Table Of ContentIntervalley scattering and localization behaviors of spin-valley coupled Dirac fermions
Hai-Zhou Lu,1 Wang Yao,1 Di Xiao,2 and Shun-Qing Shen1
1Department of Physics and Centre of Theoretical and Computational Physics,
The University of Hong Kong, Pokfulam Road, Hong Kong, China
2Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
(Dated: September 12, 2012)
We study the quantum diffusive transport of multi-valley massive Dirac cones coupled by in-
tervalley spin-orbit scattering. We show that the intervalley spin-orbit scattering and intravalley
spin-conserved scattering can be distinguished from the quantum conductivity that corrects the
2 semiclassical Drude conductivity, due to their distinct symmetries and localization trends. In im-
1 mediate practice, it allows transport measurements to estimate the intervalley scattering rate in
0 hole-doped monolayers of group-VI transition metal dichalcogenides (e.g., molybdenum dichalco-
2 genidesandtungstendichalcogenides),anidealclassofmaterialsforvalleytronicsapplications. The
resultscanbegeneralizedtoalargeclassofmulti-valleymassiveDiracsystemswheretime-reversal
p
symmetry demands opposite spins in opposite valleys.
e
S
PACSnumbers: 73.63.-b,75.70.Tj,85.75.-d
1
1
Valleyindexassociatedthedegenerateenergyextrema
] of Bloch bands in momentum space is a well defined de- conduction bands
l
l gree of freedom for low-energy carriers which can encode
a
h information just like the spin index in spintronics. This K K'
- hasledtotheconceptofvalleytronics,anewtypeofelec-
s
e tronics based on manipulating the valley index of carri-
m ers [1–5]. The timescale of intervalley scattering deter-
. mineshowlongtheinformationrepresentedbyvalleypo-
t valence bands
a larization can be retained. For valleytronics applications
m to be practical, this timescale shall be long as compared
- to the typical time for the control of valley dynamics. FIG. 1: Low-energy effective band structures of MX mono-
d 2
Measurement of intervalley scattering time is thus criti- layer in K and K(cid:48) valleys. The Fermi surface (at the top
n
cal for identifying potential valleytronics materials. edges of shadowed areas) intersects the highest two valance
o
bandsofoppositespinorientations(markedby↑and↓). Hor-
c Several extensively studied monolayer 2D crystals are
[ izontaldottedarrowsguidetheintervalleyspin-flipscattering
promising materials to host valley-based electronics, in- assisted by spin-orbit interaction.
1 cluding graphene, and graphene-like crystals such as sil-
v
icene [6–10] and monolayer group-VI transition metal
3
dichalcogenides [11–14]. In these hexagonal 2D crystals,
6 modelofmonolayerdichalcogenidesMX (M =Mo, W;
2
both the conduction and valance band edges are at the
2 X =S,Se)foraconcretediscussion. Thisproblemisthe-
2 twoinequivalentvalleysatKpoints(cornersoffirstBril-
oretically unique from the conventional 2D electron gas
. louinzone),whicharerelatedbytimereversal. Inmono-
9 systems or the massless graphene systems without spin-
0 layer dichalcogenides, ground breaking theoretical and orbit coupling. We show that the spin-valley coupled
2 experimental progresses on dynamical control of valley
bandedgeresultsindistinctsymmetriesandlocalization
1 are recently achieved [15–18]. Unlike graphene, mono-
behaviors in intra- and inter-valley scattering dominant
:
v layer dichalcogenides are described by massive Dirac regimes. Namely,thespin-conservingintravalleyscatter-
i fermions, and intrinsic spin-orbit coupling (SOC) gives
X ing leads to negative quantum conductivity and positive
rise to a spin splitting which must have opposite sign
magnetoconductivityfromweaklocalization(WL)effect,
r
a at the two valleys by the requirement of time reversal whereas the intervalley scattering assisted by spin-orbit
symmetry (see e.g. Fig. 1). This effective coupling be-
interaction gives rise to positive quantum conductivity
tweenthespinandvalleyindicescanhavetwosignificant
and negative magnetoconductivity from weak antilocal-
consequences for valleytronics: (i) the interplay between
ization (WAL). The quantum conductivity and magne-
spin and valley degrees of freedom; (ii) a unique form of
toconductivity have logarithmic dependence on the ratio
intervalley scattering which must be accompanied by a
between intra- and inter-valley scattering rates. Such
simultaneous spin flip.
