Intervalley scattering and weak localization in Si-based two-dimensional structures

0.46 MB·English

by  A. Yu. Kuntsevich

Intervalley Scattering and Weak Localization in Si-based Two-Dimensional Structures
A.Yu.Kuntsevicha, N.N.Klimovb, S.A.Tarasenkoc, N.S.Averkievc,
V.M.Pudalova, H.Kojimab, and M.E.Gershensonb
a P.N.Lebedev Physics Institute, 119991 Moscow, Russia
b Department of Physics and Astronomy, Rutgers University, New Jersey 08854, USA and
c A.F.Ioffe Physico-Technical Institute, 194021 St.Petersburg, Russia
7
(Dated: February 6, 2008)
0
0 We have measured the weak localization magnetoresistance in (001)-oriented Si MOS structures
2 with a wide range of mobilities. For the quantitative analysis of the data, we have extended the
n theory of weak-localization corrections in the ballistic regime to the system with two equivalent
a valleys in electron spectrum. This theory describes the observed magnetoresistance and allows the
J extraction of the phase breaking time τϕ and the intervalley scattering time τv. The temperature
dependences τ (T) for all studied structures are in good agreement with the theory of electron-
5 ϕ
2 electron interaction effects in two-dimensional systems. The intervalley scattering is elastic and
rather strong: τv is typically only an order of magnitude greater than the transport time, τ. It is

] found that the intervalley scattering rate is temperature-independent and the ratio τv/τ decreases
n with increasing the electron density. These observations suggest that the roughness of the Si-SiO2
n interface plays themajor role in intervalley scattering.
-
s
i PACSnumbers: 72.10.-d,73.20.Fz,73.40.Qv
d
.
t
a I. INTRODUCTION ley scattering time, τ , and the time of dephasing of the
v
m
electron wave function, τ . For weak intervalley scat-
ϕ
- tering (τ >> τ ), two valleys are independent at the
d Two-dimensional semiconductor structures with a de- v ϕ
timescaleτ relevanttothediffusionregimeofWL.Itis
n generate ground state attract a great deal of attention ϕ
thereforeexpectedthatthemagnitudeoftheWLcorrec-
o becauseoftherichnessoflow-temperaturetransportand
c tion is doubled in comparison with its value in a system
thermodynamic phenomena that are often absentinsys-
[ with strong intervalley scattering (τ << τ ); the latter
temswithasimplebandstructure. Theenergyspectrum v ϕ
correctionisthesameasinasingle-valleysystembecause
1 of a two-dimensionalelectronsystemin the metal-oxide-
v semiconductor (MOS) structures grown on the (001)- the valleys are completely mixed at the τϕ time scale.
5
oriented Si surface consists of six subbands (valleys). At In numerous measurements of the WL magnetoresis-
1
low temperatures and low electron densities, only two of tanceinSiMOSstructures[7,8,9,10],theexperimental
6
1 them are occupied (the other four valleys are consider- datawerefittedusingtheHikami-Larkin-Nagaoka(HLN)
0 ably higher in energy) [1]. The low-energy valleys are theory[11]. Interestingly,the factor-of-twoenhancement
7 almost equivalent: the valley splitting ∆ caused by an oftheWLcorrectionwasneverobserved,indicatingthat
V
0 asymmetry of the confining potential is typically negli- intervalley scattering is rather strong. In order to ex-
t/ gible in comparison with the Fermi energy [1]. When tract the intervalley scattering time from the WL mag-
a the intervalley scattering is weak, the valley degeneracy netoresistance,themeasurementsshouldbeextendedto-
m
strongly affects both electron-electron interaction and wards higher magnetic fields. However,the HLN theory,
- weak localization (WL) effects in the conductivity. In which is used for fitting the WL magnetoresistance, was
d
particular, the interaction effects in Si MOS structures developed within the diffusive approximation (i.e small
n
o are strongly amplified by the valley degeneracy [2]; this magnetic fields, see below). Therefore, for an adequate
c accounts for the anomalous “metallic” temperature de- description of the effect of intervalley scattering on WL
: pendenceoftheresistivityinhigh-mobilitySiMOSFETs in Si MOS structures, a theory applicable over a wider
v
i at intermediate temperatures [3, 4]. Accordingly, the in- range of magnetic fields should be developed.
X
tervalley scattering plays an important role in the low-
Inthispaper,weextendtheballistic(i.e. applicableto
r temperature phenomena in Si MOS structures: it de-
a arbitrary classically-weak magnetic fields) theory of WL
termines the low-temperature cut-off of the metallic-like
correctionstothecaseoftwodegeneratevalleys. Thisen-
transport and could also modify the 2D metal-insulator
ables the detailed quantitative analysis of the WL mag-
transition observed in these structures at low electron
netoresistance measured for several Si MOS structures
densities [5, 6]. However, to the best of our knowledge,
with the electron mobility varying over an order of mag-
there were no systematic studies of the intervalley scat-
nitude. Rathersmallextractedvaluesofτ ( 10τ)indi-
v
tering in Si MOS structures. ∼
cate that the intervalley scattering in Si MOS structures
The measurements of weak localization corrections to is strong. We have found that (i) intervalley scattering
theconductivityoftwo-valleysystemsallowonetostudy is temperature-independent, i.e. elastic, (ii) the scatter-
intervalleyscattering. Theeffectofintervalleyscattering ing rate depends monotonically on the electron density,
onWLdependsontherelationshipbetweentheinterval- and(iii) thereis nosimple correlationbetweenthe inter-
2
valley scattering and the mobility for different samples.
