Impurity-driven intervalley spin-flip scattering-induced 2D spin relaxation in silicon
IImpurity-driven two-dimensional spin relaxation induced by intervalley spin-flipscattering in silicon
Yang Song
1, 2, ∗ and S. Das Sarma
1, 2 Condensed Matter Theory Center, Department of Physics,University of Maryland, College Park, MD 20742, USA Joint Quantum Institute, University of Maryland, College Park, MD 20742, USA
Through the theoretical study of electron spin lifetime in the two-dimensional electron gas (2DEG)confined near the surface of doped Si, we highlight a dominant spin relaxation mechanism inducedby the impurity central-cell potential near an interface via intervalley electron scattering. At lowtemperatures and with modest doping, this Yafet spin flip mechanism can become more importantthan the D’yakonov-Perel’ spin relaxation arising from the structural Rashba or Dresselhaus spin-orbit coupling field. As the leading-order impurity-induced spin flip happens only between twonon-opposite valleys in Si, 2DEG systems in Si MOSFETs or SiGe heterostructures are a naturalplatform to test and utilize this spin relaxation mechanism due to the valley splitting near theinterface and the tunability by electrical gating or applied stress. Our proposed new spin relaxationmechanism may explain a part of the spin relaxation contribution to Si-based 2DEG systems, andshould have spintronic applications in Si-based devices.
PACS numbers:
I. INTRODUCTION
Silicon takes the unique position in both the conven-tional main-stream electronic industry (i.e. C-MOS) andthe emerging fields of quantum information science andtechnology such as spintronics [1–3] and quantum com-putation [4, 5]. Its strength derives from the maturedcapacity of extreme high purity and low cost materialgrowth, and perhaps more crucially, the orders of mag-nitude tunability in electrical conductivity enabled bydoping and gating. Silicon (Si) has continued to revealanother crucial property, that is, its long spin lifetimedue to the relatively small atomic spin-orbit coupling(SOC), bulk inversion symmetry, and zero nuclear spin inthe abundant isotope Si. Two-dimensional electron gas(2DEG) occupying the few lowest quantized 2D subbandsof various Si surfaces and quantum wells has long becomean important playground for fundamental science [6, 7]and more recently, quantum computing qubit platformsthrough gate-defined or donor-defined quantum dots [5].In particular, the long spin lifetime and the ability tocontrol the confined electrons through externally appliedelectrical voltage (i.e. fast gates) near the Si surface arethe main drivers of the great interest and activity on Si-based spintronics and quantum computing architectures.With respect to spin relaxation, there are some keydifferences between 3D and 2D Si systems worth empha-sizing right at the outset. In particular, the confiningpotential at the interface, where the 2DEG resides, maybreak the inversion symmetry of the Si crystal. Thisgenerally results in spin splitting in the band structureand induces an effective momentum-dependent magneticfield for conduction electrons [8], often referred to as the ∗ Electronic address: [email protected]
Rashba field arising purely from the structural asymme-try in real space. In addition, quantum wells with oddnumber of Si layers or broken rotoinversion symmetry atthe Si-Ge interface [9–12] induce generalized Dresselhausfield [13, 14].Such structural SOC effects obviously are not presentinside the 3D bulk Si and can exist only in the2DEG. When electrons undergo momentum scattering(e.g. by impurities or phonons) in the presence ofRashba/Dresselhaus effect, their spins precess randomlyover time and relax [15]. This D’yakonov-Perel’ (DP)process has been the only main spin relaxation mecha-nism studied so far in Si 2DEG [16–20]. This has led tothe general belief, questioned in the current work, thatthe DP mechanism is the only spin relaxing mechanismin Si 2DEG that needs to be considered theoretically.In this work, we bring in a fundamentally differentspin-relaxation mechanism, which may gradually dom-inate over the DP mechanism with increasing impu-rity densities or doping. This new impurity-inducedspin relaxation mechanism does not rely on the effec-tive Rashba/Dresselhaus magnetic field between scatter-ing events, but rather flips spins right at the scatteringevents, through the contact spin-orbit interaction at theimpurity core. Therefore our present mechanism can betermed a
Yafet process [21].While the scattering is driven by impurities in bothmechanisms at low temperatures, the difference is thatscattering serves to interrupt the spin precession in theDP process whereas it facilitates spin flip in the Yafetprocess. As a result, instead of weakening with higherimpurity density or lower mobility as in the DP spin re-laxation, our mechanism grows stronger with increasing(decreasing) impurity density (mobility), and thereforecan be distinguished experimentally from the DP pro-cess.We believe that some contributions of this new a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n impurity-induced Yafet process to Si 2D spin relaxationmay have already been detected experimentally as wediscuss later in this article. We note that in 3D Si, ourprocess is already known to be the experimentally domi-nant electronic spin relaxation mechanism in high dopingsituations [22].The other aspect we highlight is the tunability of spinlifetime. While the charge transistor builds on the tun-able conductivity (i.e. tunable carrier momentum relax-ation time), it is desirable that the spin relaxation timecan be controlled as well for spintronic applications. Aswe will show in detail, the leading-order spin flip occursonly during intervalley scattering and between two non-opposite valleys among the six Si conduction valleys (socalled “ f -process” [23]), whereas it vanishes during in-travalley scattering or intervalley scattering between twoopposite valleys (“ g -process”). For various 2DEG planeorientations relative to the Si crystallographic direction(e.g. 100, 110, 111 or arbitrary orientation), it is wellknown that the resulting 2D electronic ground states havedifferent valley configurations, with the ground state val-ley degeneracy varying from 1 to 6 depending on surfaceorientations and details [6, 24]. As such, the spin lifetimedetermined by this mechanism in 2DEG will be distin-guishable just owing to different plane orientations of the2D system, producing substantial anisotropy in the 2Dspin relaxation. Moreover, the tunability of the relativevalley energies by stress and especially, by gate voltagein Si MOSFETs, can enable fast on-chip spin lifetimecontrol. Spin-orientation dependence of the spin relax-ation, absent for the charge mobility, can also be simi-larly controlled. Since valleys do not play a central role inthe DP process, which is governed entirely by the struc-tural asymmetry, such orientation or gate voltage depen-dence of spin relaxation is qualitatively different in theDP mechanism [17, 18]. This difference can also distin-guish our proposed mechanism from the DP mechanismwith respect to Si 2D spin relaxation.In 3D bulk Si, as we mentioned this novel Yafet processhas been shown to be the dominant spin relaxation mech-anism when the scattering is caused by donor impurities[22]. It is caused by the spin-dependent interaction withthe impurity core, and is far more important than thespin flip during intravalley scattering by the long-rangeCoulomb interaction, or during intervalley scattering bythe spin-independent part of the impurity core potential,neither of which exhibits the empirically strong donordependence [25–29]. Our goal here is to introduce thisimportant mechanism into the Si 2DEG, build up its pri-mary trend qualitatively and quantitatively, and discussits experimental relevance and applications. Since theDP process and our process are completely independentspin relaxation mechanisms, generically both should bepresent in Si 2DEG, and their relative quantitative im-portance will depend on all the details of the specificsystem and samples being studied.We briefly discuss the scenarios where our proposedmechanism can be of importance and utilized. The first apparent criterion is low temperature and moderate-to-high impurity density, so that