a r X i v :c o n d -m a t /0104347v 1 [c o n d -m a t .s t r -e l ] 19 A p r 2001
Valley Splitting in Si-Inversion Layers at Low Magnetic Fields
V.M.Pudalov a ,A.Punnoo b ,G.Brunthaler c ,A.Prinz c ,G.Bauer c
a
P.N.Lebedev Physics Institute,Moscow,Russia.b
川字组词Weizmann Institute of Sciences,Rehovot,Israel
c
Institut f¨u r Halbleiterphysik,Johannes Kepler Universt¨a t,Linz,Austria
We report novel manifestation of the valley splitting for the two valley electron system in (100)Si-inversion layers at low carrier density.We found that valley splitting caus almost 100%modulation如同近义词
of the Shubnikov de Haas oscillations in very low magnetic fields ,almost on the bound of the quantum interference peak of the negative magnetoresistance.From the interference pattern of oscillations we d
etermined the valley splitting in the B =0limit which appears to vary within a factor of 1.3over the density range (3−7)×1011cm −2.Within the same range of densities,level broadenings in both electron valleys differ only by ≤3%.The latter result shows that the inter-valley scattering is not responsible for the strong (six fold)‘metallic-like’changes of the resistivity with temperature.PACS:71.30.+h,73.40.Hm,73.40.Qv
鼬的图片The apparent ‘metallic-like’temperature dependence of the resistivity is in the focus of current rearch inter-est.A number of microscopic mechanisms have been pro-pod for explaining the two major features of the resis-tivity:(i)strong metallic-like changes with temperature [1–6],and (ii)strong changes with in-plane magnetic field [7–11].One of the promising models intensively discusd in this connection is bad on scattering between different sub-bands [12–14].If the carriers mobility strongly de-pends on the Fermi energy E F ,the minority subband(s)may have a mobility esntially lower than that for the majority subband(s).The intersubband scattering may thus cau strong changes in the resistivity with temper-ature and magnetic field.In addition to the spin-related subbands,n −(100)-Si system has two minima in the con-duction band.The minima originate from six equivalent valleys located clo to the X -points in the Brillouin zone.Four of them shift up in energy by about (20-40)meV due to the confinement potential.At low densities,only the two lowest valleys ar
e filled;the sharp Si/SiO 2interface caus their additional splitting by ∆v >∼1K [15].There-fore,apriory,the remarkable strength of the metallic-like conduction in Si-samples might be related to the valley multiplicity.
The ‘metallic’temperature changes in the resistivity and the ‘metal-insulator’transition (MIT)in 2D sys-tems [1,2]are most pronounced in high-mobility samples where they take place in the regime of low carrier density and strong electron-electron interaction (the ratio of the Coulomb interaction energy to Fermi energy,r s ,is of or-der of 3-10).The interaction is expected to enhance both,spin-and valley-splitting [15];a possibility of a sponta-neous polarization,caud by interactions,was discusd earlier [16]and was recently recalled again in connection with the problem of the MIT in 2D [17].
In strong quantizing magnetic fields B ⊥(ωc τ≫1),valley splitting is known to be enhanced by exchange in-teraction between energy levels [15]and was directly mea-sured from Shubnikov-de Haas (ShdH),quantum Hall (QHE)effect,and chemical potential oscillations.In par-ticular,in Ref.[18],valley splitting was found to be al-most density independent and to increa linearly with magnetic field,∆v (B )=∆0v +αB ,where α=0.6K/T and the sample dependent ∆0v ≈2K.The latter renor-malization is intrinsic only to the strong field regime of well parated energy levels and is irrelevant to the prob-lem of metallic conduction at B =0;valley splitting in Si-structures at B →0remai
狂涛巨浪ned so far unexplored.In the current work we have found that valley splitting manifests itlf in high mobility Si-samples not only at high fields,but also at low magnetic fields,B <0.8T ,giving ri to beatings in the ShdH oscillations.Such beatings were discusd theoretically in Ref.[15]b but have not been obrved experimentally.Due to very high mobility of the samples we traced the oscillations down to the fundamental limit of ≈0.2T,given by the ont of the quantum interference (i.e.,weak negative magne-toresistance)[19].From this novel beating pattern we determined the zero-field valley splitting as a function of the carrier density.We found that ∆v varies only weakly in the range of low densities,n =(3−7)×1011cm −2.Al-most 100%amplitude of modulation in the beating pat-tern evidences that the scattering times in the two val-leys are about equal .This obrvation demonstrates that the inter-subband scattering mechanism is not responsi-ble for the strong ‘metallic-like’temperature dependence of the conduction in Si-inversion layers.