Thephaserelaxationtimeinallstudiedstructuresiswell
described by the theory of electron-electron interaction
Dimensionless magnetic field b
effects [12].
396 -0.9 -0.6 -0.3 0.0 0.3 0.6 0.9
The paper is organizedas follows. In section II we de-
T=500 mK
scribe the samples,measurementtechniques and provide Si6-14
T=400 mK
examplesofexperimentaldata. InsectionIII,tocompare 394
T=200 mK
our data with previously reported results, we fit the WL
a)
magnetoresistance with the HLN theory. The results of
392
this analysis indicate that the WL correction is approx-
imately a factor of 2 smaller than one could expect for
a system with two independent valleys. The theory of
390
the WL magnetoconductance for systems with degener- -90 -60 -30 0 30 60 90
ate multi-valley spectrum is presented in section IV and 381
Si39 T=3.6 K
the data are analyzed using this theory in section V. In ) T=3 K
section VI we discuss the results of the data analysis, in m/380 T=1.9 K
h
particular, the intervalley and phase breaking times. (O b) T=1.3 K
379
II. EXPERIMENT
378
-300 -200 -100 0 100 200 300
BelowwepresentdataforthreerepresentativeSiMOS-
FET samples: (i) high-mobility (µ 2 m2/Vs at 0.1K) 1020 Si40 T=4.2 K
≈ T=3 K
sample Si6-14 [4] which demonstrates a strongly pro-
1010 T=2.25 K
nounced “metallic-like” dependence σ(T) and a metal-
T=1.5 K

insulator transition with decreasing electron density n, 1000 c)
(ii) sample Si39 with an intermediate mobility µ
0.45 m2/Vs [13], and (iii) low-mobility sample Si40 wit≈h 990
µ 0.18 m2/Vs at T < 4.2K. The transport times
≈ 980
for these samples within the studied range of n were -2000 -1000 0 1000 2000
τ 2 ps, 0.6 ps, and 0.2 ps, respectively. The resistance Magnetic field (Gauss)

≈
wasmeasuredusingthe standardfour-terminalACtech-
nique by a resistance bridge LR700 (for sample Si6-14)
and by SR830 lock-in amplifier in combination with a FIG.1: Examplesofthemagnetoresistanceρ(B)dataforSi6-
14, n=1×1012cm−2(a), Si39, n=3.6×1012cm−2 (b), and
differential preamplifier (for samples Si39 and Si40). A
Si40, n = 3.5×1012cm−2(c). The data within the hatched
sufficiently small measuring current (1-3 nA for Si6-14
regions have been used to extract the τ value. Upper axes
and 50–100 nA for Si39 and Si40) was chosen to avoid showthemagneticfieldinunitsofBtr =ϕΦ0/2πl2,loweraxes
overheatingofelectrons. Theelectrondensity,controlled
show the field in Gauss. The data points were chopped for
by the gate voltage, was found from the period of the clarity.
Shubnikov-deHaasoscillationsandthedependenceofthe
Hall resistance on magnetic field. Both results were con-
sistentwitheachotherwithin2%accuracyinthestudied
thin aluminum film. For reliable extraction of the phase
rangeofdensities;thisuncertaintyofnisinsignificantfor
relaxation time τ from the WL magnetoresistance, we
the further analysis. To ensure that only the lowest size ϕ
havechosenasmallfieldstepsize: 1GforSi6-14and3G
quantizationsubbandisfilled[1],weperformedmeasure-
ments at n<4 1012cm−2. for Si39 and Si40. The examples of magnetoresistance
× ρ(B)dataforSi6-14,Si39andSi40atafixeddensityand
various temperatures are shown in Figs. 1a, 1b and 1c,
The WL magnetoresistance was measured at T =
respectively. Hereafter throughout the paper we will use
0.05 0.6K for high-mobility sample Si6-14, and at
− magnetoconductance (MC) ∆σ ρ(B)−1 ρ(B =0)−1.
T =1.3 4.2KforsamplesSi39andSi40. Atthesetem- ≡ −
−
peratures,thephasebreakingtimeexceedsthetransport
time by one to two orders of magnitude. The magnetic
fieldalignedperpendiculartothe planeofSiMOSstruc-
III. FITTING THE DATA WITH THE
tures was varied from -1 to +1kG (Si6-14) and from -3
HIKAMI-LARKIN-NAGAOKA THEORY
to +3kG(Si39andSi40). When sampleSi6-14wasmea-
sured at T < 1K, an additional in-plane field 200G
∼
was applied to quench the superconductivity in the cur- It is a common practice to extract the phase breaking
rent/voltagecontactpadsandthegateelectrodemadeof time from the WL magnetoconductance using the HLN
3
tem for various values of τ /τ and magnetic fields b. It
ϕ
was found that both approaches agree with each other
within a limited range of fields b < 0.15 provided that
2.0
τ /τ > 30 and the conductivity is much greater than
ϕ
Experiment e2/2π2¯h. Thus, within these limits, Eq. (1) can be used

for extraction of τ from the WL magnetoresistance in
) ϕ
1.5
2 a single-valley system, and the adjustable parameter α
2 is aproximately equal to 1. For a system with two val-
/
2 e leys and weak intervalley scattering, the prefactor α is
1 (.0 a) expectedtobetwotimeslarger,becauseeachvalleycon-
tributes the term ∆σ with α 1 to ∆σ.