phonon-driven spin relax-ation is relatively weak. Conversely, our mechanism ismost likely overshadowed by the DP spin relaxation pro-cess in intrinsic to low-doped 2DEG systems, typical forSi/SiGe quantum wells and other modulation-doped het-erostructures where interface impurity scattering is rel-atively weak [16, 30], except for symmetrically designedwells [9, 31] where the structural asymmetry can be re-duced. It can be easily shown that DP spin relaxationalone leads to rapidly diverging spin lifetime once themobility µ is lowered to a few m /Vs [18]. Also, as thepresent mechanism relies on the SOC between the freeelectron and the impurity core, for a given impurity den-sity its quantitative effect is ranked according to the signof impurity charges: positively charge impurities > neu-tral impurities (cid:29) negatively charged impurities, the lastof which repel the electrons and render little central-cellcorrection [32–34]. As such, both n-type SiGe quantumwell and accumulation layer in n-type MOSFET are goodcandidates for studying our mechanism. In low-mobilitySi MOSFET samples, however, inversion as well as accu-mulation layers can be both relevant due to the dominantinterface oxide charges [6, 35], which could scatter carri-ers strongly leading to a strengthening of our mechanism.Noticeably, this mechanism is more effective in 2D thanin 3D Si, as the former retains ionized donors under mostrelevant experimental conditions [36].In Sec. II we develop our basic theory of 2D spinrelaxation, obtaining detailed results for the relaxationtime for different surface orientations and applied exter-nal stress in Sec. III. Section IV is devoted to a discussionof our results in the context of experimental implicationsand the existing 2D spin relaxation experiments. Weconclude in Sec. V with a summary and an outlook. Theintervalley scattering physics and its relevant symmetryanalysis is reviewed in the Appendix. II. THEORETICAL FORMULATION
Our spin relaxation mechanism arises directly from theimpurity SOC, in contrast with the structural SOC ef-fects that emerge from a combination of atomic SOC andbroken structural symmetry. The electron spins flip uponscattering. Moreover, the spin flip is governed not by thespin mixing in the conduction electron states but by theSOC of the scattering potential. Our approach to formu-late the scattering of electron states in the 2DEG sub-bands by an impurity potential is to take doubly-appliedeffective mass approximation (EMA) [37–39]. One of theEMA is conventionally applied with the envelope func-tions of subband states confined in a quantum well ornear the surface [6, 40] (as discussed in more detail be-low, we do not consider the opposite-valley coupling dueto interface in our leading-order theory). Assuming thescattering matrix element for conduction states in a 3Dbulk Si of volume V is U dv , s ; v , − s between valley v , spin s and valley v , spin − s , the EMA connects it to that ofthe 2DEG with an area S , U dv ,n ; v ,n ( z, s ) = ξ v ,n ( z ) ξ v ,n ( z ) S V U dv , s ; v , − s , (1)for a given impurity located at z along the width direc-tion of the 2DEG, where ξ v,n ( z ) is the envelope functionin valley v and quantized 2D subband n , and normalized (cid:82) dzξ v,n ξ v,m = δ n,m . In the following, we first elabo-rate the physics of the spin-flip matrix element U dv , s ; v , − s ,where another use of the EMA is crucial to relate a scat-tering problem with a donor-state problem. Then westudy in detail the different specific confinements and re-sulting subbands.The bulk spin-flip scattering is treated rigorously interms of the general symmetry of the impurity poten-tial [22]. Since the initial and final conduction statesare the eigenstates of the bulk Si (one-body) Hamilto-nian V , the scattering potential is the difference betweenthe substitutional impurity and the original Si atom, U = V imp − V Si , and breaks the O h space symmetry ofthe diamond lattice structure. Without going into the de-tails of U (including the screened Coulomb potential andshort-ranged central-cell correction), U obeys the tetra-hedral T d point group symmetry [41] and one can derivethe matrix element form ( U dv , s ; v , − s = (cid:104) ψ v , − s | U | ψ v , s (cid:105) )with the correct dependence on valleys and spin orienta-tion of the involved conduction states [22]. We summa-rize the relevant intervalley scattering in bulk Si and itssymmetry analysis in the Appendix. Between conduc-tion states at the valley centers, it turns out that spinflip survives only in intervalley f -process scattering. Itscounterparts in intravlley and intervalley g -process scat-tering are forbidden by the C rotation symmetry of the T d group and the time reversal symmetry, respectively.We denote the bare spin angular dependence of U dv , s ; v , − s without the quantitative prefactor as ˆ U v , s ; v , − s , and theexpression between + x and + y valleys for arbitrary spinorientation s = (sin θ cos φ, sin θ sin φ, cos θ ) ≡ ( s x , s y , s z )reads [22],ˆ U + x, s ;+ y, − s = i θe iφ + η (1 − i )2 √ θ − i sin θ e iφ ) ≡ is x − s y η (1 − i )4 √ (cid:18) s z − i ( s x + is y ) s z (cid:19) , (2)where the dimensionless constant η is the ratio betweenthe two symmetry-allowed terms (in particular, from the¯ F -symmetry states of the T d group; see Appendix fordetails). This leads to the anisotropic dependence of spinrelaxation on spin orientation.In order to determine the magnitude of the prefac-tor in U d , we make an important connection betweenit and the spin-split spectrum of the localized impuritystates, using the essence of EMA. By comparing thescattering problem and the localized eigenenergy prob-lem of the same impurity, one can realize that the po-tential is exactly the same for the two problems and the only difference between the localized state and theconduction state comes from the envelope function inthe former due to the Coulomb confinement. As a re-sult, the prefactor in U d can be related to the spinsplitting ∆ so of the bound impurity states such that U dv , s ; v , − s = ( πa B /V )∆ so ˆ U v , s ; v , − s , where a B is the im-purity Bohr radius and V the bulk volume, an EMA ef-fect not too different from that of Eq. (1) applied for the2DEG confinement. When the experimental spectrum isavailable for ∆ so , such as those in group V donors [42, 43],this method is most efficient and also likely more accu-rate than numerical calculations that may miss part ofthe microscopic contributions. The ratio constant η isestimated to be about 2 from spin relaxation data inhighly doped n-type Si [22]. For other types of substi-tutional impurities η is expected to have a value of theorder of unity. In principle, an estimate of η can be ob-tained by first principles calculations which are out ofscope for the current work. Again it is much preferableby empirically comparing with experiments since a pre-cise quantitative calculation of η is essentially impossibletheoretically, particularly in the context of spin relax-ation in the 2DEG. As discussed in the Introduction, ingeneral, the SOC scattering strength depends on the typeof impurities being considered, and would in general besmaller for neutral and negatively charged impurities.