We performed measurements on three high mobility n −Si-MOS samples lected from three different wafers (who plane coincide with (100)-crystal plane to within 1o ):Si11(peak mobility µpeak =3.9m 2/Vs at T =0.3K),Si12(µpeak =3.4)and Si15(µpeak =4.0).All samples exhibited strong (six-fold)metallic-like fall in the resistivity with temperature,and an apparent MIT at a ’critical density’n c ≈0.9×1011cm −2.
Figure 1shows a picture of the oscillations typical for
1
high magneticfields and for a relatively high carrier den-sity,n=10.24×1011cm−2.As the magneticfield in-creas,ShdH oscillations evolve into the quantum Hall effect.In quantizingfields the energy spectrum is
ε=¯hωc(N+1
2
g∗µB B±
1
δρxx(B⊥)with Lifshitz-Kovich(LK)formulae[21,22] for spin-degenerate carriers:
δρxx(B⊥)
2eB⊥
−πs +
A−s cos 2πs hn v−
ωcτ±q
2π2skT/¯hω∗c
τq=(τ+q+τ−q)/2for quan-tum lifetime.The former parameter determines en-tirely the Fourier spectrum of oscillations and nodes location,whereas the latter one describes the absolute amplitude of oscillations and its monotonicfield depen-dence.The difference of partial lifetimes for each valley, (τ+q−τ−q)/
∆v,open symbols are for ∆0v.The uncertainty is reprented either by error bar or the symbols size.
Our measurements were performed in very low though finitefields,B≈0.3T.It is not clear whether the em-piricfield dependence Eq.(2)remains valid down to B=0or∆v decreas more rapidly as levels starts overlapping in lowfields.The accuracy of our data was insufficient to determine this experimentally though we found thefitting to be better withfield dependent ∆v(B)(as in Eq.(2)).Given Eq.(2)is applicable at B→0,it contributes about20%to the obtained∆v values.We prent therefore in
Fig.4the results of bothfitting,withfield independent
The sought-for∆(B=0)value is thus within the inter-
Graduate School of Weizmann Institute of
Science. val from∆v and∆0v exhibit a weak
non-monotonic dependence who origin is unclear.Our
analysis of the ShdH oscillations in the two-valley system
performed up to r s=5didn’t reveal any deviation from
the LK-formulae.This justifies the assumption of the
conventional Fermi-liquid behavior which we ud in the
analysis above.
An additional important result follows from ourfit-
ting.We found the quantum lifetimeτq to be almost
the same in the two electron valleys,the maximal dif-
ference being only(τ+q−τ−q)/
ρ(T)which the sample exhibit at
the same density[23].From the proximity of the quan-
tum life times in both valleys,we conclude therefore that
the inter-valley scattering mechanism[12]is not respon-
sible for the strong metallic-like temperature variation of
the resistivity in Si-inversion layers.
To summarize,in high mobility(100)Si-inversion lay-
ers,the system which exhibits very strong‘metallic-like’
features in conduction,we obrved a novel manifesta-
tion of valley splitting:it caus unexpected beatings
in Shubnikov-de Haas oscillations in low magneticfields,
(0.15–0.4)T,right on the bound of the negative mag-
netoresistance peak.From the beatings pattern of os-
cillations we determined the valley splitting∆v in the
B=0limit.We found that∆v varies with density
rather weakly and doesn’t display a critical behaviour
in the range of densities(3−7.5)×1011cm−2or r s=
3–5.We determined also the individual quantum life-
times,τ±q in both valleys,which appear to differ by less
than3%.This insignificant difference inτq demonstrates
that the miclassical mechanism of mobility changes re-
lated to the inter-valley scattering is not the origin of
the strong metallic-like temperature dependence of the
resistivity in Si-inversion layers.Thefindings t novel
constraints on the microscopic models developed to ex-
plain the‘metallic’-conduction.