a dependence makes possible measurement of interval-
In this Letter, we study the quantum diffusive trans- ley scattering rate in a vastly broad range in monolayer
port of multi-valley massive Dirac fermions with spin- dichalcogenides,fromthreeordersofmagnitudebelowto
valleycoupling. Withoutlossofgeneralitywechoosethe three orders of magnitude above the intravalley scatter-
2
ingrate,whichcanbereadilytestedinavailabledichalco-
genide field effect transistors [19–21]. Experimental veri- t 0/t I = 0
ficationsofthesephenomenawillrevealthenatureofim-
puritiesandprovidegeneralguidanceonthesuppression (a ) 1 .0 (b )
V = 0 .0 1 e V
of intervalley scattering for a longer valley life time nec- ] 0 .8
essary for valleytronic applications. Our approach and 2/h 0 .6
results can also be readily extended to spin-valley cou- [e 0 .0 5
pledDiracfermionswithmuchsmallermassesinsilicene Ds 0 .4 0 .1
V
or monolayers of germanium and tin [6]. F e rm i 0 .2
0 .1 5
The band edge electrons and holes in monolayer E n e rg y 0 .0
-0 .4 -0 .2 0 .0 0 .2 0 .4
MX are well described by the massive Dirac fermion
2 B [T e s la ]
model [15]: V = 0 .1 e V Z
H =(cid:126)v(±k σˆ +k σˆ)+ ∆σˆ ±λsˆ ⊗ ˆ1−σˆz , (1) 0 .4 (c ) t 0/t I = 0 00 ..00 00 46 (d ) t 0/t I = 0 .1
x x y y 2 z z 2 /h] 0 .2 0 .0 0 1 0 .0 1 0 .0 0 2
where ± stands for K and K(cid:48) valleys, respectively. The 2 [e 0 .0 0 .0 0 0 1
Pauli matrices σx,y,z act on the pseudo√spin indexing the Ds -0 .2 1 0 0 -0 .0 0 2 1 0
A ≡ dz2 and B ≡ (dx2−y2 ± idxy)/ 2 orbitals while -0 .4 1 0 0 0 -0 .0 0 4
sz =↑,↓ is the z-component of real spins. (kx,ky) is the -0 .6 1 0 0 0 0 -0 .0 0 6
wavevectormeasuredfromK (K(cid:48))points. TheHamilto- -0 .4 -0 .2 0 .0 0 .2 0 .4 -0 .4 -0 .2 0 .0 0 .2 0 .4
nianisdescribedbythreeparameters: theSOCstrength B [T e s la ] B [T e s la ]
Z Z
λ, the band gap ∆, and the effective velocity v. The
resulting band structure consists of four sets of massive
Dirac cones, two in K and two in K(cid:48) valleys. An impor- FIG. 2: (a) Sketch of band structure, where V ∈ [0,2λ] eV
measures the Fermi energy from the valence band top. [(b)-
tant feature is the large SOC splitting [15, 22](2λ∼ 0.15
(d)]Magnetoconductivity∆σunderaperpendicularmagnetic
eV in molybdenum dichalcogenides and 2λ ∼ 0.4 eV in
fieldB ,andwhentheFermienergyintersectsthetwohigher
tungsten dichalcogenides) between the spin-up and spin- z
valence bands. The positive (negative) logarithmic magne-
down states at the valence band top. The conduction toconductivity is the signature of the weak localization (an-
band bottoms remain degenerate. The band dispersion tilocalization). (b) No intervalley scattering (τ /τ = 0) for
0 I
is schematically shown in Fig. 1. different V. (c) At V = 0.1 eV and for different τ /τ , the
0 I
Becauseofthelargespinsplittinginthevalencebands, ratiobetweenintervalleyandintravalleyscatteringrates. (d)
Zoom-in of (c) for τ /τ = 0.1,1,10. Parameters: ∆ = 1.66
inthefollowingwefocusonthelocalizationeffectinhole- 0 I
eV and λ = 0.075 eV [15], mean free path (cid:96) = 10 nm, and
dopedsamples. Itisinstructivetofirstconsidertwolim-
phase coherence length (cid:96) =300 nm.
itingregimeswheneitherintra-orinter-valleyscattering φ
dominates.