ˆ U + x, s ;+ y, − s in Eq. (2) describes the leading-order inwavevector spin-flip matrix element in one of all the 24paths of the f -process scattering among six Si conductionvalleys. In this work we group ˆ U v , s ; v , − s into 12 time-reversal (TR) related pairs, and then connect them toˆ U + x, s ;+ y, − s by specific spatial symmetry operations inthe T d group: | ˆ U v , s ; v , − s | TR = | ˆ U − v , s ; − v , − s | , (3) | ˆ U x, s ; − y, − s | C x = | ˆ U x, s (cid:48) =( s x , − s y , − s z ); y, − s (cid:48) | , | ˆ U y, s ; x, − s | σ x − y = | ˆ U x, s (cid:48) =( − s y , − s x , − s z ); y, − s (cid:48) | , | ˆ U − y, s ; x, − s | S = | ˆ U x, s (cid:48) =( s y , − s x ,s z ); y, − s (cid:48) | , | ˆ U x, s ; z, − s | σ y − z = | ˆ U x, s (cid:48) =( − s x , − s z , − s y ); y, − s (cid:48) | , | ˆ U x, s ; − z, − s | σ y + z = | ˆ U x, s (cid:48) =( − s x ,s z ,s y ); y, − s (cid:48) | , | ˆ U z, s ; x, − s | C = | ˆ U x, s (cid:48) =( s z ,s x ,s y ); y, − s (cid:48) | , | ˆ U − z, s ; x, − s | C = | ˆ U x, s (cid:48) =( − s z ,s x , − s y ); y, − s (cid:48) | . | ˆ U z, s ; y, − s | σ x − z = | ˆ U x, s (cid:48) =( − s z , − s y , − s x ); y, − s (cid:48) | , | ˆ U − z, s ; y, − s | σ x + z = | ˆ U x, s (cid:48) =( s z , − s y ,s x ); y, − s (cid:48) | , | ˆ U y, s ; z, − s | C = | ˆ U x, s (cid:48) =( s y ,s z ,s x ); y, − s (cid:48) | , | ˆ U y, s ; − z, − s | C = | ˆ U x, s (cid:48) =( s y , − s z , − s x ); y, − s (cid:48) | , where the vector subscript of the reflection operator ( σ )marks the normal direction of the reflection plane, and C and S denote the usual proper and improper rotationsrespectively with the given axes (the unspecified axis ofthe C rotation is along one of the cubic body diagonals).These individual U v , s ; v , − s expressions are important inevaluating spin relaxation in Si 2DEG, where not all val-leys are always equally occupied and can be subsequentlysummed together. More specifically, the anisotropy of ef-fective mass and strain may split the energy degeneracyof the six valleys in different ways, but always keep en-ergy the same for the two opposite valleys. This is truewithout including the small effects from SOC and short-wavelength perturbation beyond the EMA, which couldinduce splitting of the order of 1 meV or less [5, 6, 44–47].This splitting effect is negligible compared with typicalFermi levels and in the context of the leading-order cal-culation of the spin relaxation time. We therefore do notinclude such small interface coupling between oppositevalleys in the current work. All in all, we are always al-lowed to group the 24 f -process paths into three parts,(I) ± x ↔ ± y , (II) ± x ↔ ± z and (III) ± y ↔ ± z , andsum | U v , s ; v , − s | over each group which shares the sameelectron statistical distribution factor. Utilizing Eqs. (2)and (3), we have, (cid:88) ∈ I | ˆ U v , s ; v , − s | = 29 [1 − s z + 3 η (1 + s z )] ≡ S ( s z ) , (4) (cid:88) ∈ II | ˆ U v , s ; v , − s | = S ( s y ) , (5) (cid:88) ∈ III | ˆ U v , s ; v , − s | = S ( s x ) . (6)We will see that Eqs. (4)-(6) directly lead to the strongspin angular dependence of the spin relaxation in the(110) and (001)-oriented 2DEGs, as well as in the (111)2DEG under external stress in the next section.Next we address the specific confinements and subbandenvelope functions in Eq. (1). Before doing that, we com-bine Eq. (1) and U dv , s ; v , − s = ( πa B /V )∆ so ˆ U v , s ; v , − s togive U dv ,n ; v ,n ( z, s ) = ξ v ,n ( z ) ξ v ,n ( z ) S πa B ∆ so ˆ U v , s ; v , − s . (7)We stress that the EMA suits our problem especiallywell, even for the relatively narrow 2DEG: As we haveshown in Ref. [22], the relevant intervalley spin scatter-ing potential comes from the core region of the impu-rities, evidenced by the strong dependence of the spinrelaxation times on the donor species. The overall 2Dconfinement is much smoother than the impurity corepotential whose linear dimension is much less than a lat-tice constant, and Eq. (7) can be safely used for mostof the randomly or uniformly distributed impurities inthe 2DEG. The 2DEG system is essentially of 3D naturewith respect to the short-range scattering in the immedi-ate impurity core region since the 2D confinement lengthscale ( ∼
10 nm or larger) is much larger than the atomiccore size ( ∼ ξ v,n ( z ), but otherwise the scattering interaction and the conduction Bloch func-tions ψ v, s near the impurity core region are unchangedto this order of perturbation. Thus the spin-flip selec-tion rules are still dictated by the bulk symmetry, wellretained near the impurity core region. Under this levelof approximation, we also neglect any small change in∆ so and η , and in ˆ U v , s ; v , − s , in going from the 3D bulkto the 2DEG.We define an effective width d v ,n ; v ,n for the scat-tering between v , n and v , n states, in terms of theenvelope functions in Eq. (7),1 d v ,n ; v ,n ≡ (cid:90) dz | ξ v ,n ( z ) ξ v ,n ( z ) | , (8)which, together with the 2DEG area S , yields an effec-tive volume of the 2DEG Sd v ,n ; v ,n (taking the roleof V in 3D bulk) for a given subband transition. To bespecific, we choose two representative confinements forthe 2DEG. The first one is a square well, correspondingto the typical 2D heterostructure quantum well (such asSiGe/Si/SiGe). Focusing on the lowest few levels, weapproximate the well potential as an infinite barrier for0 < z < d (where d here is the physical well width) andobtain simple analytical solutions, ξ sq v,n ( z ) = (cid:114) d sin ( n + 1) πzd , (9)where n = 0 , , , ... denotes various 2D confined sub-bands. The corresponding energies at subband bottomsare E v,n = [ π (cid:126) ( n + 1)] m z,v d , (10)where the effective mass m z,v along the z direction de-pends on the valley v and the 2DEG plane orientation aswe will describe in detail below. Note that here E v,n ismeasured from the bottom of the v th valley, E v , in the3D bulk, as opposed to the lowest subband bottom. Dif-ferent valley bottoms may shift relatively to each otherupon various stress configurations (either deliberately ap-plied from outside or present because of intrinsic inter-face strain). Following these ξ v,n ( z ), we can obtain theeffective width parameter [Eq. (8)] for square wells as d sq n ; n = d δ n ,n / , (11)which is independent of the involved valleys.The second representative confinement we use for pro-ducing numerical results is a triangular well potential atthe interface V ( z ) = eF z for z > ∞ for z < F is the electric field including built-in potential gradient).It corresponds approximately to the inversion (accumu-lation) layer of hole(electron)-doped Si MOSFET, whenthe 2DEG density is smaller than the saturated chargedensity of depletion layer per area, N depl [6]. We takethe inversion layer as an example in Sec. III, while both From Eq. (8) max[ 𝐸 𝑣 ,0 𝑒𝐹 , 𝐸 𝑣 ,0 𝑒𝐹 ]
111 surface (a)
001 surface, 𝑥, 𝑦 ↔ 𝑧 (b)
001 surface, 𝑥 ↔ 𝑦 (c) F ( V/cm) 10.8
110 surface, 𝑥, 𝑦 ↔ 𝑧 (d) F ( V/cm) 1
110 surface, 𝑥 ↔ 𝑦 (e) F ( V/cm) 10.80.60.40.2 F ( V/cm) F ( V/cm)
FIG. 1: d v , v , defined in Eq. (8) (the blue curves) asa function of effective electrical field F for five representa-tive cases of the triangular shape wells: f -process scattering(a) near the [111] surface, (b) between the 4 and 2-valleygroups near the [001] surface, (c) within the 4-valley groupsnear the [001] surface, (d) between the 4 and 2-valley groupsnear the [110] surface, and (e) within the 4-valley groups nearthe [110] surface. In comparison, we also plot side by sidemax[ E v , , E v , ] /eF (the yellow curves). types of 2DEG layers are treated in Sec. IV with the vari-ational approach. The envelope function in this model isanalytically solved [48], ξ tr v,n ( z ) = α v,n Ai (cid:34)(cid:18) m z,v eF (cid:126) (cid:19) (cid:18) z − E v,n eF (cid:19)(cid:35) θ ( z ) , (12) E v,n ≈ (cid:18) (cid:126) m z,v (cid:19) / (cid:20) πeF (cid:18) n + 34 (cid:19)(cid:21) / , (13)where Ai denotes the Airy function, and α v,n is the nor-malization factor [ y = Ai( x ) satisfies the original Airyequation y (cid:48)(cid:48) − xy = 0]. E v,n are the asymptotic val-ues for large n , but fall within 1% of the exact valueeven for n = 0. The Airy function depicts the oscilla-tion of state envelopes within the classical turning point( z t = E v,n /eF ) and the decay beyond it. The step func-tion θ ( z ) [=1(0), for z > ( < )0] arises from the one-sidedinfinite barrier in the model. For the triangular well, d v ,n ; v ,n does not have a simple analytical form. InFig. 