Valley splitting of SiSiGe heterostructures in tilted magnetic fields
a r X i v :c o n d -m a t /0601637v 1 [c o n d -m a t .m e s -h a l l ] 27 J a n 2006
Valley splitting of Si/Si 1?x Ge x heterostructures in tilted magnetic ?elds
https://www.wendangku.net/doc/859451675.html,i,1T.M.Lu,1W.Pan,2D.C.Tsui,1S.Lyon,1J.Liu,3Y.H.Xie,3M.M¨u hlberger,4and F.Sch¨a ?er 4
1
Department of Electrical Engineering,Princeton University,Princeton,New Jersey 08544
2
Sandia National Laboratories,Albuquerque,NM 87185
3
Department of Material Science and Engineering,UCLA,Los Angeles,CA 90095
4
Institut f¨u r Halbleiterphysik,Universit¨a t Linz,A-4040Linz,Austria
(Dated:February 6,2008)
We have investigated the valley splitting of two-dimensional electrons in high quality Si/Si 1?x Ge x heterostructures under tilted magnetic ?elds.For all the samples in our study,the valley splitting at ?lling factor ν=3(?3)is signi?cantly di?erent before and after the coincidence angle,at which energy levels cross at the Fermi level.On both sides of the coincidence,a linear density dependence of ?3on the electron density was observed,while the slope of these two con?gurations di?ers by more than a factor of two.We argue that screening of the Coulomb interaction from the low-lying ?lled levels,which also explains the observed spin-dependent resistivity,is responsible for the large di?erence of ?3before and after the coincidence.
PACS numbers:73.43.Fg,73.21.-b
The study on the valley splitting of the two-dimensional electron gas (2DEG)con?ned in (001)Si surface has been highlighted by recent research e?ort on Si-based quantum computation[1].For a Si 2DEG,only the two out-of-plane valleys are relevant since the other four in-plane valleys are lifted from the conduction band edge.To realize a functional Si quantum computer us-ing spins as quantum bits,a large valley splitting that lifts the remaining two-fold degeneracy is desirable since the existence of two degenerate states associated with the ±k z valleys is believed to be a potential source of spin decoherence [1].In the single-particle picture,theories [2,3,4]in the early period of the 2D physics proposed that the surface electric ?eld in the presence of 2D inter-face breaks the symmetry of these two valleys,resulting in an energy splitting proportional to the carrier density.The understanding of the valley splitting in real Si sys-tems,however,is not a trivial task and requires much beyond such non-interacting band picture.In fact,the many-body e?ect [2,4]was speculated to account for the enhancement over the bare valley splitting under strong magnetic (B)?elds,while a detailed calculation is not yet available.
Experimental research on the valley splitting,on the other hand,was conducted mainly on the Si metal-oxide-semiconductor ?eld-e?ect transistors (MOSFETs),in which the disorder e?ect is strong and direct measure-ment of the valley splitting proves to be di?cult [5,6].More than a decade ago,the introduction of the graded bu?er scheme signi?cantly improved the sample quality of the Si/SiGe heterostructures [7].To date,the val-ley splitting has been studied by various experimental techniques,including thermal activation [8],tilted ?eld magnetotransport [9,10],magnetocapacitance [11],mi-crowave photoconductivity [12]and magnetization [13].However,as pointed out by Wilde et al .in Ref.[13],results reported by di?erent groups are ambiguous and
inconsistent with previous band calculations.The na-ture of this valley splitting,especially its behavior under strong B-?elds,stays as an unsettled problem.
Of the various methods used to study the valley split-ting,tilted ?eld magnetotransport,also known as the coincidence method [14],is frequently utilized.In a B-?eld tilted by an angle θwith respect to the 2D plane,the ratio of the cyclotron energy E C =ˉh ωC =ˉh eB ⊥/m ?,where B ⊥is the perpendicular ?eld and m ?the e?ective mass,to the Zeeman energy E Z =g ?μB B tot ,where g ?is the e?ective g-factor,μB the Bohr magneton and B tot the total ?eld,can be continuously tuned by adjusting θ=cos ?1(B ⊥/B tot ).In particular,the so-called coin-cidence happens when the energy levels from di?erent Landau levels (LLs)are aligned at the Fermi level.In a recent experiment [15],the inter-valley energy gaps at the odd-integer quantum Hall (QH)states were studied and found to rise rapidly towards the coincidence.In this work,we show that the anomalous rise was not observed in the even-integer QH states,whose energy gaps close as θapproaches the single-particle degenerate points.For all the samples in our study,the ν=3valley splitting be-fore the coincidence follows a linear density dependence that extrapolates to about -0.4K at zero density,which is probably due to level broadening.The ν=3gap after the coincidence also depends linearly on density,while the slope increases by more than a factor of two.We ar-gue that screening of the Coulomb interaction from the low-lying ?lled levels,which also explains the observed spin-dependent resistivity,is responsible for the change of the observed ν=3gaps on di?erent sides of the coin-cidence.
