Multiscale theory of valley splitting
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Multiscale theory of valley splitting
Sucismita Chutia, S. N. Coppersmith, and Mark Friesen
Department of Physics, University of Wisconsin-Madison, Wisconsin 53706, USA
The coupling between z valleys in the conduction band of a Si quantum well arises from phenomenaoccurring within several atoms from the interface, thus ruling out a theoretical description basedon pure effective mass theory. However, the complexity and size of a realistic device precludesan analytical atomistic description. Here, we develop a fully analytical multiscale theory of valleycoupling, by combining effective mass and tight binding approaches. The results are of particularinterest for silicon qubits and quantum devices, but also provide insight for GaAs quantum wells. PACS numbers: 73.21.Fg,73.20.-r,78.67.De,81.05.Cy
The two-dimensional electron gas formed at a siliconheterointerface underpins the modern electronics indus-try. But in spite of its ubiquity, the silicon interfaceexhibits phenomena that are not fully understood, andcannot be explained by the conventional effective masstheory. In the emerging field of nanoelectronics, quan-tum degrees of freedom like spin form the basis for noveltechnologies, such as spintronics [1] and quantum com-putation [2, 3]. An alternative degree of freedom is asso-ciated with the low-lying features in the conduction bandstructure, known as valleys, for Si and other indirect gapsemiconductors [4]. The degeneracy of the low-lying val-ley states is 2-fold for both Si[001]/SiO and Si[001]/SiGeinterfaces, due to mass anisotropy and strain effects, re-spectively. For quantum devices, where valley physicshas been studied most extensively [5], the valley statesmay either be “frozen out,” in favor of the spin degreeof freedom [6], or utilized as qubits [7]. In either case, acomplete understanding of the valley physics is essential.One main concern is the origin and the magnitudeof valley coupling, which lifts the valley degeneracy.The two approaches previously applied to this prob-lem involve continuum theories like effective mass (EM)[8, 9, 10, 11, 12, 13], and atomistic theories like tightbinding (TB) [14, 15]. While the former provides in-tuition because of its analytical nature, it cannot fullyaccount for the fundamentally discrete and atomistic na-ture of the valley coupling. On the other hand, atom-istic theories give an accurate description of the valleysplitting from first principles, but they cannot provideanalytical results, except in the simplest geometries [14].In this letter, we bridge the gap between microscopic andmacroscopic theories through a multiscale technique, andwe provide theoretical justification for an intuitive ex-tended EM theory. We show how atomic scale correctionsnear an interface lead to significant improvements in theEM description, even for direct gap semiconductors likeGaAs.It is well known that the conventional EM theory ofelectron confinement in a semiconductor crystal breaksdown near a sharp confining potential, like a quantumwell [4]. For direct gap materials, the resulting errors are typically small and may be treated perturbatively in thelong-wavelength EM theory [16]. For indirect gap ma-terials like Si, the atomic scale physics of the interface,which is absent from the EM theory, is also responsi-ble for valley splitting, corresponding to an energy scale( ∼ r ) = X n = ± z α n e ik n z u k n ( r ) F ( z ) , (1)where α n are the valley composition factors (of no im-portance here), k ± z = ± k are the positions of the val-ley minima in the Brillouin zone, u k n ( r ) are periodicBloch functions, and F ( z ) is the long wavelength enve-lope. Note that we have only included contributions fromthe two z valleys, as appropriate for [001] strained quan-tum wells. The short wavelength physics is contained inthe fast phase oscillations and the Bloch functions. Notethat the silicon valley minima are located near the Bril-louin zone boundaries, with [14] k = 0 . π/a ) for aSi cubic unit cell of width a = 5 .