Authors are grateful to A.M.Finkelstein,M.E.Ger-
shenson and H.Kojima for discussions.Two of us(V.P.
and A.P.)acknowledge the hospitality of the Lorenz Cen-
ter for theoretical physics at Leiden University,where a
part of this work was done in June,2000.The work was全国企业信息公示系统官网
supported by NSF DMR-0077825,FWF(13439),INTAS,
German Ministry of Science(DIP),RFBR,the Programs
‘Physics of nanostructures’,‘Statistical physics’,‘Inte-
gration’and‘The State support of the leading scientific
schools”.A.P.acknowledges the support from Feinberg
军事类书籍
I.APPENDIX:ENERGY SPECTRUM IN
SI-INVERSION LAYER AT LOW MAGNETIC
FIELDS
The results on the valley splitting prented above shed a light on the energy spectrum of the two-va
lley elec-tron system in(100)Si-inversion layers in lowfields.We plotted in Fig.5the energy for three lowest Landau lev-els,N=0,1,2,according to Eqs.(1)and(2)(from the main ction).In highfields,the cyclotron splitting is the largest and the quence of energy levels corresponds to the int to Fig.1.Counting from the lowest Fermi energy,the1st gap in the energy spectrum(filling factor ν=1)corresponds to valley splitting;ν=2is for the reduced Zeeman splitting,ν=3is again for the valley splitting andν=4is the reduced cyclotron
gap.This
picture was verified in numerous experiments[1,2].
红楼梦每回读后感
Due to the nonzero valley splitting at B=0,as mag-
neticfields decreas,the energy levels start crossing each
other at B<∼1T[3].In the region of crossing,this single-
electron picture fails and the repulsion between levels
should be taken into account.There is almost no ex-
perimental data available on the details of the energy
spectrum in Si in lowfields.It is well known only[4–6]
that in lowfields/low density regime quantum oscilla-
tions are missing forν=3,4,5and remain pronounced
forν=1,2and forν=6and10;this result was recently
confirmed in ref.[7].
Ã
@
r
t
Ã
F
7Ã U
FIG.5.Schematic magneticfield dependence of the en-
ergy spectrum(in the single-electron approximation)for
two valley electron system in Si and for three lowest Lan-
dau levels.The parameters of the spectrum correspond
to Eqs.(1)and(2);∆v=2K,as for the sample Si12.
Infields B<0.7T,the quence of energy levels
changes as shown in the int to Fig.5.For example,
at B=0.6T,the1st splitting,ν=1,corresponds to
the Zeeman gap,ν=2is the cyclotron gap(reduced by
the spinflip energy),ν=3is a combination of the val-
ley and cyclotron splitting etc.In this simplified picture,
at somefield values(B=0.45T and0.25T in the int
to Fig.5)a number of energy levels coincide.We don’t
think though that the simplified single-electron picture
of Fig.5may predict exact location of the level crossing
regions.
The identification of each energy splitting in low
field/low density regime is not transparent and requires
more thorough theoretical calculations of the spectrum
and much more detailed experiments.As an example,
on the n−B plane in Fig.6we reproduce from Ref.[3]
a Landau fan diagram of oscillations in the low density
limit;similar data were reported recently in Ref.[7].In
Fig.6,theρxx-minima atν=6can be traced with-
out interruption down to thefield B=0.55T,through
the‘metal-insulator’boundary,however the origin of this
splitting may change from‘Zeeman gap’at highfields to
any other ‘valley gap’or‘cyclotron gap’at low
fields.By now,there is nofirm physical background to
identifyν=6oscillation in lowfields with spin split-
ting.For example,one can e from the int to Fig.5,
that in the single-electron approximation theν=6gap
at B=0.6T corresponds to the transition with chang-
ing valley(but is reduced by the spinflip energy)and
ν=4gap is the combined gap with changing both,the
cyclotron number and the spin projection.
FIG.6.Landau fan diagram for the four different
ρxx-minima,reproduced from Fig.2of Ref.[3].Full dots
show MIT-boundary,defined according to vanishing ac-
沙县是哪个省的
tivation energy.Other symbols designate location of the
ρxx-minima on the n−B-plane.
In Ref.[7],from the estimated ratio of the two gaps,at
ν=6(which authors of Ref.[7]assumed to be the spin-
gap)and atν=4(was assumed to be the cyclotron gap),
a conclusion was drawn on the unexpectedly strong en-
hancement of the g∗-factor at low electron densities.Tak-
ing into account vanishing∆c in the crossing region and
ill-defined splittings at lowfield,wefind this argument
5
unjustified.Recent direct measurements of the Zeeman energy in lowfields vs carrier density[8]show that the Zeeman energy increas smoothly as density decreas down to n c,in agreement with the anticipated Fermi liq-uid renormalization and with recent results by Okamoto et al.[9],without any une
xpected deviations or diver-gence in the range of densities n=(1−90)×1011cm−2. It is noteworthy that for the weakfield interference pattern which we analyzed above,the interpretation of each oscillation is not important and does not influence on the conclusion about the valley origin of the interfer-ence and on the results of our analysis.