HLN
≈
Figure2ashowsafitofourtypicalWLMCcurvewith
Eq.(1). The fitting performed over the magnetic field
0.5
range b = 0 – 0.2 gives τ /τ = 133 and, contrary to the
ϕ
expectation for a two-valley system without intervalley
scattering, α = 1. Changing the magnetic field range,
0.0
wherethedataarefitted,causesonlyminorvariationsof
0.00 0.05 0.10 0.15 0.20 0.25
these parameters (see Fig. 2b). An attempt to analyze
Dimensionless magnetic field b
the MC curve using Eq.(1) with a fixed prefactor α = 2
1.04 results in a much worse fit (dashed line in Fig. 2a).
136
or Severalreasonsforthe reductionofαinasinglevalley
ct1.02
Prefa1.00 b) 132 sMyastkeim-Thhoamvepsboenencocrornecstidioenre,d(iii)nDReenfs.ity[1-4o]f,-sitnactleusdcinorgre(ci)-
tion, (iii) higher order corrections in (k l)−1, where k
F F
0.98 128 is the Fermi wave vector, and (iv) low τϕ/τ ratio. The
0.10 0.15 0.20 0.25 corrections (i) and (ii) were shown to be small [14]. To
b-range us ed for fitting ensure that the higher order corrections are also small,
wehavestudiedthe WL MConly forlargeconductances
100 e2/2π2¯h. When the ratio τ /τ decreases, the
FIG.2: a)ExampleoftheMCdata(points)forsampleSi40, ∼ × ϕ
T = 1.45K and n =3.34×1012cm−2. Solid line is calculated HLN theory becomes inadequate, and the fitting proce-
dure results in an artificially reduced prefactor. Corre-
using Eq.(1) with two fitting parameters α = 1 and τ /τ =
ϕ
spondingly,weperformedmeasurementsatsuchlowtem-
133. The dashed curve is an attempt to fit the same data
over the same range of b with a fixed prefactor α = 2 and peraturesthattheinequalityτϕ/τ >30wassatisfied. We
τ /τ =61. b) Dependences of the fittingparameters α (solid conclude therefore that the aforementioned reasons can-
ϕ
squares)andτ /τ (opensquares)onthemagneticfieldrange not accountfor a low value of the prefactor α 1 in the
ϕ
≈
0 – b which was used for fitting. studied multi-valley structures. It should be noted that
α 1 in Si MOS structures was obtained in numerous
≈
previous experiments [7, 8, 9, 10]. We show below that
theory[7, 8, 9, 10, 14]: the prefactorreductioncanbe welldescribedby the the-
ory which explicitly takes the intervalley scattering into
τ αe2 1 τ bτ
∆σ b, ϕ = ψ + +ln ϕ . (1) account.
HLN τ 2π2¯h 2 bτ τ
(cid:16) (cid:17) (cid:20) (cid:18) ϕ(cid:19) (cid:21)
Hereψ isthedigamma-function,eistheelectroncharge,
IV. THEORY
¯h is the Planck constant,b=B/B is the dimensionless
tr
magnetic field, B = Φ /2πl2, Φ = π¯h/e, and l is the
tr 0 0 A consistent theory of weak localization is developed
transport mean free path [15]. The prefactor α and the
in the framework of the diagram technique. The weak-
dimensionless ratio τ /τ are treated as fitting parame-
ϕ localization corrections to the conductivity arise in the
ters. Note that with an increase of the magnetic field, first order in the parameter (k l)−1, where the mean
the crossover from the diffusive regime (b << 1) to the F
free path l is governedby the scattering time τ, which is
ballistic regime (b 1) is expected in the WL correc-
tions. Equation (1∼) with prefactor α = 1 is the exact controlled by both intra-valley (τv) and inter-valley (τi)
scattering processes:
result for a single-valley system in the diffusive regime,
i.e. at τ τ and for sufficiently small magnetic fields 1/τ =1/τ +1/τ . (2)
ϕ v i
≫
b 1[11]. Ontheotherhand,theexperimentaldataare
≪ Theweak-localizationcorrectiontotheconductivityin
oftenobtainedbeyondtheselimitsand,therefore,should
the magnetic field has the form
be described by more general ballistic theories [16, 17].
In Ref. [18], the HLN theory was numerically com-
pared with the ballistic theory for a single-valley sys- ∆σ(B)=∆σ(a)+∆σ(b), (3)
4
where the terms ∆σ(a) and ∆σ(b) correspond to the
standarddiagrams,whichhavebeen consideredindetail
in Refs. [17, 19, 20]. We neglect both valley and spin- + Wαν P(ρ,ρ′′) (2)νβ(ρ′′,ρ′)dρ′′,
γµ Cµδ
orbit splitting in Si MOS structures. [21] Then, for the
νµ Z
X
short-range scattering potential [22], one obtains
(3)αβ(ρ,ρ′)= (2)αβ(ρ,ρ′) WανWνβP(ρ,ρ′),
∆σ(a) = ¯h J2(ρ,ρ′) (3)αβ(ρ′,ρ)dρdρ′, (4) Cγδ Cγδ − νµ γµ µδ
π x Cβα X
αβ Z
X where P(ρ,ρ′)=GA(ρ,ρ′)GR(ρ,ρ′).