43 ˚A, consisting of fouratomic planes along [001].We have previously argued that the atomic scalephysics of valley coupling can be incorporated into thelong wavelength theory by means of a δ -function poten-tial at the quantum well interface [11]. Similar argumentshave also been put forth in Refs. [17, 18], leading to acoupled set of envelope equations of the form [13, 19, 20] X n = ± z α n e ik n z (cid:20) − ~ m l ∂ ∂z + V ( z ) (2)+ X m Λ δ ( z − z m ) − E (cid:21) F ( z ) = 0 . Here, m l = 0 . m is the longitudinal effective mass, V ( z ) is the vertical confinement from the conductionband offset of V , and z m = ± L/ z valleys.Since V ( z ) is constant away from the interface, the so-lutions for the ground state envelope function are givenby F ( z ) = (cid:26) A cos( qz ) ( | z | < L/ B e − pz ( | z | ≥ L/ . (3)Conventionally, the unknown parameters in Eq. (3) aredetermined by matching the envelope function and itsfirst derivative on either side of the interface [21]. How-ever, the latter matching condition must be modified inthe presence of a δ -function interface potential. Integrat-ing Eq. (2) over an infinitesimal range about the inter-face, we obtain the new matching condition,Λ = ~ m l F ′ + ( L/ − F ′− ( L/ F ( L/ , (4)where F ′ + ( F ′− ) correspond to right-hand (left-hand)derivatives. Note that a similar discontinuity in F ′ ( z )also occurs at any heterojunction with two different ef-fective masses [21]. This effect is unrelated to valley cou-pling, and we ignore it below. Indeed, for silicon-richSiGe materials, the mass variations are small and incon-sequential for our main results.We now construct a multiscale theory for Λ. To be-gin, we note that the conventional EM theory remainsvalid and accurate, except within about one atom dis-tance from the quantum well interface. In the vicinity ofthe interface, the EM theory should be replaced by anatomistic one. The simplest TB theory that can describevalley coupling was derived in Ref. [14]. The model in-volves two bands, with nearest and next-nearest neighbortunneling parameters, t and t , respectively. An addi-tional onsite parameter describes the confinement poten-tial V ( z ) of the quantum well. Although these parame-ters may vary with position, depending on the alloy com-position, the variations are small for SiGe and we ignorethem here. The tight-binding coupling parameters arethen given by [14] t sin ( k a/
4) = 2 ~ /m l a , t = 4 t cos( k a/ . (5)Near the top interface, the TB Hamiltonian is given by H = . . . t t t t t t t t t t V t t
00 0 0 t t V t t . . . . (6)(Diagonal elements are highlighted in bold, for clarity.)The eigenstates of H correspond to vectors of TB coef-ficients ( . . . , C − , C − , C , C , . . . ). Here, we consider a FIG. 1: (Color online) Differences between the effective mass(EM) and tight binding (TB) wavefunction solutions areshown for a GaAs quantum well. (Only the right half of thewell is shown.) The dashed line shows the conventional EMtheory, while the solid line includes multiscale corrections.The vertical dotted line marks the quantum well boundary.Inset: Multiscale ansatz for envelope functions near the inter-face. For Si, 7 atoms are needed. For GaAs only 4 atoms areused. Open circles correspond to sites in the barrier region. quantum well of size L = a ( N +1) /
2, containing (2 N +1)atoms, and centered at atomic position N = 0.To implement a multiscale theory, we require that theTB eigenstates match the EM solutions away from theinterface. From Eq. (1), the two lowest-energy TB wave-functions can be expressed as C j = ( − j √ jk a/ F j , (7) S j = ( − j √ jk a/ F j , (8)where the actual ground state depends on the width ofthe quantum well [13]. Here, ( − j / √ α + z , α − z ) = (1 , ± / √
2, respectively. Theenvelope coefficient F j = F ( ja/
4) is determined fromthe correspondence with Eq. (3). The physics of valleycoupling is captured by the sudden change of slope ( i.e. ,the kink) in the envelope at the interface. To facilitatecalculations, we consider the model parameters shown inthe inset of Fig. 1. Immediately adjacent to the interface,the envelope function exhibits a change of slope, with F ′− = (4 /a ) β and F ′ + = (4 /a ) γ on either side of the kink,and amplitude F N +1 = f right at the interface. Thus, F N = f + β , F N +2 = f − γ , and so on.Before solving the multiscale theory, it is illuminat-ing to gauge the accuracy of the EM ansatz of Eqs. (7)-(8). This is accomplished in Fig. 2, using the two lowest-energy TB eigenstates to infer the envelope function fromthe relation 2 F j = C j + S j . The result exhibits resid-ual short wavelength structure, arising from the fact that