1, we plot d v ,n ; v ,n ( F ) in the typical range ofelectrical field F (10 − V/cm) numerically in thequantum limit, n = n = 0, for all representative sur-faces and valley configurations. We find that for all thesecases d v , v , can be well approximated by d tr v , v , ≈ max[ E v , eF , E v , eF ] . (14)The results in both Eqs. (11) and (14), derived fromthe general definition of effective width in Eq. (8), can bephysically interpreted as follows. First, the volume nor-malization of the initial and final states scales the scat-tering matrix element U d inversely with (cid:112) d v ,n d v ,n where d v,n is the effective spread of the given subbandstate in the z direction. Second, the relaxation ratescales with the number of impurities in the overlapping region of the two states, ∝ min[ d v ,n , d v ,n ]. Com-bining these two factors, the spin relaxation should beroughly proportional to 1/max[ d v ,n , d v ,n ] [which is ef-fectively Eq. (8)]. This is a generic prediction of ourtheory for impurity-induced 2DEG spin relaxation in Si,which could be directly tested experimentally. Finally, d v,n is basically d for the square well and around theclassical turning point E v,n /eF for the triangular wellwith slope eF .With the core factors U v , s ; v , − s and d v ,n ; v ,n elabo-rated, in the following section we present our calculatedspin relaxation results in all typical Si 2DEG orientationsand stress configurations. By standard time-dependentperturbation theory, one integrates out the periodic timefactors of the states resulting in the effective energy con-servation in the large time limit, i.e., the Fermi goldenrule [49, 50], a standard application for relaxation rates,1 τ ds ( s ) = 4 π (cid:126) (cid:28) (cid:88) v ,n (cid:90) d k π /S N i S (cid:90) dz | U dv ,n ; v ,n ( z, s ) | δ [ ε v ,n ( k ) − ε v ,n ( k )] (cid:29) k (15)where k and k are the 2D wavevec-tors for initial and final states, N i is theimpurity density per volume, (cid:104)O(cid:105) k ≡ (cid:80) v ,n (cid:82) d k O [ ∂ F /∂ε v ,n ( k )] (cid:14)(cid:80) v ,n (cid:82) d k [ ∂ F /∂ε v ,n ( k )]denotes the shortcut for the normalized integrationover k with F being the Fermi-Dirac distribution(see, e.g., Ref. [21], p. 73). In our calculation, weneglect the dependence of U dv ,n ; v ,n on the smallwavevector measured from its respective valley center( k ) [51] which only renders a higher-order relativeerror [ ∼ | k − k | / (2 π/a ) (cid:28) a being the Si latticeconstant], and thus U d depends only on the valleysand subbands of the involved states. Our leading-orderin wavevector theory establishes the first quantitativeanalysis for impurity-induced 2D spin relaxation beyondthose arising from the DP mechanism. III. NUMERICAL RESULTS FOR DIFFERENT2DEG ORIENTATIONS AND APPLIED STRESSA. Without external stress
In the limit of large well width, the number of occu-pied subbands for a given Fermi level is proportional tothe width d , and 1 /τ ds reduces to the 3D limit indepen-dent of d [22], which we have explicitly verified numer-ically. In this section, we give concrete quantitative re-sults for the opposite 2D limit of only one (“the quantumlimit”) or few lowest subbands being populated, in thelow-temperature limit of our interest. As we stressed, theEMA is well applied in this limit for our problem wherethe interaction occurs within the impurity core regions.For more details on the general justification of applyingEMA to tightly-confined quantum structures, the read-ers can refer to Ref. [40]. We also note that this the-ory treats on equal footing the “weak-field” and “strong-field” limits that arise from the study of structural SOCand oscillation of valley splitting [11, 12]. For a squarequantum well under strong electric field, the 2DEG maybe modeled in a triangular confinement (or some specificvariations) for our mechanism, as exemplified in Sec. II.We remark that since our spin-flip mechanism draws on f -process intervalley scattering, it is easier to be seen forthe (111) and (110)-oriented 2DEGs where non-oppositevalleys coexist in the ground states, than for the (001)one. For the latter case, multi-subband occupation is re-quired in order for our leading-order spin relaxation toplay a key role.The situations with potential confinement but no stressare studied first. τ ds as a function of 2DEG electrondensity N d for a given well potential V ( z ), as well as τ ds versus V ( z ) for a given N d , are computed. Depending on N d and the subband splitting (or the corresponding wellwidth), different subbands may be populated. Unlike the3D bulk case, in 2DEG the electron density and the Fermilevel are decoupled from the impurity density N i as theformer can be controlled by the gate voltage. The generalrelation between N d and the Fermi level ε F reads, N d = 12 π (cid:126) (cid:88) v (cid:88) n √ m ,v m ,v ( ε F − E v,n ) θ ( ε F − E v,n ) , (16)where m ,v and m ,v are the in-plane effective massesin the v th valley (the effective mass is anisotropic in Sidue to the ellipsoidal forms of the bulk conduction bandminima), and θ ( x ) = 0 or 1 for x < x > m z = 3 m t m l / ( m t + 2 m l ) is the same in every valley,so the six energy “ladders” of subbands remain degener-ate among different valleys (neglecting any small valleysplitting correction beyond the effective mass approxima-tion). The spin relaxation rate, Eq. (15), then becomes,1 τ (111) s ( s ) = π a B ∆ N i (cid:126) G (111) , (17)with the orientation-specific factor assuming the low-temperature limit, G (111) = (cid:80) n ,n η ) √ m m d n ,n θ ( ε F − E n ) θ ( ε F − E n )3 (cid:80) n θ ( ε F − E n ) , (18)where m = m t and m = ( m t + 2 m l ) /
3. As in the 3Dcase, τ (111) s ( s ) is isotropic in spin orientation.We quantify the spin relaxation time τ s for the twobasic well types introduced in Sec. II, (1) the infinitesquare well and (2) the triangular well. For a square wellwith width d , E n and d n ,n follow Eqs. (10) and (11),and Eq. (18) reduces to G (111)sq = 4(1+6 η ) √ m m d (cid:20) N (cid:18)(cid:114) ε F E (cid:19) + 12 (cid:21) θ ( ε F − E ) , (19) where N ( x ) returns the integer part of x . We plot τ s asa function of ε F and relate it to the corresponding N d inFig. 2 for three different representative well widths, (a)15 nm, (b) 30 nm, and (c) 45 nm. We also plot τ s as afunction of d for three fixed ε F = 10 ,
20 or 30 meV inFig. 2(d). Since one can simply scale τ s with ∆ − and N − i as shown in Eq. (17), we choose typical parame-ters ∆ so = 0 . N i = 10 cm − . The clear kinks in N d versus ε F and the jumps in τ s versus ε F or d reflect the onset of (de)populating moresubbands. τ s decreases as ε F or N d increases for a fixed d , since more subbands are available for states at theFermi level to be scattered into. As d increases towardsthe bulk limit, denser subbands gradually evolve towardsthe density of state for 3D bulk at a given ε F [see insetof Fig. 2(d)]. On the other hand, at small d , τ s decreasescontinuously with decreasing d within the window of afixed number of occupied subbands, n occ . This is a gen-eral trend dominated by the volume normalization of theinvolved state captured in Eq. (8). When n occ reduces byone, the number of available final states decreases and τ s increases again. (a) (c) (d) d=15 nm d=45 nm
10 meV 𝜀𝜀 𝐹𝐹
20 meV
30 meV (nm) (b) d=30 nm
10 8 6 4 . d (nm)
200 400 ( 𝝁𝝁 s) 𝜺𝜺 𝑭𝑭 (meV) 𝜺𝜺 𝑭𝑭 (meV) 𝜺𝜺 𝑭𝑭 (meV) FIG. 2: τ s in the (111) square well. We plot τ s as a functionof ε F and the corresponding N d for three representative wellwidths d , (a) 15 nm, (b) 30 nm, and (c) 45 nm, and τ s as afunction of d for three fixed ε F = 10 ,
20 or 30 meV in (d).Inset of (d) shows the large d behavior approaching the bulklimit. Note that the energy of the lowest subband bottomis E = π (cid:126) / m z d relative to the zero reference energy.∆ so = 0 . N i = 10 cm − are chosen here and forall the following figures. For the triangular well V ( z ) = eF z , with its slopecontrollable by the gate voltage in an inversion layer,the solution of ξ n ( z ) becomes the Airy functions givenin Eq. (12), and d tr n ,n for n = n = 0 can be esti-mated by Eq. (14). The general changes from the squarewell are (1) d n ; n in Eq. (17) becoming n , dependent,and (2) the different dependence of E n on n . However,we emphasize that the triangular model is more valid as N d /N depl decreases (and N d < N depl ) [6]. A realisticacceptor density we choose is N A = 10 cm − . As aresult, this usually corresponds to the situation whereonly the lowest subbands in the inversion layer are occu-pied. Therefore the numerical results in Fig. 3 is givenin this practical energy window. In this case, τ s becomesindependent of ε F or N d due to the constant 2D den-sity of states per subband, and we only need to showthe dependence of τ s on F . For higher N d , V ( z ) is notindependent of but largely determined by N d . F ( V/cm)7654 FIG. 3: τ s in the triangular well V ( z ) = eF z to the (111)surface for F ≤ V/cm. This is at the quantum limitwhere only the lowest subbands are occupied. In this limit, τ s is independent of ε F and N d . We choose the realisticacceptor density in the depletion as well as in the inversionlayer, N