The specimens in our study are modulation-doped n-type Si/SiGe heterostructures grown by molecular-beam epitaxy.Important sample parameters,such as the electron density (n ),mobility (μ)and width of the quantum well (W ),are listed in Table. 1.For the
2 samples labeled as LJxxx,relaxed Si0.8Ge0.2bu?ers
provided by Advanced Micro Devices(AMD)were used
as substrates,followed by a1μm Si0.8Ge0.2bu?er layer
prior to the growth of the strained Si channel.On top
of the Si quantum well,a20nm Si0.8Ge0.2spacer,a
delta-doped Sb layer,a25nm Si0.8Ge0.2cap,and a4nm
Si cap layer are subsequently grown.The carrier density
is controlled by the amount of Sb dopants.The high
mobility sample labeled as1317is the same specimen
as that used in Ref.[15]and its density and mobility
can be tuned by controlling the dose of low temperature
illumination by a light-emitting diode(LED).
Sampleμ(m2/Vs)Illumination
3.110
LJ1269.8Saturated
2.110
LJ13912Saturated
1.415
1317-II22Unsaturated
2.415
3
E n e r g y (a r b . u n i t )
E n e r g y g a p (K )
B-?elds.The LL (N),spin (↑or ↓)and valley (+or –)indices are indicated for each level.The positions of the 1st and 2nd coincidences are indicated.(b)Measured energy gaps at ν=4(B ⊥=1.5T)and (c)ν=6(B ⊥=1.0T)of sample 1317-I as a function of 1/cos θor B tot /B ⊥.The solid lines correspond to g ?=2.
the level diagram in Fig.2a,we label the valley split-ting as ?3(N=0,↓)and ?3(N=1,↑)before and after the coincidence,respectively.
In Fig.3,we plot the measured ?3(N=0,↓)and ?3(N=1,↑)gaps for all 7samples as a function of the carrier density.
The band calculation of valley split-ting in a Si 2DEG [2,3,4]based on the e?ective-mass approximation,showing a linear dependence ?v (K)～0.17n (1011cm ?2)at B =0,is also plotted (solid line)for comparison.Despite some scattering in the data,the measured ?3(N=0,↓)gaps essentially fall on a straight line that extrapolates to -0.4±0.2K at zero density.We note that this energy of -0.4K is within the order of the sample-dependent disorder broadening (Γ～ˉh /τ=
ˉ
h e/m ?
μ,where τis the transport scattering time),which lies between 0.3K and 1.1K in our samples.Interestingly,the detailed sample structure,e.g.,the well width W ,seems less important here.The ?3(N=1,↑)gaps of the same set of samples also fall onto a line extrapolating to -0.7±0.3K at n =0,again within the order of level broadening.On the other hand,the slope of the linear density dependence di?ers by more than a factor of 2(0.5K vs.1.4K per 1011cm ?2)before and after the coin-cidence.And both are signi?cantly higher than that of the band calculation at B=0.
The linear density dependence of the valley gaps and strong enhancement over the bare valley splitting were recently reported in a Si-MOSFET system using magne-tocapacitance method [11].The authors pointed out that the electron-electron (e-e)interaction,especially the ex-change interaction,is likely to account for the observed large valley gaps.In order to shed some light to the
0123
3
(K )
n (1011cm -2
)
FIG.3:Density dependence of the valley splitting at ν=3.The empty symbols (triangles for samples 1317and circles for LJxxx)stand for ?3(N=0,↓)and the ?lled symbols for ?3(N=1,↑).Dashed lines are linear ?ts to the data and ex-trapolate to ?nite values at zero density.The solid line shows the band calculation of valley splitting in Ref.[2].The inset shows the ?3gap of sample LJ127as a function of B tot .The coincidence occurs around B tot =7T.