(1) Intravalley scattering dominant regime. In the va-
symmetry because of the opposite spin splitting at the
lence bands, when holes are confined to a single valley,
two valleys (Fig. 1). For two possible spin-flip mecha-
the SOC acts like a strong Zeeman field, which fully po-
nisms, the magnetic scattering and spin-orbit scattering,
larizes the spins. Therefore, the intravalley scattering
thelatterislikelytoexistinthematerials. Thespin-orbit
can only happen between the same spin species. At the
scattering preserves time-reversal symmetry, along with
same time, because of the large band gap (∆∼1.6 eV),
thebrokenspin-rotationalsymmetry,ithasthesymplec-
the orbital pseudospin is almost fully-polarized and does
tic symmetry [24] which leads to WAL [25].
not play a role. The frozen pseudospin can be quanti-
To test the above picture, we calculated the WL and
tatively seen from the Berry phase [23], which for the
WAL using the diagrammatic techniques [26–30]. Be-
valence bands is given by
cause the results for gapless graphene without spin-orbit
V +∆−λ coupling can not be applied directly, we generalize the
φv =2π2V +∆−λ ∼2π (2) calculation to this multi-valley massive Dirac cones cou-
pled by intervalley spin-orbit scattering. The method is
where V measures the Fermi energy from the valence
basedontheperturbationexpansionoftheKuboformula
band top. ∆ is much larger than λ and allowed V
for conductivity. The Hamiltonian in Eq. (1) is taken as
(∈[0,2λ] eV) in MX [15]. The spin-conserved intraval-
2 theunperturbedpartandscatteringpotentialsaspertur-
leyscatteringthatalsoconservestime-reversalsymmetry
bations. The scattering potentials are given as follows.
belongtotheorthogonalsymmetryclass[24],whichgives
The spin-conserved intravalley scattering is modeled by
rise to WL [25].
(2) Intervalley scattering dominant regime. The in- U0 =(cid:88)uRei(k(cid:48)−k)·R, (3)
k,k(cid:48)
tervalley scattering in MX must break spin-rotational
2 R
3
where uR is the potential of an impurity at position R.
The subscript k,k(cid:48) means the potential scatters an elec- 1 .5
tronofwavevectork(cid:48) tok. Thespin-flipintervalleyscat- t t
0/ I= 1 0 0 0 0
tering is given by spin-orbit scattering 1 .0
UA 0 UA 0 0 .5 1 0 0 0
UkI,,sko(cid:48) = U0z+A U0zB −U0−zA U0−B , 2 [e/h] 0 .0 11100 0
0 UB 0 −UB F
+ z
s -0 .5 0 .1
UA/B = i (cid:88) uRei(k(cid:48)−k)·Rk×k(cid:48)·zˆ,
z 0 .0 1
R∈A/B -1 .0
0 .0 0 1
U±A/B = i (cid:88) uRei(k(cid:48)−k)·Rk×k(cid:48)·(xˆ±iyˆ), (4) -1 .5 0
R∈A/B 0 .0 1 0 .1 1
T /T *
wherethematrixisspannedbythebasisfunctionsinthe
order of |+↑A(cid:105), |+↑B(cid:105), |−↓A(cid:105), |−↓B(cid:105), and xˆ,yˆ,zˆ stand
for the unit vectors. The extra i in UkI,,sko(cid:48) protects time- FIG. 3: Zero-field quantum conductivity σF as a function
reversalsymmetryinspin-orbitscatteringasi,k,k(cid:48),and of temperature T for different τ /τ . Positive (negative) σF
0 I
spin change sign under time reversal. For mathematical correspondstoweakantilocalization(localization). Otherpa-
simplicity, we assumed short-range and delta correlation rameters are the same as those in Figs. 2 (c) and (d).
for both U0 and UI,so. Although intravalley scattering
should be related to long-range potential, the practice
analytic expressions for C and X [33]. Due to its
by the delta potential was justified [31]. 0,I 0,I
quantum interference origin, σF can be suppressed by a
From the diagrammatic techniques, the total conduc-
perpendicular magnetic field B , leading to a magneto-
tivity is given by z
conductivity
σ =σD+σF, (5)
e2
∆σ(B )≡σF(B )−σF(0)= (C F +C F ), (8)
where σD is the semiclassical (Drude) conductivity. Ac- z z πh 0 0 I I
cording to the Einstein relation, σD is given by
whereF =Ψ((cid:96)2/(cid:96)2+1)−ln((cid:96)2/(cid:96)2),1/(cid:96)2≡1/(cid:96)2+1/X2,
i B i 2 B i i φ i
σD =e2N D (6) Ψisthedigammafunction,and(cid:96) ≡(cid:112)(cid:126)/(4e|B |)isthe
F B z
magnetic length [33].
