on a change in
the degree of coupling between bulk and surface states,
whichwouldtake place inthe presence of largebackgate
voltages. A quantitative account for the suppression of
L by V must awaitfurther theoreticalinvestigations.
ϕ BG
IV. CONCLUSION
FIG. 4. (Color online) BT-24 microflake at T = 0.3 K. (a)
MRdatameasuredinperpendicularB fieldandunderseveral We have measured the resistances and magnetoresis-
VBG values, as indicated. The data are vertically offset for tancesoftwoBi2Te3microflakesatlowtemperaturesand
clarity. Thesolidcurvesareleast-squaresfitstoEq.(2). Note underappliedbackgatevoltages. Low-temperatureresis-
that resistance fluctuations are visible. (b) Variation of the tancecorrectionsduetothe2DEEIeffectinthepresence
extracted parameter −α and electron dephasing length Lϕ of weak disorder are observed. The extracted Coulomb
with VBG. The dashed curvesare guides tothe eye. screening parameter F is negative, which is in line with
thesituationofstrongspin-orbitscatteringasisinherited
in the TI materials. Positive MR dips in small perpen-
is 65 nm thick and highly doped, and hence the top sur-
dicular magnetic fields are measured, which can be sat-
facestate,ifanyexists,wouldbetoofarawaytobetuned
isfactorily described by the existing 2D WAL theory by
by V . Therefore, we tentatively ascribe the observed
BG taking into account a multichannel-conduction model in
conductive surface states to be associated with the bot-
aTImaterial. Ourextractedα(T,V )values,whichin-
BG
tomsurfaceofthemicroflake. Itisalsoimportanttonote
creasebyafactorof≈2from≈0.35to≈0.7,suggestthe
that we have ascribed the additional coherent transport
emergenceof two coherentconduction channelsas either
channel to the TI bottom states, rather than to an elec-
the temperature decreases to subkelvin temperatures or
tron accumulation layer. To rule out the latter scenario,
the backgate voltage increases to several tens of volts.
we have measured the low-T resistance as a function of
This doubling of the conduction channels seems to sig-
backgate voltage in a BT-23 microflake which possessed
nify the (partial) decoupling of the surface and the bulk
very similar electronic properties to those of the BT-24
states. In short, our overall results provide qualitative,
microflake. OurmeasuredresistancecurveR(V )atev-
BG butnotyetquantitative,evidenceforthelikelyexistence
ery temperature always revealed a maximum at a char-
of Dirac fermion states in the 3D Bi Te material. To
acteristic backgate voltage V∗ , which signified that the 2 3
BG achieve a clear-cut separation of the contributions from
Fermi energy was indeed shifted to become aligned with
theindividualsurfaceandbulkstatesundoubtedlywould
the Dirac point at this particular backgate voltage (see
requirefurthertheoreticalformulationsandexperimental
further discussion in the Supplemental Material15).
measurements.
Finally, the reasonfor the markedeffect ofV onL
BG ϕ
[Fig. 4(b)] is theoretically less clear. Under a large pos-
itive V , the carriers are slightly depleted away from
BG
ACKNOWLEDGMENTS
the bottom surface. The depletion would certainly lead
to deteriorated Coulomb screening between the carriers,
leadingtoanenhancedelectron-electron(hole-hole)scat- The authors thank Zhao-guo Li, Tai-shi Chen and
tering rate and thus a shortened L . However, as men- Feng-qi Song for providing us with the samples used in
ϕ
tioned, the hole concentrationis reducedonly by 2%un- this study andfor their reading throughanearly version
der a V = +60 V in this BT-24 device. It is thus con- of the manuscript. This work was supported by the Tai-
BG
jectured whether such a small reduction in p could have wan National Science Council through Grant No. NSC
already caused a notable change in L . There could be 100-2120-M-009-008and the MOE ATU Program.
ϕ
∗
Electronic address: [email protected] (2006).
1 M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 7 G. Tkachov and E. M. Hankiewicz, Phys. Rev. B 84,
(2010). 035444 (2011).