To calculate the weak-localization correction to the
¯h
∆σ(b) = J (ρ,ρ′)J (ρ′′,ρ)Wαβ (2)δα(ρ′,ρ′′) conductivity in multi-valley structures, we assume that
π x x γδ Cβγ the impurity potential is the same for particles in differ-
αβγδZ
X entvalleysand,thus,theelectronscatteringinthevalleys
[GR(ρ,ρ′)GR(ρ′′,ρ)+GA(ρ,ρ′)GA(ρ′′,ρ)]dρdρ′dρ′′. isstronglycorrelated. Inparticular,inthe(001)-oriented
×
Si-based structures the nonzero correlatorsare
Here α,β,γ,δ =1,2 are the valley indices; J (ρ,ρ′) is
x
the x-component of the current vertex, which is defined
as W11 =W22 =W22 =W11, W21 =W12. (9)
11 22 11 22 12 21
k τ ρ ρ′ Othercorrelatorsvanishdue to averagingoverthe im-
J(ρ,ρ′)=ie F − [GA(ρ,ρ′)+GR(ρ,ρ′)], (5) puritypositionsbecausethe lowestconduction-bandval-
m∗ ρ ρ′
| − | leysinsiliconarelocatedinthe∆-pointsoftheBrillouin
GA(ρ,ρ′)andGR(ρ,ρ′)aretheadvancedandretarded zone and, therefore, the Bloch functions contain oscilla-
tory factors. We note that the intravalley and interval-
Green functions,
ley scattering times are determined by these correlators:
1/τ =m∗W11/¯h3, 1/τ =m∗W21/¯h3 .
GR(A)(ρ,ρ′)= ψs,ky(ρ)ψs∗,ky(ρ′) , (6) canUiseinxgpatnhde1sP1ta(nρd,ρar′)d pavnrodcetdhuereC1(2osoepeeRroenfss. [i1n9,th2e3])s,eroinees
E E i¯h/(2τ) i¯h/(2τ )
F s ϕ
sX,ky − ± ± of eigenfunctions of a particle with the double charge
in a magnetic field and derive equations for the weak-
ψ (ρ) is the wave function of an electron subject to
s,ky localization corrections ∆σ(a) and ∆σ(b). Calculations
an external magnetic field B, which is perpendicular to
show that the corrections have the form
the 2D plane, s and k are the quantum numbers (s is
y
theLandaulevelnumberandk isthein-planewavevec-
y
tor), Es = h¯ωc(s+1/2) is the energy of the sth Landau e2b ∞
level, ωc = eB/m∗ is the cyclotron frequency, m∗ is the ∆σ(a) =−2π2¯h CNPN2 , (10)
effectiveelectronmass;Cγ(2δ)αβ(ρ,ρ′)andCγ(3δ)αβ(ρ,ρ′)are NX=0
the Cooperons which depend on four valley indices. The
parameters Wγαδβ are determined by intervalley and in- e2b ∞
travalley correlators of the scattering potential and are ∆σ(b) = 2π2¯h (CN +CN+1)Q2N/2. (11)
defined by N=0
X
hVαk1,βk2Vγk3,δk4iNimp =Wγαδβδk1+k3,k2+k4 , (7) N = 2(1−τ/τv)3PN + PN (1−2τ/τv)3PN ,
C 1 (1 τ/τ )P 1 P −1 (1 2τ/τ )P
v N N v N
where Vαk1,βk2 is the matrix element of scattering be- − − − − − (12)
tweenelectronstates(β,k2)and(α,k1)inzeromagnetic where PN and QN are coefficients which are given by
field, k (j = 1...4) are wave vectors in the 2D plane,
j
N is the two-dimensional density of impurities, and
imp ∞
the angle brackets stand for the averagingover impurity 2 2 τ x2
spatial distribution. PN = exp x 1+ LN(x2)dx,
The Cooperons (2)αβ(ρ,ρ′) and (3)αβ(ρ,ρ′) repre- rb Z0 "− rb (cid:18) τϕ(cid:19)− 2 #
sentthe sums ofintCeγrδnalpartsoftheCfaγnδ diagramsstart- (13)
ing with two and three lines, respectively, [17, 19, 20].
∞
They can be found from the following equations: 2 2 τ x2 L1 (x2)x
Q = exp x 1+ N dx,
N
rb Z "− rb (cid:18) τϕ(cid:19)− 2 #√N +1
0
(2)αβ(ρ,ρ′)= WανWνβP(ρ,ρ′) (8)
Cγδ γµ µδ and L and L1 are the Laguerre polynomials.
Xνµ N N
5
completeness, we also depicted α and τ /τ for a single-
ϕ

valley system at τ /τ = 1. The main results of the fit
v
4 are as follows: (i) the extracted phase breaking time τϕ
a) coincides with its true value within a few percent (this
uncertainty is insignificant for further analysis), and (ii)
) 3 the observed prefactor increases from 1 to 2 as τv
2 increases and becomes greater than τ≈. Ther≈efore, the
ϕ
2
2 e/ 2 approximate equality α≈ 1 is simply a consequence of a
( large ratio τϕ/τv ≫1.
1
V. FITTING THE DATA WITH THE
BALLISTIC THEORY
0
0.0 0.2 0.4 0.6 0.8 1.0
Dimensionless m agnetic field b It is intuitively clear and will be discussed in more
detailbelow,thatthe MCinlowfieldsb 1ispredomi-
ctor1.8 b) 102 nantly determined by τϕ. In principle, on≪e could use the
Prefa11..25 91800 “rabnaglleistoifc”mtahgenoertyicfofireldfisttianngdtthheusMfCorddaettaerimninthinegwbhootlhe
τ and τ from a single fit. However the series Eqs. (10)
ϕ v
0.9 96 and (11) converge very slowly in small b region. There-
fore, to determine τ , it is more practical to use in low
1 10 100 1000 10000 ϕ
fields Eq.(1), that was shown(see Fig. 3) to provide the
v
correct τ value.