i = N A = 10 cm − . Next we study the (001) well orientation, where addi-tionally we have the relative shift between different sub-bands belonging to the 2-valley group ( ± z valleys in the3D limit) and 4-valley group ( ± x and ± y valleys in the3D limit). The subband edges in these two groups aredetermined by the different m z ’s, m z = m l ( m t ) for the2(4)-valley group. This symmetry breaking between the 6otherwise equivalent ladders of subbands results in spin-orientation dependence, absent in the (111) well case. Inthe quantum limit, only the 2-valley group is occupiedand the spin relaxation due to impurities vanishes in theleading order. The general spin relaxation rate followsEq. (17) with G (111) replaced by G (001) ( s ) = (cid:88) n ,n θ ( ε F − E x,n ) (cid:26) S ( s z ) √ m t m l θ ( ε F − E x,n ) d x,n ; y,n + [ (1 + 6 η ) − S ( s z )] m t θ ( ε F − E z,n ) d x,n ; z,n (cid:27)(cid:30)(cid:88) n (cid:20)(cid:114) m t m l θ ( ε F − E z,n ) + 2 θ ( ε F − E x,n ) (cid:21) , (20)after utilizing the spin-orientation form factors inEqs. (4)-(6). We have used the anisotropic in-plane effec-tive masses: m = m = m t for the 2-valley group, and m = m t , m = m l for the 4-valley group. For square well with width d , we have G (001)sq ( s ) = θ ( ε F − E x, ) d (cid:20) N (cid:18)(cid:114) ε F E x, (cid:19) + 12 (cid:21)(cid:26) S ( s z ) √ m t m l N (cid:18)(cid:114) ε F E x, (cid:19) +[ 49 (1 + 6 η ) − S ( s z )] m t N (cid:18)(cid:114) ε F E z, (cid:19)(cid:27)(cid:30)(cid:20)(cid:114) m t m l N (cid:18)(cid:114) ε F E z, (cid:19) + 2 N (cid:18)(cid:114) ε F E x, (cid:19)(cid:21) . (21) 𝜃𝜃 𝑧𝑧 = 𝜋𝜋 /6 𝜋𝜋 /3 𝜋𝜋 /2 𝜀𝜀 𝐹𝐹 = 10 meV (a) d (nm)15 25 35 45 𝜀𝜀 𝐹𝐹 = 20 meV (b) d (nm) 𝜀𝜀 𝐹𝐹 = 30 meV (c) d (nm) 𝑑𝑑 = 15 nm (d) 𝜺𝜺 𝑭𝑭 (meV) 𝑑𝑑 = 30 nm (e) . 𝑭𝑭 (meV) 𝑑𝑑 = 45 nm (f) 𝑭𝑭 (meV) FIG. 4: τ s ( s ) in the (001) square well, anisotropic in spinorientation s . τ s depends on s ’ polar angle, θ z , with respectiveto the well normal, z . In (a)-(c), we vary the well width d from 15 to 45 nm for three Fermi levels ε F =(a) 10, (b) 20and (c) 30 meV. In (d)-(f), we plot τ s ( s ) as a function of ε F , E x, < ε F <
30 meV (and also a function of N d by relating ε F with N d ), for three different well widths, d = (d) 15, (e)30 and (f) 45 nm. In each subplot we exemplify four spinpolar angles θ z = 0 , π/ , π/ π/ Plots of τ s ( s ) with G (001)sq ( s ) in Eq. (21) as functionsof d and ε F (and the corresponding N d ) are given inFig. 4. An apparent new feature is the generally smallersteps in comparison with the (111) well results, result-ing from the consecutive fillings of the 2-valley subbandswhich have smaller interband splitting. As mentioned,the important consequence of the inequivalency betweenthe 2-valley and 4-valley groups is the spin-orientationdependence of the spin lifetime. The anisotropy is thestrongest when the occupied states in the two groupsdiffer the most, which happens right before one more4-valley subband begins to be filled. We need to notethat merely n occ,z > n occ,x is not enough to guaran-tee spin anisotropy, but it has to be n occ,z /n occ,x > (cid:112) m ,x m ,x /m ,z m ,z = (cid:112) m l /m t based on Eq. (21).This is most appreciable preceding the filing of the sec-ond subband in the 4-valley group, n occ,z /n occ,x = 4 : 1.In the large ε F (or large d ) limit, on the other hand, n occ,z /n occ,x → (cid:112) m z,z /m z,x = (cid:112) m l /m t ≈ .
27. Wesee that this ratio exactly cancels out the effective massdifference in the 2D ( x - y ) plane, owing to the fact that m m m z ≡ m l m t is orientation-independent for a givenellipsoid. 𝜃𝜃 𝑧𝑧 = 𝜋𝜋 /6 𝜋𝜋 /3 𝜋𝜋 /2 𝐸𝐸 𝑥𝑥 , < 𝜀𝜀 𝐹𝐹 < 𝐸𝐸 𝑧𝑧 , (a) F ( V/cm) 10.80.60.40.2 𝐸𝐸 𝑧𝑧 , < 𝜀𝜀 𝐹𝐹 < 𝐸𝐸 𝑧𝑧 , (b) F ( V/cm) 10.80.60.40.281216 𝐹𝐹 = 5 × 10 V/cm (d) 𝜺𝜺 𝑭𝑭 (meV) . . . = 10 V/cm (c) . . 𝜺𝜺 𝑭𝑭 (meV) . . FIG. 5: τ s ( s ) in the triangular well V ( z ) = eF z to the (001)surface. In (a) and (b), τ s ( s ) is plotted as a function of electricfield F < V/cm for the Fermi level ε F (a) just above theonset of f -process scattering ( E x, < ε F < E z, ), and (b) inthe next energy window ( E z, < ε F < E z, ), where τ s ( s ) is in-dependent of ε F . In (c) and (d), τ s ( s ) is plotted as a functionof ε F and N d for a range E x, < ε F < E z, , at electric field F = (c) 10 and (d) 5 × V/cm. At F = 10 V/cm, theonset of f -process scattering already requires N d ≈ × cm − , a comparable value to the typical depletion layer im-purity density N depl = (cid:112) E G κN A /e [6] where E G and κ arethe Si band gap and permittivity ( N depl ∼ × cm − at N A = 10 cm − ). For the triangular well, Eqs. (8), (12) and (13) are sub-stituted into Eq. (20), and τ s ( s ) is shown in Fig. 5. Afterthe onset of f -process scattering ( ε F > E x, ) follows thesecond subband of the 2-valley group, as shown clearlyin Fig. 5(c) and (d). ( E z, − E x, ) /E x, is a small fixedratio for any electric field F according to Eq. (13). Sincewe work in the regime where the triangular well modelis valid and multiple subbands are occupied, ε F and F should not be too large. As a result, we focus on twoenergy intervals, E x, < ε F < E z, and E z, < ε F < E z, below F = 10 V/cm (in practice, our mechanism workswell under higher electrical field, as long as the Fermilevel can reach the x and y valleys and the triangle wellapproximation is relaxed).For scattering involving subband n = 1, we obtain d z, x, ≈ . E z, /eF using Eqs. (8) and (12). An inter-esting behavior is the large anisotropy ( ∼ τ s ( s )right at the onset of the f -process scattering [Fig. 5(a),(c)and (d)], which drops ( ∼ E x, < ε F < E z, , the number of occupied subbands ineach valley is one for both 2-valley and 4-valley groups.The large anisotropy here is the sole consequence of ef-fective mass anisotropy in the 2D plane and is opposite in sign to that in the square well before filling E x, .The change of anisotropy may be sharply tuned by thegate voltage in Si inversion layer. Since E z, − E x, and ε F − E x, depend on the electric field F , by just tuning F the chemical potential (i.e., Fermi energy) can cross E z, and therefore induce a switch between Fig. 5(a)and (b). An even more important application of our re-sults may be the sharply gate-voltage modulated spin life-time in Si inversion layer following a similar reasoning:the crossover of chemical potential to E x, , which is thethreshold of finite leading-order spin relaxation, could beachieved solely by the top gate voltage. This is the on-chip real-time electrical switch for a substantial change ofthe spin lifetime. We emphasize that both gate-voltagemodulations envisioned above should be very robust ingeneral Si inversion layers, not relying on the triangularwell approximation we used in producing Fig. 5.Last, we study the case of the (110) well. The dif-ference of the (110) well from the (001) one lies in theeffective masses, which result in a quantitative differencein the valley splitting, the subband splitting and the den-sity of states, all playing roles in determining τ s . For the2-valley group, m z = m t , m = m t and m = m l ; for the4-valley group, m z = 2 m t m l / ( m t + m l ), m = m t and m = ( m t + m l ) /
2. The most important distinction isin the quantum limit where the lowest subbands in the(110) case are from the 4-valley group, rather than fromthe 2-valley group in the (001) case. Consequently, thespin lifetime due to the leading-order spin relaxation isfinite even for the lowest electron density in the (110)well. 𝜃𝜃 [ ] = 𝜋𝜋 /6 𝜋𝜋 /3 𝜋𝜋 /2 𝜀𝜀 𝐹𝐹 = 10 meV (a) d (nm) 15 25 35 45 𝜀𝜀 𝐹𝐹 = 20 meV (b) d (nm) 15 25 35 45 𝜀𝜀 𝐹𝐹 = 30 meV (c) d (nm) 15 25 35 45 𝑑𝑑 = 15 nm (d) 𝜺𝜺 𝑭𝑭 (meV) 15 10 5 . 𝑑𝑑 = 30 nm (e)
30 20 10 . 𝜺𝜺 𝑭𝑭 (meV) 𝑑𝑑 = 45 nm (f) 𝜺𝜺 𝑭𝑭 (meV) 50 30 10 FIG. 6: τ s ( s ) in the (110) square well, anisotropic in spinorientation s . τ s depends on s ’ polar angle, θ [001] , with re-spective to the [001] crystallographic direction. In (a)-(c), wevary the well width d from 15 to 45 nm for three Fermi levels ε F = (a) 10, (b) 20 and (c) 30 meV. In (d)-(f), we plot τ s ( s )as a function of ε F <