apparent large di?erence between the ?3(N=0,↓)and ?3(N=1,↑)gaps,we scrutinize the many-body e?ect for the two con?gurations of ν=3,shown in Fig.4.For the relevant perpendicular B-?elds in this work,the e-e inter-action energy E e ?e ～e 2/4π?l B (l B =(ˉh /eB ⊥)?1/2is the magnetic length)is larger than the LL spacing so mix-ing between di?erent LLs has to be taken into account.Consequently,we explicitly include the lower two ?lled levels (N=0,↑,±),which are kept intact for all tilt an-gles,into the analysis.Before the coincidence,electrons in these two low-lying levels have the same LL but op-posite spin indices comparing to the ones near the Fermi level (E F ).Since the Pauli exclusion principle does not prevent the opposite spins from approaching each other,these low-lying electrons can come close to the electrons at E F and strongly screen the Coulomb interaction.The enhancement of the ν=3gap due to the electron-electron interaction is thus much reduced and the gap is close to the bare value at this LL.On the other side of the coincidence,however,such screening is much less e?ec-tive.First,the electrons near E F are from the N=1LL and their wave function is di?erent from the N=0lev-els.The o?-diagonal matrix element of this Coulomb energy between the two di?erent LLs should be consid-erably smaller than that from the same LL.Second,even in the presence of LL mixing e?ect,the exclusion princi-ple limits the screening between the same up-spin levels.As a result,the ?3(N=1,↑)gap is greatly enhanced over the bare valley splitting.We nevertheless emphasize here that in the last few LLs,the shape of the wave function
4
FIG.4:Level diagram atν=3before(left)and after(right) the coincidence.E F resides in the gap between the lowest empty levels and the top?lled levels.The level occupation, as well as the spin orientation,is indicated in the plot.Before the coincidence,the low-lying(N=0,↑,±)electrons,separated by E Z=g?μB B tot from the Fermi level,strongly screen the Coulomb interaction for electrons near E F,resulting in a less enhanced?3(N=0,↓)over the bare valley splitting.The same screening,on the other hand,is less e?ective from the like-spin charges in a di?erent LL,giving a large?3(N=1,↑).
is completely di?erent from the plane wave at B=0. So even the bare valley splitting here could be di?erent from the results obtained by Ohkawa and Uemura[2], who only consider high LLs by using simple average over the in-plane k-vector.
Finally,we note that the spin-dependent resistivity,?rst reported by Vakili et al.[17]and successfully ex-plained by screening from the?lled LLs,is also observed in our samples.In Fig.1,after the1st coincidence,the overallρxx amplitude is lower(dashed red curves)when the spins at the Fermi level orient opposite to the ma-jority up-spins in the system and higher when the two are aligned(solid blue curves),which was attributed to screening from the low-lying?lled LLs.Due to the ex-clusion principle,electrons with same spins cannot ap-proach each other to e?ectively screen the disorder po-tential,resulting in a higherρxx comparing to the oppo-site case.Interestingly,the same alternating pattern is also observed in the strengths of the odd-integer valley states.
In summary,we have carried out a titled?eld study of the Si/SiGe heterostructures and measured the energy gaps of integer QH states as a function of the tilt angle. The gaps at the even-integer?llings follow qualitatively the independent-electron picture,while the odd-integer states show rapid rise towards the coincidence angles.For all the samples we studied,theν=3valley splitting on both sides of the coincidence shows linear density depen-dence with signi?cantly di?erent slopes.The di?erence of the?3(N=0,↓)and?3(N=1,↑)gaps,as well as the observed spin-dependent resistivity,can be qualitatively explained by screening of the Coulomb interaction from the low-lying?lled levels.
This work is supported by the NSF,the DOE and the AFOSR and the AFOSR contract number is FA9550-04-1-0370.We thank Dr.Qi Xiang of AMD for supplying us with the high quality relaxed SiGe substrates.San-dia National Labs is operated by Sandia Corporation,a Lockheed Martin Company,for the DOE.The experi-ment performed in NHMFL is under the project number 3007-081.We thank E.Palm,T.Murphy,G.Jones,S. Hannahs and B.Brandt for their assistances and Y.Chen and D.Novikov for illuminating discussions.
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