ϕ
Consequently,wehaveusedthe followingprocedureof
FIG. 3: a) WL magnetoconductance calculated for a two- extractingτ fromtheWLmagnetoconductance. Firstly,
v
valleysystemusingEqs.3,10,11(solid lines)andforasingle-
we analyzed the MR data in sufficiently weak magnetic
valleysystem[17](dottedline). Forsolidcurvesfrombottom
fields and at low temperatures. In this regime (the
totop,τv/τ =10,25,50,100,1000,10000,respectively. Two
hatchedregionsinFig.1),thedephasingoccursatatime
upper curves are indistinguishable by eye. For all the curves
scalemuchgreaterthanτ , andwe canapplyEq.(1) for
τ /τ = 100. The hatched region was used for fitting with v
ϕ
extractingτ ;thesecondadjustableparameter,prefactor
Eq. (1). b) Dependences of the fitting parameters, α and ϕ
τϕ/τ, on τv/τ. The data points at τv/τ =1 correspond to a α,appearstobe closeto1. Atthe nextstage,wesubsti-
single-valley system. tute τϕ into the “ballistic”formulae[Eqs.(3), (10),(11)]
and calculate the MC curves in a wide range of fields
(b<1)forvariousτ . Figure4 illustratesthis procedure
v
Equations(3), (10), and(11) describe the WL magne- using as an example the same MC data as in Fig. 2. We
toconductance over the whole range of classically weak calculate ∆σ(b) using the summation technique similar
magnetic fields ω τ µB < 1. In the limit of vanishing to that described in Ref. [24] for single-valley systems.
c
≡
intervalley scattering (1/τ = 0), Eqs. (10) and (11) are Figure 4 shows that the experimental MC (circles) is
v
reduced to the conventional expressions for the WL cor- smaller than the MC for a system with two unmixed de-
rectionstothe conductivity ofasingle-valleysystem[19] generatevalleys(curve1)andlargerthanMCforasingle
and, in particular, to the HLN formula [11] in the diffu- valleysystem(curve5). Thisobservationagainindicates
sionregime. Theonlydifferenceisaprefactorof2,which thattheMCinthestudiedSiMOSstructuresisaffected
accounts for the valley degeneracy. byvalleymixing. Curves2,3,and4inFig.4correspond
to τ /τ = 15, 12, and 9, respectively. Note that in the
v
To illustrate the effect of intervalley scattering on the magnetic field range b < 0.15, these three curves, the
magnetoconductance, we calculated the ∆σ(b) depen- experimental data, the HLN formula, and the ballistic
denceusingEqs.(3),(10),and(11)forafixedτ /τ =100 resultfora singlevalleysystemarealmostindistinguish-
ϕ
and various values of τ /τ. The results are shown in able fromeachother (see the inset to Fig.4). Therefore,
v
Fig. 3a by solid lines. For comparison, we also calcu- τ cannot be reliably found from the MC in low fields.
v
lated the MC using a similar theory [17] developed for a Figure4showsthatthediscrepancybetweenthecurve
single-valley system (dotted line). We then fitted these 5 for a single-valley system and the curves 2,3,4 for two
dependences over the range b<0.15 using the HLN the- mixedvalleys(τ /τ =15,12,and9)growsasbincreases.
v
ory[Eq.(1)]withtwofitting parameters,theprefactorα This observationhas atransparentphysicalexplanation:
andτϕ/τ. Inotherwords,wefitted the theoreticalcurve with increasing b, the typical size of electrontrajectories
the same way as the experimental data have been fitted which contribute to the WL correction diminishes, and
above in Sec III. the valley mixing overthe time of travelalong these tra-
Figure 3b shows the resultant fitting parameters; for jectories becomes small when b > Φ /Dτ . As a result,
0 v
6

3
2
1
3
4
15
)
2 2 5 /v
2 10
2 e/ ) 6 3
( 2 1.0 2
2 5
2 /
e
( 4
1
0.5 5 0
0.0 0.2 0.4 0.6 0.8 1.0
Dimensionless magnetic field b
0.0
0.00 0.05 0.10 0.15
0
0.0 0.2 0.4 0.6 0.8 1.0
Dimensionless magnetic field b FIG. 5: Intervalley scattering time determined from fitting
the difference, ∆σ(b1)−∆σ(b2), for the same magnetoresis-
tancecurveasinFig.4. Thefittingranges(b1,b2)areshown
FIG. 4: Comparison between the WL magnetoresistance for bythe horizontal bars. τ /τ =133.
sample Si40, n = 33.4 × 1011cm−2, T = 1.45K and the ϕ
ballistic theory. Different MC curves are calculated using
Eqs. (3),(10),(11): 1 – for two unmixed valleys (τv = ∞); A question arises therefore what range of magnetic
2 – τv/τ = 15; 3 – τv/τ = 12; 4 – τv/τ = 9; 5 – MC for
fields shouldbe chosenfor τ extraction? To answerthis
a single-valley system (equal to curve 1 divided by 2). In- v
question we have analyzed errors of our method; the re-
set blows up the data in the range b < 0.15. Curve 6 is the
HLN theory (Eq. (1)) with a prefactor α = 1 (see Fig. 2a). sultingroot-mean-squaresumofallerrorsisshownbythe
τ /τ =133 for all calculated curves. errorbarsinFig.5. Theerroranalysisispresentedinde-
ϕ
tailintheAppendix(Sec. IX),whereitisshownthatnei-
ther small fields (b<0.1) nor large fields (b 1) should
∼
beusedforτ extraction. Inweakfields,theMCisinsen-
the WL magnetoconductance in strong magnetic fields v
sitive to τ , whereas in strong fields one approaches the
approachesthetheoreticalpredictionforatwo-valleysys- v
limits ofapplicabilityofthe theorydevelopedinSec.IV.
tem with no intervalley scattering.