30 meV and N d , for three different wellwidths, d = (d) 15, (e) 30 and (f) 45 nm. We show the key results for the (110) square well inFig. 6 and the triangular well in Fig. 7. We use similarparameters (Fermi level ε F , well width d , electric field F ) as those in the (001) case, so that we can focus onthe differences between (110) and (001) wells. First, τ s is finite in the (110) 2DEG even for the lowest ε F inFig. 6 (d)-(f) and Fig. 7 (c) and (d), as mentioned above.Due to the smaller number of available f -process paths(two as opposed four for each state), though, τ s is largerin the (110) case than the longest finite τ s in the (001)case. This is a clearly verifiable sharp prediction of ourtheory. Note that here the plane normal ( z ) is alongthe crystallographic direction [110] (not [001]). To avoidambiguity, we use scripts 001 or 100 rather than z or x to denote directions.The most significant feature in this quantum limit for(110) wells is the nearly 50% variation of τ s ( s ) on spinorientation (more specifically, on s ’ projection along [001]crystallographic direction). This is the extreme case ofonly one type of f -process scattering [Eq. (4)] with zeroweight from the other two. This feature is clearly seenin the left side of Fig. 6(d)-(f), Fig. 7 (c) and (d), andthe entire range in Fig. 7(a). Therefore, the idea of gate-tuned anisotropic τ s ( s ), discussed in the context of (001)wells, is even more prominent in (110) wells. Note thatthis dependence on the polar angle around [001] direc-tion, θ [001] , is opposite in sign to that of the (001) squarewell when the anisotropy is the strongest and the same asthat in the (001) triangular well. The anisotropy and theorientation dependence of 2D Si spin relaxation arising inthe impurity-induced spin-flip is an important predictionof our theory. [001] = 𝜋/6 𝜋/3𝜋/2 (a) 𝐸 < 𝜀 𝐹 < 𝐸 F ( V/cm) 8642 (b) 𝐸 < 𝜀 𝐹 < 𝐸 F ( V/cm)0.80.6 (c)
𝐹 = 10 V/cm 𝜺 𝑭 (meV) (d) 𝐹 = 5 × 10 V/cm 𝟎.𝟖
10 8 9 10 𝜺 𝑭 (meV) FIG. 7: τ s ( s ) in the triangular well V ( z ) = eF z to the (110)surface. In (a) and (b), τ s is plotted as a function of theelectric field F for the Fermi level ε F (a) in the lowest energyinterval ( E , < ε F < E , ) with F < V/cm , and(b) in the next energy window ( E , < ε F < E , ) with F < V/cm , where τ s ( s ) is independent of ε F . We workin a smaller field in (b) for the same reason in Fig. 5 (a) and(b). In (c) and (d), τ s is plotted as a function of ε F and N d for a range ε F < E , , at electric field F = (c) 10 and (d)5 × V/cm.
B. With external stress
When external stress is applied, different ladders ofsubbands may undergo relative shift with each other [52].On the other hand, no leading-order effect comes from theslight variation of interband splitting within each ladder,as the effective masses are fixed under stress to the lead-ing order [52]. We study τ s individually for all three wellorientations under uniaxial stress as described below. (111) well . Out-of-plane [111] stress does not in-duce additional symmetry breaking or shift subband lad-ders, while the in-plane stress in [1¯10] (¯1 ≡ −
1) or[1 − √ , √ , −
2] direction does. These latter twostress directions are in the plane of the 2D well and re-alizable experimentally [24]. The [1¯10] (or [11¯2]) stressresults in a relative shift ∆ V = E − E between the2-valley and 4-valley groups, and the [1 − √ , √ , − V = E − E = E − E .Under the [1¯10] stress, the form factor G changes fromEq. (18) for the unstrained (111) well to, G (111)1¯10 ( s ) = (cid:114) m t ( m t + 2 m l )3 (22) (cid:88) n ,n θ ( ε F − E ,n ) d n ; n (cid:26) S ( s z ) θ ( ε F − E ,n )+ (cid:20)
49 (1 + 6 η ) −S ( s z ) (cid:21) θ ( ε F − E ,n − ∆ V ) (cid:27)(cid:30)(cid:88) n [ θ ( ε F − E ,n − ∆ V )+2 θ ( ε F − E ,n )] . Focusing on the near quantum limit with only n = 0subbands occupied, d , = 2 d/ π (cid:126) ) / / [4(2 m eF ) / ] for the triangular well,by Eqs. (11) and (14) respectively. Equation (22) hasonly a few discrete outcomes for a given well width orelectric field: (1) when ε F > { E , , E , + ∆ V } , lines2-4 of Eq. (22) reduce to 4(1+6 η ) / d and τ s recoversthe no-strain result (Figs. 2 and 3); (2) when E , <ε F < ∆ V + E , , the same factor decreases to [1 − s z +3 η (1 + s z )] / d with a strong s z dependence; (3) when∆ V + E , < ε F < E , , 1 /τ s = 0.Under the [1 − √ , √ , − ≡ γ ) stress, threegroups of f -process scattering vary independently. Uti-lizing Eqs. (4)-(6), the form factor G (111) γ reads, G (111) γ ( s ) = (cid:114) m t ( m t + 2 m l )3 (cid:80) n ,n ,i S ( s i ) d n n θ ( ε F − E i +1 ,n ) θ ( ε F − E i +2 ,n ) (cid:80) n,i θ ( ε F − E i,n ) . (23)where i denotes the 3 cyclic directions 100 ,
010 and 001, E ,n = E ,n + ∆ V and E ,n = E ,n − ∆ V . Al-though three f -process groups depend on spin projec-tion along different directions, shown in Eqs. (4)-(6),0 τ s ( s ) can be associated with a fixed projection direc-tion in each energy window of ε F , thanks to the con-stant density of state per subband. Focusing on the n = 0 limit, the second line in Eq. (23) has severalpossible outcomes: (1) 4(1 + 6 η ) / d , when ε F > { E , , E , , E , } , recovering the no-strain result;(2) S ( s ) / d , when { E , , E , } < ε F < E , ;(3) S ( s ) / d , when { E , , E , } < ε F < E , ;and finally (4) 0, when E , < ε F < { E , , E , } or E , < ε F < { E , , E , } . (001) well . Out-of-plane [001] stress keeps the 2-valleyand 4-valley degeneracy and tunes the energy distance E z − E x between them. In-plane [100](or [010]) stressbreaks the 4-valley degeneracy into two groups and tunes E x − E y while keeping the splitting E y ( x ) − E z > E z − E x to be tunable.In the n = 0 limit, d v , v , = 2 d/ π (cid:126) ) / / [4(2 min[ m ,v , m ,v ] eF ) / ] forthe triangular well. In this limit we have the additionalpossibility that E x, < ε F < E z, which results in strong s -anisotropic spin lifetime.Under the [100] (or [010]) stress, one may have f -process scattering available between n = 0 subbands, bypushing x ( y ) valleys lower towards z valleys and y ( x )valleys further away. This may require a relatively largecompressive stress for narrow wells. In this situation, theform factor goes to G (001)100 ( s ) = S ( s y ) m t / (1 + (cid:112) m t /m l )and s y ↔ s x for G (001)010 ( s ). (110) well . The unique feature for (110) well understress is that the 2-valley and 4-valley groups remainrespectively degenerate, for the out-of-plane [110] stressand all the in-plane stress directions including [001] and[¯110]. Therefore, the spin relaxation rate expression fol-low the unstrained one, only with the energy distancebetween E z and E x tunable by stress.The stress dependence of 2D spin relaxation, arisingentirely from the valley-dependent subband structure ofthe 2DEG, is a characteristic feature of the Yafet impu-rity process being considered in our current work, whichis completely absent in the Rashba/Dresselhaus-basedDP relaxation mechanism. IV. EXPERIMENTAL IMPLICATIONS
In this section, we discuss the potential experimentalimplications of the impurity-driven spin relaxation mech-anism. Variational calculations for the 2DEG subbandstructure are carried out which are valid for a broaderrange of MOSFET parameters, including both inversionand accumulation layers, and use variables that are con-venient to compare with experiments. Possible experi-mental proposals are discussed to differentiate major con-tributions to the spin relaxation in Si 2DEG. In particu-lar, the few existing 2D Si spin relaxation measurements available in the literature (see our discussion below) haveall been interpreted using the DP relaxation mechanismalthough the quantitative agreement between theory andexperiment is in general not satisfactory.Far fewer spin relaxation measurements have beenmade on the Si 2DEG than on the bulk Si. In n-typebulk Si, with the inversion symmetric lattice structure,Elliott-Yafet (EY) spin relaxation is the dominant mech-anism for conduction electrons. It is established that thescattering is caused mainly by the electron-phonon inter-action at elevated temperatures [21, 53–57], or by variousprocesses involving impurities under high impurity den-sity and low temperatures (see, e.g., the review in [1] andFig. 