0 10 20 30 40 500.150.10.050
0 20 400.20.10-0.1 E n v e l ope , F j ( a / ) / C j ( a / ) / Atom position
Atom position
FIG. 2: (Color online) Dimensionless envelope for the groundstate wavefunction in a 10 nm Si/SiGe quantum well. (Onlythe right half of the quantum well is shown.) The verticaldashed line marks the quantum well boundary. The discretepoints are obtained from TB theory, as described in the text.Spurious short wavelength structure is cause by limitations inthe EM theory. Inside (outside) the quantum well, the blue(red) solid lines are fits to Eq. (3). Inset: the full, dimension-less TB wavefunction, including the fast oscillations. the exact k values of the fast oscillations of the two low-est eigenstates are nearly (but not quite) identical [15].This is a signature of the incomplete separation of thelong and short wavelength physics, and it places a fun-damental limit on our ability to match the TB and EMtheories. In the present work, the spurious “jitter” ev-ident in Fig. 2 leads to errors in the evaluation of thekink.The multiscale theory involves just two equations fromthe full TB Hamiltonian, and we can choose which equa-tions to use. The spurious jitter in Fig. 2, could be miti-gated by an averaging procedure. Indeed, in this way, weobtain excellent agreement with previous estimates of thevalley splitting [13]. However, such techniques detractfrom the simplicity of the multiscale approach. Here, wetake a different tack, noting that the alternating behaviorof the TB coefficients in Fig. 2 can be partially mitigatedsimply by using alternating TB equations. We considerthe following equations centered symmetrically aroundthe interface: t C N − + t C N − + t C N +1 + t C N +2 = E TB C N ,t C N + t C N +1 + V C N +2 (9)+ t C N +3 + t C N +4 = E TB C N +2 . Note that either the cosine (7) or sine (8) functions canbe used here.It is necessary to take into account the curvature ofthe envelope functions in system (9), in order to avoidunphysical solutions. Near the interface, the cosine en-velope in Eq. (3) has a vanishing second derivative, butthe exponential envelope does not. To leading order, wecan express the latter in terms of parameters f and γ as follows: F N +2 = f − γ , F N +3 = f − γ + γ /f , and F N +4 = f − γ + 3 γ /f . Evaluating system (9), we nowobtain F ′ + ( L/ − F ′− ( L/ F ( L/
2) = ( β − γ ) f a (10) ≃ γ f = (cid:16) pa (cid:17) ≃ mV a ~ , where we have dropped higher order terms in the smallparameter a/L . The anticipated linear dependence of Λon V [13] emerges from Eq. (4):Λ ≃ V a/ . × − ) V , (11)where Λ is in units of eVm when V is in eV.Eq. (11) is our main result, obtained through a mul-tiscale analysis of the kink of the envelope function. Wecan compare this with the apparent kink obtained by fit-ting Eq. (3) to the full TB envelope function, as shownin Fig. 2. In spite of the spurious jitter, the two esti-mates agree to within 20%, in the wide quantum welllimit. We can also compare Eq. (11) to the estimateΛ ≃ (7 . × − ) V obtained in Ref. [13], by fitting the-oretical EM predictions to TB numerical solutions for thevalley splitting. The latter differs from Eq. (11) by abouta factor of two, which we attribute to the fundamentallimitations of the EM ansatz of Eqs. (7)-(8). Neverthe-less, it is clear that the EM theory and the multiscaleanalysis, described here, capture the essential physics ofvalley splitting, and enable semi-quantitative predictions.Thus justified, the more accurate numerical estimate forΛ in Ref. [13] forms the basis for a fully quantitative EMtheory. Further improvements in the EM ansatz and thekink analysis should lead to better correspondence be-tween the estimates for Λ.We now turn to direct gap materials, such as GaAs.Although a sharp confinement potential does not causevalley coupling in this case, there are still atomic scalecorrections to the EM theory. As in the Si case, the cor-rections tend to be more significant for narrow quantumwells [22]. Foreman has also noted that the correctionscan be treated within the EM theory by introducing a δ -function at the interface [17]. Here, we apply the mul-tiscale theory developed for Si to the GaAs quantum well,obtaining an analytical expression for the GaAs interfacepotential, Λ. We also compare the improved wavefunc-tion solutions to those obtained from TB theory.The simplest TB theory for a single (Γ)-valley mate-rial involves just the nearest-neighbor tunneling param-eter t = − ~ /m ∗ a , where a = 5 .