Therefore, we conclude that an intermediate range of
We also note that all calculated curves deviate from
magneticfieldsismostsuitableforextractingτ . Forthe
the experimental data. As Fig. 4 shows, curve 4 calcu- v
further analysis, we choose the range b = 0.2 – 0.4. We
lated for τ /τ = 9 at b > 0.4 is approximately paral-
v haveverifiedthatourconclusionsonthetemperatureand
lel to but lower than the experimental data in magnetic
densitydependencesofτ arenotaffectedifthisrangeis
fields b>0.4. On the other hand, curve 2 calculated for v
changed.
τ /τ =15almostcoincideswiththedatainlowmagnetic
v
fields b < 0.4, though deviates substantially from them
in higher fields. The minimal mean-square deviation of
VI. RESULTS AND DISCUSSION
the calculated curve from the data is realized for τ /τ
v
=12 (curve 3).
Thus, the value of τ depends on the magnetic field A. Phase breaking time
v
interval (b ,b ) within which the MC data is fitted.
1 2
The (τ /τ) values, obtained from fitting the difference As we have already mentioned, at the first stage of
v
∆σ(b ) ∆σ(b ) asa function of(b ,b ), decreaseasb= the analysis we estimated the phase breaking time τ .
1 2 1 2 ϕ
−
(b +b )/2increases(see Fig.5). This monotonicdepen- Comparison of the τ (T) dependences with the theory
1 2 ϕ
dence has been reproduced for all samples and tempera- of interaction effects [12] is shown in Figs. 6 a-c. The
tures. We believe that this apparentτ (b) dependence is uncertainty in the values of α and τ (shown as the
v ϕ
an artifact of the fitting procedure. In all above calcula- error bars in Fig. 6) reflects mainly the uncertainty in
tions we assumed τ to be field-independent. However, σ(b) in the weak fields b < 0.01. The magnitude of
ϕ
τ shoulddepend ona perpendicular magnetic field[12]. thephasebreakingtimeanditstemperaturedependence
ϕ
To the best of our knowledge, there are neither experi- are in good agreement with the theory for all samples
mental nor theoretical systematic studies of this depen- within the studied ranges of electron density [Si6-14:
dence beyond the diffusive limit. Ignoring this depen- n=(0.28 1.5) 1012 cm−2, Si39: n=(2 2.5) 1012
denceinourfittingcouldleadtotheobservedmonotonic cm−2, Si4−0: n =×(3 4) 1012 cm−2]. N−ote th×at no
− ×
variation in τ with b. adjustable parameters are involved in this comparison,
v
7
5% (dashed line in Fig. 6). The condition h¯/τ < k T
v B
is violated at temperatures lower than 0.3 K for sample
Si6-14. Still, the τ (T) data agree better with theory
ϕ
400 800 when all 15 rather than 3 triplet terms are taken into
1.0
Si6-14 s) account. Theobservedquantitativeagreementofthe ex-
300 0.8 60 0 (p perimental values of τϕ with the theory suggests that
τ is weakly affected by intervalley scattering near the
200 a) 0.6 0.2 0.4 0.6 400 cϕrossoverkBTτv/¯h∼1.
100 200
0 0.1 0.2 0.3 0.4 0. 5 0.6 0 B. Prefactor α
1. 0
120 90
Si39 By fitting the weak-field MC data with the HLN the-
s)
0.8 p ory, we obtained the prefactor α that is close to 1 for
90 (
60 all samples (see the insets to Fig. 6); this suggests that
0.6
60 b) 2 4 the valleys are intermixed on the τϕ time scale. The
30 decrease of α from 0.9 to 0.6 with increasing tempera-
30 ture, obtainedfor sample Si39 (see the inset to Fig.6b),
we believe, is an artifact, because relatively small values
0 0
1 2 3 4 5 τ /τ 30, observed for this sample at high tempera-
ϕ
∼
tures, make Eq. (1) inadequate. The complete theory
200 1.2
Si40 s) described in Sec. IV explains that small value of α is
150 1.1 40 (p a consequence of a fast phase relaxation. For the same
1.0 reason, there is a larger scattering in the values of α(T)

100 c) for samples Si6-14 and Si40 at the highest temperatures
2 4 20
(Fig. 6), where τ is small.
ϕ
50
Itisworthnotingthatthesmallnessofprefactorαhas
0 0 been attributed to the intervalley scattering in Ref. [8].
1 2 3 4 5
However, the MC data in this experiment were fitted
Temperature (K)
with the theory [23], which does not take into account
the non-backscattering correction Eq. (11). Our esti-
mates show that for the parameters of samples studied
FIG. 6: Temperature dependence of the extracted τ value
ϕ
in Ref. [8] (τ /τ = 20, τ /τ = 4, and b = 0.5–7), the
in units of τ (left axes) and in picoseconds (right axes): a) ϕ v
Si6-14, n = 9.98×1011cm−2, b) Si39, n = 29.4×1011cm−2, non-backscattering correction contributes about 50% to
c) Si40, n = 33.4×1011cm−2. Solid lines show the τϕ(T) the extracted value of τv.
dependencepredictedbythetheoryofinteractioncorrections
[12] with 15 triplet terms, dashed line - with 3 triplet terms.