2 in [57]). Despite the weak SOC in Si, spin lifetime τ s is only of the order of 10 ns at room temperature, and0.1-100 ns at low temperatures and high donor concen-trations depending on the specific donor type [22, 25–29, 58]. In comparison, in 2DEG several additional fea-tures emerge with respect to spin relaxation. Aside fromthe tunability of spin lifetime and anisotropy showed inSec. III, the DP spin relaxation becomes relevant, causedby the inversion-breaking structure and interfaces andthe associated Rashba/Dresselhaus field. A third featureis the large amount of interface disorder, especially inMOSFETs, which may produce valley-spin-flip scatter-ing. As the 2DEG quantum limit is studied typicallyat low temperatures, the spin relaxation is determinedby scattering with impurities, where EY and DP mecha-nisms happen respectively during and between the scat-tering events. In the following, we quantify the spin life-time in the quantum limit based on our studied Yafetmechanism, taking into account the experimentally mea-sured parameters and the uncertainty in the interfacedisorder.To make comparison with experiments, we take the ac-ceptor concentration N A and 2D electron density N d astwo independent variables. We further take the effectiveimpurity density N i in the 2DEG as a separate variableas it can be significantly different from that in the bulk.To apply to a wide range of parameter values and toboth inversion as well as accumulation layers, we relaxthe triangular approximation and adopt the variationalsubband wavefunction [48, 59, 60], ξ ( z ) = (cid:114) b z exp( − bz . (24)with a variational variable b . After the numerical en-ergy minimization to find b taking into account both thedepletion and 2DEG layer potentials (the band bendingtakes 1.1 eV and 45 meV for inversion and accumulationrespectively), simple expressions for the effective width d v , ,v , [see Eq. (8)] can be obtained in terms of b : d v , ,v , = ( b + b ) b b , (25)where b and b are respectively from valleys v and v which may have different m z masses. Substituting the1 𝑵 ( 𝐜𝐦 −𝟐 ) 𝑵 ( 𝐜𝐦 −𝟐 ) 𝑵 ( 𝐜𝐦 −𝟐 ) 𝑵 ( 𝐜𝐦 −𝟐 ) 𝑵 ( 𝐜𝐦 −𝟐 ) 𝑵 ( 𝐜𝐦 −𝟐 )
105 302515 (b) (a) (c) (d) (e) (f)(111) (111)(001) (001)(110) (110)Inversion Accumulation 𝐜𝐦 −𝟑 𝑁 𝐴 [001] 𝜋/6𝜋/3𝜋/2 𝐜𝐦 −𝟑 𝑁 𝐴 FIG. 8: Normalized spin relaxation time independent of im-purity density, ˜ τ s = ( N i / cm − ) τ s , for an inversion [(a),(c) and (e)] or accumulation [(b), (d) and (f)] layer in theSi MOSFET setup. All three 2DEG plane orientations areshown: (111) in (a) and (b), (001) in (c) and (d) and (110) in(e) and (f). Only the lowest subbands are occupied [for the(001) case, the lowest subbands in the 4 higher valleys arealso occupied]. They are plotted as functions of 2D electrondensity, N d , and for acceptor density (related to potentialfrom the depletion layer), N A = 10 , , for the in-version layer and 10 , , for the accumulation layercase. As previous calculations, we also give spin orientationdependence for the (001) and (110) cases. obtained effective widths into Eqs. (4), (17), (18) and(20), one can obtain τ s for three 2DEG orientations,(111), (001) and (110), for any arbitrary spin orienta-tion, as a function of N A , N d and N i . We collectall useful information in Fig. 8. For visual clarity, weleave out the reciprocal linear dependence on N i and plot˜ τ s = N i cm − τ s . For the inversion (accumulation) layercase, we choose the majority (minority) acceptor den-sity N A as 10 , and 10 cm − (10 , and 10 cm − ), and 10 < N d < cm − , covering typicalexperimental choices. As expected, ˜ τ s decreases slightlywith increasing N d due to the steeper confinement. Sim-ilar trends occur with N A . More importantly, of course,the absolute time τ s ∝ ˜ τ s /N i decreases much faster with N i . For clean interfaces, N i ≈ N A ( N D ) for the inver-sion (accumulation) layer. However, for some highly dis-ordered Si/SiO interfaces, including oxide charges onthe SiO side, N i may be much larger than the major-ity dopant density. In the high N d limit, the potentialconfinement is dominated by N d over N depl ∝ √ N A [6],and as a result ˜ τ s converges for different N A ’s.Finally, we should note that the SOC parameter ∆ so used here is 0 . .
03 or 0 . so and η [see Eq. (2)] need to bestudied separately. This can be an important study inthe future, and we will come back to this issue at theend. In fact, the effective SOC parameters of the inter-face impurities are an important unknown in the theory,which can be adjusted to get agreement between our the-ory and all existing experimental data. We refrain fromdoing so, however, emphasizing that if the measured 2Dspin relaxation time shows a positive correlation with thequality of the interface (i.e. improving interface qualityleads to longer spin relaxation time), then it is likely thatthe impurity induced Yafet mechanism discussed in thispaper is playing a dominant role in contrast to the DPmechanism which mostly leads to a lower spin relaxationtime with higher mobility.Now we briefly discuss some available experimentallymeasured T ( ≡ τ s in our notation) in Si 2DEG. We stressthat these samples are not particularly highly doped andtherefore our mechanism is not expected to be dominantunless extrinsic disorder associated with interface impuri-ties are playing a crucial role. Reference [30] measured T in Si/SiGe (001) quantum wells with relative high qual-ity interface and mobility µ . Their device I with µ = 9m /Vs, 2D electron density N d = 3 × cm − andwell thickness d = 20 nm, yields a T = 2 . µ s. Theirdevice II with µ = 19 m /Vs, N d = 1 . × cm − and d = 15 nm, yields a T = 0 . µ s. These N d and d combinations indicate that only the ground subbands inthe ± z valley are likely occupied.We show in the following that our mechanism cannotquantitatively account for the measured T , even if weassume the lowest subbands in the ± x and ± y valleysare occupied. Using our calculation that leads to Fig. 4,one needs a 3D impurity density N i ≈ . × cm − oran effective 2D impurity density n i ≈ N i d ≈ . × cm − for device I, and N i ≈ × cm − or n i ≈ N i d ≈ . × cm − for device II. The precise n i de-pends weakly on the detailed impurity distribution inthe 2DEG. To estimate the experimental impurity den-sity residing in the quantum well, we use the theoreticalresult of Ref. [61] which relates mobility with the chargedimpurity. From Fig. 1 of that paper, for the (001) wellorientation with two ground valleys, one needs n i ≈ cm − for device I and n i ≈ × cm − for device II.These n i ’s make our spin relaxation mechanism too weakto yield the measured T time. The spin anisotropy in ourcalculation has the same sign as in the measurement butnot as large in magnitude [ T ( θ = π/ /T ( θ = 0) = 1 . T values within an order of magnitude of the measured T values. Given the uncertainties associated with the im-purity SOC parameters, the possibility that the Yafetmechanism is perhaps playing a (minor) role in the ex-periment cannot be ruled out although it does appearthat the main spin relaxation mechanism in these high2mobility Si/SiGe quantum wells is likely to be the DPmechanism.Reference [62] measured T time for 2DEG in a Si/SiO (001) accumulation layer doped with 10 P donors. Atthe gate voltage of 2 V, T = 0 . µ s while µ = 1 m /Vsand N d = 4 × cm − [63]. Once again we checkthe effect of our mechanism by assuming for a momentthat the ± x and ± y valleys are reached. From Fig. 8,one needs about N i ≈ × cm − or n i ≈ × cm − . Note the unknown N A value affects the result onlyslightly (a factor less than 2). From the measured mo-bility at 5 K, we can deduce n i ≈ cm − [61]. As aresult, the impurity density is again too small to inducethe measured T time by our mechanism, even if the finitemobility is entirely caused by impurities in the 2DEG re-gion and ∆ so = 0 . T which is within a factor of 4 of the measured value,indicating that for spin relaxation in disordered Si MOS-FETs, perhaps our impurity-driven mechanism is playinga more important quantitative role. This is not unex-pected since Si MOSFETs typically have larger impuritydensities than Si/SiGe quantum wells, leading to possi-bly stronger spin relaxation due to the Yafet mechanism.Note that the DP mechanism does not find agreementwith the experimental data either, which can be verifiedby the calculation in Ref. [18] in combination with theexperimental parameters. It is possible that in Si MOS-FETs both DP and Yafet mechanisms are operationalin producing the observed low value of T in the exper-iment. Obviously, more experimental measurements areessential in understanding