64 ˚A is the width ofthe GaAs cubic unit cell. For Al x Ga − x As, used in thebarriers, the effective mass m ∗ depends on the composi-tion x to a greater degree than silicon alloys. Hence, t depends on the atomic position. The onsite parameter ǫ ( z ) = 16 ~ /m ∗ a + V ( z ) also depends on composition.However, for the sake of transparency, we will ignore ef-fective mass variations, taking ǫ ( z ) = V ( z ), and setting m ∗ to an appropriate average of the effective masses nearthe interface. Indeed, m ∗ eventually drops out of theleading order expression for Λ, and a more careful treat-ment provides only small corrections.The matching condition, Eq. (4), also hold for GaAs.However, because there is only one valley, and the TBmodel has only one band, the TB envelope function andwavefunction are now identical: C j = F j . Near the inter-face, we parametrize the envelope as shown in the insetof Fig. 1. The TB eigenstates are smooth (in contrastwith Fig. 2), so we may now use adjacent TB equations: t C N − + t C N +1 = E TB C N ,t C N + V C N +1 + t C N +2 = E T B C N +1 . (12)In this case, there is no need to consider curvature of thewavefunction. Solving for β and γ directly, and notingthe relation E TB = 2 t + E between the TB and EMenergies, we obtain the GaAs result,Λ = a ( V − E ) / . (13)In the physically relevant limit of E ≪ V , the GaAs andSi interface potentials have identical forms. For GaAsmaterials parameters, we obtain Λ ≃ (1 . × − ) V .The effect of the interface potential on the wavefunc-tion is shown in Fig. 1, where we plot the differencesbetween the approximate (multiscale) and exact (TB)results for a narrow quantum well. We also show resultsfor the conventional (Λ = 0) EM theory. Deviations fromthe EM theory are small in both cases. However becausethere is no jitter in the GaAs TB envelope function, wefind that Eq. (13) captures the atomic scale correctionswith great accuracy.In conclusion, we have demonstrated that leading cor-rections to the effective mass theory at a sharp quantumwell boundary arise from the atomic scale physics nearthe interface. The corrections appear as a small kink inthe envelope function. A multiscale approach, combin-ing effective mass and tight binding theories, leads to ananalytical expression for the effective interface potentialin silicon, which determines the valley splitting. Simi-lar corrections apply to GaAs quantum wells, althoughthere is no valley coupling. For other device geometries,including graded interfaces and electric fields, the presentapproach remains robust. These situations may also betreated by a multiscale analysis.This work was supported by NSA/ARO contract no. W911NF-04-1-0389 and by NSF grant nos. DMR-0325634, CCF-0523675, and CCF-0523680. [1] I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. , 323 (2004).[2] D. Loss and D. P. DiVincenzo, Phys. Rev. A , 120(1998).[3] B. E. Kane, Nature (London) , 133 (1998).[4] T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. , 437 (1982).[5] S. Goswami, K. A. Slinker, M. Friesen, L. M. McGuire, J.L. Truitt, C. Tahan, L. J. Klein, J. O. Chu, P. M. Mooney,D. W. van der Weide, R. Joynt, S. N. Coppersmith, andM. A. Eriksson, Nat. Phys. , 41 (2007).[6] M. Friesen, P. Rugheimer, D. E. Savage, M. G. Lagally,D. W. van der Weide, R. Joynt, and M. A. Eriksson,Phys. Rev. B , 121301(R) (2003).[7] V. N. Smelyanskiy, A. G. Petukhov, and V. V. Osipov,Phys. Rev. B 72, 081304(R) (2005).[8] W. Kohn, in Solid State Physics , edited by F. Seitz andD. Turnbull (Academic Press, New York, 1957), Vol. 5.[9] F. J. Ohkawa and Y. Uemura, J. Phys. Soc. Japan ,907 (1977); ibid , , 917 (1977).[10] L. J. Sham and M. Nakayama, Phys. Rev. B , 734(1979).[11] M. Friesen, Phys. Rev. Lett. , 186403 (2005).[12] D. Ahn, Journ. Appl. Phys. , 033709 (2005).[13] M. Friesen, S. Chutia, C. Tahan, and S. N. Coppersmith,Phys. Rev. B , 115318 (2007) .[14] T. B. Boykin, G. Klimeck, M. A. Eriksson, M. Friesen,S. N. Coppersmith, P. von Allmen, F. Oyafuso, and S.Lee, Appl. Phys. Lett. , 115 (2004).[15] T. B. Boykin, G. Klimeck, M. Friesen, S. N. Copper-smith, P. von Allmen, F. Oyafuso, and S. Lee, Phys.Rev. B , 165325 (2004).[16] M. G. Burt, J. Phys. Cond. Mat. , 6651 (1992); M. G.Burt, Phys. Rev. B , 7518 (1994).[17] B. A. Foreman, Phys. Rev. B , 165345 (2005).[18] M.O. Nestoklon, L.E. Golub, and E.L. Ivchenko, Phys.Rev. B , 235334 (2006).[19] H. Fritzsche, Phys. Rev. , 1560 (1962).[20] W. D. Twose, in the Appendix of Ref. [19].[21] J. H. Davies, The Physics of Low-Dimensional Semicon-ductors (Cambridge Press, Cambridge, 1998).[22] F. Long, W. E. Hagston, and P. Harrison, in23rd Inter-national Conference on the Physics of Semiconductors