2 J. E. Moore, Nature (London) 464, 194 (2010). 8 Y. Takagaki, B. Jenichen, U. Jahn, M. Ramsteiner, and
3 X.L.QiandS.C.Zhang,Rev.Mod.Phys.83,1057(2011). K.-J. Friedland, Phys. Rev.B 85, 115314 (2012).
4 G. Bergmann, Phys. Rep.107, 1 (1984). 9 H. Steinberg, J.-B. Laloe, V. Fatemi, J. S. Moodera, and
5 S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor. P. Jarillo-Herrero, Phys. Rev.B 84, 233101 (2011).
Phys. 63, 707 (1980). 10 J. Chen, X. Y. He, K. H. Wu, Z. Q. Ji, L. Lu, J. R. Shi,
6 E. McCann, K. Kechedzhi, V. I. Fal’ko, H. Suzuura, T. J. H. Smet, and Y. Q. Li, Phys. Rev. B 83, 241304(R)
Ando, and B. L. Altshuler, Phys. Rev. Lett. 97, 146805 (2011).
7
11 J. Chen, H. J. Qin, F. Yang, J. Liu, T. Guan, F. M. Qu, 28 The p value determined this way is a few times smaller
G. H. Zhang, J. R. Shi, X. C. Xie, C. L. Yang, K. H.Wu, than thatinferred from theHall measurement on theBT-
Y.Q.Li,andL.Lu,Phys.Rev.Lett.105, 176602 (2010). 15 device. This difference is not critical for the present
12 H.-T. He, G. Wang, T. Zhang, I.-K. Sou, G. K. L. Wong, discussionanddoesnotaffecttheconclusiondrawninthis
J.-N.Wang,H.-Z.Lu,S.-Q.Shen,andF.-C.Zhang,Phys. work.
Rev.Lett. 106, 166805 (2011). 29 Y. Takagaki and K.-J. Friedland, J. Appl. Phys. 101,
13 J. G. Checkelsky, Y. S. Hor, R. J. Cava, and N. P. Ong, 113916 (2007).
Phys. Rev.Lett. 106, 196801 (2011). 30 P. A. Lee, A. D. Stone, and H. Fukuyama, Phys. Rev. B
14 M.Liu,C.-Z.Chang,Z.Zhang,Y.Zhang,W.Ruan,K.He, 35, 1039 (1987).
L.-L. Wang, X. Chen, J.-F. Jia, S.-C. Zhang, Q.-K. Xue, 31 P. Y. Yang, L. Y. Wang, Y. W. Hsu, and J. J. Lin, Phys.
X. Ma, and Y.Wang, Phys. Rev.B 83, 165440 (2011). Rev.B 85, 085423 (2012).
15 See Supplemental Material at [URL will be inserted by 32 S. Matsuo, T. Koyama, K. Shimamura, T. Arakawa, Y.
publisher] for (1) a detailed estimate of the error bar in Nishihara, D.Chiba, K.Kobayashi,T. Ono,C. Z.Chang,
our R(cid:3) value, and (2) a discussion of the possible band K.He,X.C.Ma,andQ.K.Xue,Phys.Rev.B85,075440
bending effect. (2012).
16 Y. S. Hor, D. Qu, N. P. Ong, and R. J. Cava, J. Phys.: 33 H.Z.LuandS.Q.Shen,Phys.Rev.B84,125138 (2011).
Condens. Matter 22, 375801 (2010). 34 N.GiordanoandM.A.Pennington,Phys.Rev.B47,9693
17 J. Wang, A. M. DaSilva, C.-Z. Chang, K. He, J. K. Jain, (1993).
N. Samarth, X.-C. Ma, Q.-K. Xue, and M. H. W. Chan, 35 S. P. Chiu and J. J. Lin, unpublished. In the case of
Phys. Rev.B 83, 245438 (2011). one and only one coherent conduction cahnnel as well
18 B. L. Altshuler, A. G. Aronov, and P. A. Lee, Phys. Rev. as in the limit of strong spin-orbit scattering (i.e., α =
Lett. 44, 1288 (1980). −1/2), the 2D WAL MR in parallel B fields is given by:
19 J. J. Lin and N.Giordano, Phys.Rev.B 35, 545 (1987). △R(cid:3)(B)/[R(cid:3)(0)]2 =(e2/4π2~)ln[1+(eB2t2)/(12~Bϕ)].
20 In the formal EEI theory, the screening parameter F 36 In Ref. 12, theauthors had studied a 50-nm thick Be2Te3
should be replaced by the parameter F˜ = (32/3F)[(1+ film at T = 2 K. They ascribed their measured MR dip
F/2)3/2 −1−3F/4]. However, F˜ equals F to within ≈ in parallel B fields to the 3D WAL effect but extracted
10% for −1.F .1. It is thus not critical to distinguish a comparatively large value of Lϕ = 331 nm. Then, their
between F˜ and F in most experiments. See, for example, bulk channel must be 2D, instead of 3D, with regard to
Ref. 22. theWAL effect.
21 B. L. Altshuler, A. G. Aronov, and A. Yu. Zuzin, Solid 37 Without verticaloffsets, ourmeasured MR curves(except
State Commun. 44, 137 (1982). the dips) at all temperatures below 10 K essentially over-
22 C. Y. Wu, W. B. Jian, and J. J. Lin, Phys. Rev. B 52, lap, dueto the metallic feature of thesample.
15479 (1995); J. J. Lin, S. Y. Hsu, J. C. Lue, and P. J. 38 J. J. Lin and N.Giordano, Phys. Rev.B 35, 1071 (1987).
Sheng, J. Phys. Chem. Solids 62, 1813 (2001). 39 In practice, the electron dephasing lengths of the sur-
23 A possible alternative explanation is that an increase in face and thebulkchannelscan not beunambiguouslydis-
R(cid:3)(T) larger than the theoretical prediction, i.e., (1 − tinguished. Nevertheless, according to the current under-
3F/4) > 1, might imply that both the bulk and surface standing(seeRef.9),thetwolengthscales[(Lϕ)surfaceand
states contribute to the measured 2D EEI effect, see Ref. (Lϕ)bulk] differ by no more than a factor of ∼ 2. This
8. Yet another explanation for negative F values is as fol- in turn could cause an error bar of . 40% in the differ-
lows. In Ref. 22, Wu et al. had reported that, under the ence,ifanyexists,betweentheeffectiveLϕandthesurface
circumstance of (intermediately) strong electron-phonon (Lϕ)surface values.
coupling (such as that in a BCS superconductor), F can 40 Analternativescenariowhichinvokedasurface-bulkscat-
take negative values. This early experimental observation tering process in low-carrier-density TIs was proposed in
mightbeinlinewiththerecenttheoreticalfindingreported Ref. 9. See also the recent theoretical calculations by I.
in Ref. 24, where Giraud and Egger have showed that the Garate and L. Glazman, Phys. Rev.B 86, 035422 (2012).
phonon-mediatedeffectiveinteractionamongDiracsurface 41 S. M. Huang, T. C. Lee, H. Akimoto, K. Kono, and J. J.
fermionscanbeattractiveatlowfrequencies,whichwould Lin, Phys.Rev. Lett.99, 046601 (2007).
then possibly allow for superconductingcorrelations. This 42 Y.L.Zhong,A.Sergeev,C.D.Chen,and J.J.Lin,Phys.
issuedeservesfurthertheoreticalandexperimentalstudies. Rev.Lett. 104, 206803 (2010).
24 S. Giraud and R. Egger, Phys.Rev.B 83, 245322 (2011). 43 B. L. Altshuler, A. G. Aronov, and D. E. Khmelnitsky,J.
25 E. Abrahams, P. W. Anderson,D. C. Licciardello, and T. Phys. C: Solid StatePhys. 15, 7367 (1982).
V. Ramakrishnan, Phys.Rev.Lett. 42, 673 (1979). 44 C. Y.Wu,B. T. Lin,Y.J. Zhang, Z.Q.Li, and J. J. Lin,
26 Strictlyspeaking,theaboveresultsdonotallowustosep- Phys. Rev.B 85, 104204 (2012).
arate the individual contributions of the surface and the 45 J. J. Lin and J. P. Bird, J. Phys.: Condens. Matter 14,
bulk states. R501 (2002).
27 J.S.Dugdale,TheElectricalPropertiesofDisorderedMet-
als (Cambridge UniversityPress, Cambridge, 1995).