The insets show the corresponding temperature dependences
C. Intervalley scattering time: independence of
of the prefactor α.
temperature
the Fermi-liquid parameter Fσ and the effective mass Following the procedure described in Sec. V, we have
0
∗
m were obtained in independent measurements [25]. extracted τ by fitting the WL MC data with “ballistic”
v
The theoretical curves (solid lines in Fig. 6) are calcu- Eqs. (3),(10),(11). Figure 7 shows that the values of τ
v
latedfollowingRef. [12]for 15triplet channels[2],which are temperature-independent within the accuracy of our
implies small valley splitting (k T > ∆ ) and relatively measurements. This observation suggests that the inter-
B v
weakintervalleyscattering(h¯/τ <k T). Asfoundfrom valley scattering is elastic, i.e. governed by static disor-
v B
the analysis of the low-temperature transport and mag- der. Similar conclusion can be also drawn from the fact
netotransportdata[26],theconditionk T >∆ wassat- thatαremainscloseto1,whiletheextractedτ muchex-
B v ϕ
isfied for samples Si6-14 and Si39 over the major part of ceedsτ andgrowswithoutsaturationasT decreases(see
v
thestudiedtemperaturerange. Whetherornotthiscon- Fig. 6). Indeed, were the intervalley scattering inelastic,
ditionisfulfilledforlowmobilitysamplesSi39andSi40is one would have observed a prefactor α 2 because the
∼
actually not important, because the measurements were dephasing would occur in two valleys independently and
performed at such high densities that the amplitude of the intervalleyscatteringwouldbejustanadditionalde-
the triplet term in the interaction corrections to τ was phasing mechanism. In the latter case, a cut-off of the
ϕ
small in comparison with the singlet term: changing the dephasingtimeatthelevelτ =τ isalsoexpected. The
ϕ v
number of triplet terms from 15 (two-valley case) to 3 two observations, the absence of the cut-off and α 1
≈
(single-valley case) caused variation of τ by less than support the self-consistency of our analysis.
ϕ
8

15 30
Si6-14 ps) 15
/1v0 20 (v Si6-14
5 10 /1v0
a)
0 0.2 0.4 0.6 0 5 a)
3 2
) 0 4 8 12 16
s 12
/v2 Si39 1 (pv Si39
1 / 8
b) v

01 2 3 40 4 b)
15
3 s) 0
/1v 0 Si40 2 (pv 15 20 25 30 35

5 c) 1 10 Si40
/
v
0 0
1 2 3 4 5
Temperature (K)
c)
0
FIG.7: Temperaturedependenceofτvinunitsofτ (leftaxes) 32 34 3611 -2 38 40
andinpicoseconds(rightaxes): a)Si6-14n=9.98×1011cm−2, Density (10 cm )
b) Si39 n=29.4×1011cm−2, c) Si40 n=33.4×1011cm−2. Solid
horizontal lines show the average τv. FIG.8: Densitydependenceoftheintervalleyscatteringtime
(averaged over temperature) for samples Si6-14 (a), Si39 (b)
and Si40 (c).
The intervalley transitions are not expected to be in-
elasticforthefollowingreason. Theintervalleyscattering
requires a large momentum transfer comparable to the proportional to Ψ 2 and increases with n [1]; this is in
0
vector of reciprocal lattice 2π/a 108cm−1 (a is the in- line with the beh|avi|or shown in Fig. 8.
∼
teratomic distance). At liquid helium temperatures only Intheexperimentsweusedsampleswiththemobilities
static disorder can cause these transitions, as the mo- whichvaryoveradecade. Wefindnocorrelationbetween
menta of electrons kF ∼ 106cm−1 for the studied range τv and the mobility for different samples. This suggests
ofdensitiesandphononskph kBT/(h¯vs)(herevs isthe thattheintervalleyscatteringisdeterminedbyasample-
∼
soundvelocity)aremuchsmallerthan2π/a. Staticdisor- specific interface disorder, namely the surface roughness
der canleadonly to elastic scatteringbecause it changes at the atomic length scale, which might be different for
momentum of scattered electrons but does not change thesamplesfabricatedondifferentwafers. Incontrastto
their energy. theintervalleyscattering,themobilityisgovernedmostly
by impurities in the bulk and by the interface roughness
at a large length scale, 2π/k .
F
∼
D. Intervalley scattering time: density and sample The measured values of τv for all samples are within
dependence theinterval(3-12)τ,whichindicatesthatthevalleyindex
remains a good quantum number at the time scale τ.
∼
Figure 8 shows the density dependence of τ values
v
averagedoverthetemperature. Forallthreesamples,the
relativerateoftheintervalleytransitions(withrespectto VII. SUMMARY
the momentum relaxation rate) increases with density.
Thispoints tothe dominant role of the Si-SiO interface To summarize, we have studied the weak localization
2
in theintervalley transitions. Theelectronwavefunction magnetoconductance in Si MOS structures over wide
Ψ in Si MOS structure is positioned mostly in the bulk rangesofthe electrondensities,mobilities, andtempera-
silicon and exponentially decays in SiO [1]. When the tures. In order to quantitatively analyze the experimen-
2
gate voltage(and, hence, the density n) is increased,the tal data, we have developed the theory of weak localiza-
electronsare“pushed”towardstheSi-SiO interface,and tion for two-dimensional multivalley systems, which is
2
the amplitude of the wavefunction at the interface, Ψ , validinboththediffusionandballisticregimes. Thethe-
0
increases. The probability of the interface scattering is ory, which explicitly takes the intervalley scattering into
9
account,allowedus toconductthe firstdetailedstudyof
the intervalley scattering in the Si MOS structures. It
was found that: ∆σ(b1)EXP ∆σ(b2)EXP = (14)
−
1. Intervalley scattering in Si MOS structures is an =∆σ(b1,τϕ/τ,τv/τ)TH−∆σ(b2,τϕ/τ,τv/τ)TH
elastic and temperature-independent process.