where NF = EF/(π(cid:126)2v2) is the density of states at the Figure 2(b) shows the magnetoconductivity in the ab-
Fermi surface with EF the Fermi energy measured from sence of intervalley scattering, and when the Fermi en-
the Dirac point, D = ηvvF2τ/2 is the diffusion constant ergy intersects with the two highest valence bands. The
with ηv the ladder diagram correction to the velocity of WLisaquantuminterferenceinducedsuppressionofcon-
Dirac fermions [32], vF the Fermi velocity, and the total ductivity,whichcanbeliftedbymagneticfieldandgives
scattering time given by τ = (1/τ0 +1/τI)−1, where τ0 positive magnetoconductivity. Figure 2(b) shows that
andτI areintravalleyandintervalleyscatteringtimes,re- themagnetoconductivityisalwayspositiveintheallowed
spectively. Therefore, σD is a function of the total scat- range of the Fermi energy, indicating the WL from each
tering time, thus can not distinguish the contributions valence band. Figures 2(c) and (d) shows the magneto-
from intervalley and intravalley scattering. Besides, σD conductivity as the intervalley scattering increases. It is
is insensitive to magnetic field. For these reasons, the convenient to define the ratio between intervalley spin-
quantuminterferencecorrectiontotheDrudeconductiv- orbit and intravalley scattering rates
ity σF (quantum conductivity for short) has to be con-
sidered, which is found to take the form of logrithmic τ /τ , (9)
0 I
functions
whichincreaseswithincreasingintervalleyscattering. In
e2 (cid:32) X−2+(cid:96)−2 X−2+(cid:96)−2(cid:33) Figs. 2(c) and (d), as τ /τ increases, the magnetocon-
σF = C ln 0 φ +C ln I φ , (7) 0 I
πh 0 X−2+(cid:96)−2 I X−2+(cid:96)−2 ductivity changes from positive to negative, correspond-
0 I
ing to a crossover from WL to WAL. The magnetocon-
whereC andC areweightfactorsfortheintravalleyand ductivity covers a wide range of τ /τ , from < 0.001 to
I 0 0 I
intervalley contributions. X and X are corresponding > 1000. When τ /τ > 10 or < 0.1, the magnetocon-
0 I 0 I
characteristic lengthes that effectively reduce the mean ductivity changes sensitively with τ /τ . This sensitivity
0 I
free path (cid:96) and phase coherence length (cid:96) . The inter- allows us to estimate even a very small τ when τ dom-
φ I 0
valley/intravalley scattering ratio is incorporated in the inates, or vice versa. Considering the low mobility in
4
conduction bands conduction bands. All four conduction bands can take
spin
part in transport as they always intersect the Fermi sur-
face together. Therefore, different from the transport
in hole-doped samples, both intravalley spin-orbit and
intervalley spin-conserved scattering are possible in the
conductionbands. Theruleisthatspin-flipscatteringas-
sisted by spin-orbitinteraction leads to WALwhile spin-
conserved scattering to WL because the pseudo-spin de-
greesoffreedomisfrozen. Alltypesofscatteringandthe
K K' valley resulting weak (anti-)localization are shown in Fig. 4.
ThisworkwassupportedbytheResearchGrantCoun-
FIG. 4: Weak localization (WL) and antilocalization (WAL)
cilofHongKongunderGrantNo. HKU705110P(H.Z.L.
in the conduction bands. Spin-up (↑) and spin-down (↓) are
and S.Q.S.) and HKU 706412P (W.Y.). D.X. was sup-
offsetforclarity. Solidanddottedarrowsrepresentspin-orbit
and spin-conserved scattering, respectively. portedbytheU.S.DepartmentofEnergy,OfficeofBasic
Energy Sciences, Materials Sciences and Engineering Di-
vision.
MX [20],thecalculationhasassumedashorttotalscat-
2 √
tering time τ as well as short mean free path (cid:96) ≡ Dτ,
then the results are not sensitive to (cid:96) and τ, which are
fixed throughout the paper. Note that similar crossover
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