HeresubscriptEXPdenotesexperimentaldata,subscript
2. The ratioτ /τ monotonicallyincreasesasthe elec- TH denotes calculation using Eqs. (3),(10),(11). Conse-
v
tron density decreases. This observation suggests quently,theuncertaintyinτv isdeterminedby(i)uncer-
that the intervalley scattering is governed by the tainty in b, (ii) uncertainty in τϕ and (iii) uncertainty in
disorder at the Si-SiO interface. the conductivity. To estimate each contribution to the
2
error, we varied the corresponding parameter (b, τ or
ϕ
3. There is no simple correlation between the inter- ∆σ) within its uncertainty and determined the variation
valley scattering rate and the sample mobility (or in τ by solving Eq. (14).
v
the momentum relaxation rate); this points to a The uncertainty δb in b = 2Bπl2/Φ value is deter-
0
sample-specificratherthanuniversalmechanismof mined by the uncertainty in the mean free path l [15].
the intervalley scattering. The latter is about 2-3 % due to the uncertainties in
electron density n and Drude conductivity. However, δb
4. The smallness of the prefactor α 1, that is ob- affects τ rather weakly for the following reason: MC
∼ v
tainedfromfitting the experimentalWL data with in the studied magnetic field range behaves approxi-
the HLN formula, is a consequence of a fast in- matelyasln(b),therefore∆σ(b ) ∆σ(b ) ln(b /b )=
1 2 1 2
tervalley relaxation rate which exceeds the phase ln(B /B ). − ∼
1 2
relaxation rate. The error related to the uncertainty in τ is essential
ϕ
inlowmagneticfieldswheremagnetoconductanceissen-
5. The temperature dependence of the phase relax-
sitive to τ . Correspondingly,the errorbarsin lowfields
ϕ
ation time in Si MOS structures is in quantitative
b < 0.15 in Fig. 5 are determined predominantly by the
agreement with the theory of electron-electron in-
uncertainty in τ .
ϕ
teractioneffectsindisorderedtwo-dimensionalsys-
Another source of errors is related to the precision of
tems [12].
the absolute value of WL MC (“calibration error”). In-
deed, the accuracy of our measurements of the absolute
We note that the approach similar to that developed
magnetoresistance value is 0.5%. Higher order cor-
in this paper can be used for studies of intervalley re- ∼
rections, Maki-Thompson and DOS corrections[14] can
laxation in other multi-valley two-dimensional electron
modifyMCbyapproximately2-3%(asshowninRef.[14],
systems, such as AlAs-AlGaAs heterostructures [27], Si
δ(∆σ)/∆σ 2e2ρ /2π2¯h 0.025). In order to estimate
MOX structures [28], and Si-SiGe quantum wells [29]. ≈ D ≈
this error we artificially changed our experimental data
by 3% and studied the corresponding change in τ . The
v
errorappearstogrowinsmallmagneticfieldwheremag-
VIII. ACKNOWLEDGEMENTS
netoconductance is weakly sensitive to τ . Therefore,
v
small fields should not beusedfor the extraction of τ . In
v
The authors are thankful to I. V. Gornyi and large magnetic fields (b 1) MC becomes again weakly
G. M. Minkov for illuminating discussions. The research sensitive to τ , and the l∼atter error grows as b increases,
v
at Lebedev Institute and Ioffe Institute was supported as shown by the error bars in Fig. 5. The calibration
by RFBR, INTAS, Programs of the RAS, Russian Min- error is minimal in intermediate magnetic fields, where
istry for Education and Science, Program “Leading sci- MC is most sensitive to τ .
v
entificschools”(grants5596.2006.2,2693.2006.2,andthe Our attempts to analyze the WL MC data in strong
State contact 02.445.11.7346)and Russian Science Sup- fields b > 1 using Eqs. (3),(10),(11) resulted in a large
port Foundation. NK and MG acknowledge the NSF uncertainty of the fitting parameter τ /τ (large scatter-
v
support under grant ECE-0608842. AK acknowledges ing of extracted τ /τ for various electron densities and
v
Education and Research Center at Lebedev Physics In- temperatures). In large magnetic fields, there are sev-
stitute for partial support.
eral other error mechanisms which are difficult to take
into account. For example, at b 1, τ differs from
ϕ
∼
its small-field value [12]. Moreover, in Ref. [30] the MC
IX. APPENDIX: ANALYSIS OF POSSIBLE for b > 1 was shown to behave in a non-universal man-
ERRORS IN τv ner: it strongly depends on details of scattering poten-
tial, whereas our theory assumes an uncorrelated short-
Wepresenthereananalysisoferrorsinthefittingpro- range disorder. Some other mechanisms of magnetocon-
cedure which determine the size of error bars in Fig. 5. ductance (such as classical memory effects, interaction
As discussedin Section V, τ was found fromthe follow- corrections, Maki-Thompson corrections etc.) may also
v
ing equation: become essential in large fields where the shape of WL
10
MC curve flattens. Therefore we believe that the in- extraction of the intervalley scattering rate.
termediate field range b = 0.2 0.4 is optimal for the
−
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