this important puzzle.We propose several experimental ways to properly in-vestigate the nature of spin relaxation in Si 2DEG. Tobegin with, (111) and (110) orientations are better suitedfor our intervalley spin-flip mechanism to have importantcontribution, as we have mentioned before. Moreover, alowering of the x and y valleys in the (001) 2DEG mayalso show a sudden jump of spin relaxation rate whichserves as a turn-on signal of our mechanism. This valleytuning can be achieved by external stress or gating, asemphasized in Sec. III. A similarly sudden change in τ s anisotropy can also occur for the (110) 2DEG due to ourmechanism.Apart from the 2DEG plane orientation, a numberof aspects are important in the experimental verifica-tion of our proposed spin relaxation mechanism. First,it is crucial (and we urge future experiments) to mea-sure a series of samples with different mobilities ( µ ) atsame carrier densities (and all other parameters). The(anti)correlation of τ s with µ is a characteristic signa-ture for Yafet (DP) spin relaxation mechanism [1]. Thecrossover occurs at modest doping levels, as the spin life-time from the DP mechanism rises rapidly past tens of µ salready around mobility 5 × cm / Vs [18]. In partic-ular, our mechanism should become dominant when the2DEG region is heavily doped.Second, there is an positive correlation between con-duction electron density and spin relaxation in the DP mechanism. The Rashba or generalized Dresselhaus fieldscales linearly with the wavevector ( k ) measured fromvalley bottom yet our spin flip matrix elements dependlittle on k . Third, for the MOSFET setup, it is usefulto measure τ s separately for both the bulk Si and the2DEG to deduce the contribution of interface disorder tothe 2DEG spin relaxation.It is also possible to deduce the distribution of 2DEGimpurities from the gate voltage dependence of τ s : fromEqs. (15) and (7), τ s ∝ d for a uniformly distributed N i ( z ) while τ s ∝ d if all impurities are concentratedat the interface [ N i ( z ) ∝ δ ( z )]. For the Si/SiGe setup,making two-sided symmetric confinement may separateout the contribution from the DP spin relaxation, as thechange of interfacial symmetry property greatly affectsthe DP mechanism through the envelope functions butleaves the Yafet one the same. In addition, as our cal-culated spin orientation dependence [rooted in Eqs. (4)-(6)] is distinct from that of the DP mechanism due tothe Rashba or Dresselhaus field, τ s anisotropy measure-ment can also help to disentangle the two contributions(for the large magnetic field limit, we note that, the DPmechanism is partially suppressed similar to the bulk case[1, 17, 64]).Parenthetically, while this work does not focus on theDP mechanism, we point out the existing studies con-cerning its various contributions [65]. Different viewshave emerged to account for the same experimental mea-surement in Si/SiGe quantum well [16], being it domi-nated by the Rashba field [16–18] or the Dresselhaus one[11, 12]. They lead to different SOC anisotropy but sim-ilar overall spin relaxation rate as both SOC fields scalelinearly in wavevector in the 2DEGs. DP spin relaxationand its anisotropy has also been studied in Si/SiGe quan-tum dots [66]. Up to the present, the relative magnitudeof the Rashba and Dresselhuas-like SOC has yet to beverified experimentally [67].Finally, our mechanism relies on the short-range in-teraction with the impurity core and directly measuresthe SOC strength of the impurity atoms. The spin re-laxation rate scales quadratically with ∆ so [see Eqs. (7)and (15)], and it increases significantly by switching fromlow atomic-number to high atomic-number dopants forthe same density. Therefore, different types of impuritiesthat lead to similar mobility may yield very different τ s times according to their SOC strengths, a unique signa-ture of this spin relaxation process.We suggest future spin relaxation measurements in 2DSi systems as a systematic function of mobility, carrierdensity, impurity type, surface and spin orientation, andapplied stress in order to develop a complete understand-ing of the mechanisms controlling spin relaxation of freecarriers near Si surfaces. The few existing measurementssimply do not have enough information for a definitiveconclusion.Last, in order to establish the relative strength of ourspin relaxation rate in comparison to the momentum re-laxation rate, which determines the device charge mobil-3ity, we calculate their relative ratio ( ν ) for a few repre-sentative cases where impurities are the dominant sourceof scattering (i.e. at low temperatures where phonons areunimportant). We take a simplified uniform distributionof the highly doped 2DEG. Define ν = τ dm /τ ds , where τ ds follows from our Eq. (15) and the momentum relax-ation time τ dm takes the form appropriate for mobilitycalculations in 2D transport studies [36, 61].The momentum scattering matrix elements are gov-erned by the well-known screened Coulomb interaction inthe intravalley scattering, as appropriate for scattering bythe random charged impurities. For this interaction, theimpurity distribution profile can be approximated as a δ function normal to the 2DEG plane. Under the 2D RPAscreening, the momentum relaxation rate in the quantumlimit is given as [36, 61],1 τ dm = 4 πe m eff n i (cid:126) κ k F (cid:90) dxx ( x + q TF / k F ) √ − x , (26)where m eff is the conductivity effective mass differentfor each specific 2DEG orientation [6, 61], n i is the 2Dimpurity density, permittivity κ is the Si permittivity, k F is the 2D Fermi wave number, 2D Thomas-Fermi wavenumber q TF = m eff e g/ (cid:126) κ , and g = g v g s is the numberof populated valleys including the spin degree of freedom( v and s denoting valley and spin).The dependence on Fermi level (similarly, on k F or N d ) is very slow for both τ ds and τ dm within a givenFermi energy window between 2D subbands [36]. Thiscan be clearly seen for τ ds over many order of magni-tudes of N d from Fig. 8. For τ dm , we plot its explicitdependence on k F in Fig. 9 for (111) and (110) 2DEGorientations for which both leading-order momentum andspin relaxation rates are nonvanishing in the quantumlimit. They are both nearly constant over the large re-gion k F ≤ . A − , i.e., about 10% of the length of theBrillouin zone.This near independence of Fermi level allows us to ob-tain simple estimation for ν for each specific 2DEG orien-tation. To get the leading-order estimate, it is sufficientto substitute τ (111) s and τ (110) s with the square well resultsthrough Eqs. (17), (19) and a variation of (21). We havethe following results, ν = 400 a B ∆ (cid:126) d (cid:114) m t ( m t +2 m l )3 (cid:14)(cid:18) m t + 3 m t +2 m l (cid:19) , (27) ν = 128 a B ∆ (cid:126) d √ m t m l (cid:14) (cid:18) m t + 2 m t + m l (cid:19) , (28)for (111) and (110) 2DEGs respectively. With re-spect to typical ∆ so and d parameters, ν =1 . × − ( ∆ so . ) ( d ) and ν = 5 . × − ( ∆ so . ) ( d ) . Obviously, this ratio dependsquadratically on the impurity SOC constant ∆ so . Forthe expected typical values of the impurity SOC ∆ so inSi and the 2DEG width d , this ratio ν varies between10 − and 10 − . In comparison, we note that for intrin-sic phonon-induced spin and momentum relaxation rates in 3D bulk Si, this ratio is around 10 − [54], which isdetermined completely by the host Si SOC. 𝒌𝒌 𝑭𝑭 ( / Å ) 1.00 0.98 0.96 0.06 0.08 0.1 0 0.94 0.04 0.02 (111) (110) FIG. 9: The k F -dependent factor in momentum scatter-ing rate [Eq. (26)], I m ≡ π ( k F q TF ) (cid:82) dxx ( x + q TF / k F ) √ − x ,for (111) and (110) 2DEG orientations respectively, over0 < k F < . A − . V. SUMMARY AND OUTLOOK
We have introduced in Si 2DEG a previously over-looked yet important spin relaxation mechanism due toelectron-impurity scattering. This mechanism dominatesover other spin relaxations in the multi-valley Si con-duction band as impurity density increases, and can besignificantly suppressed when electrons are transferredinto two opposite ground valleys by specific 2DEG ori-entations and stress configurations. We provide the gen-eral expression for obtaining the leading-order spin relax-ation rate under arbitrary confinement potential, appliedstress, and subband occupation. We calculate quantita-tively the ( T ) spin relaxation time τ s ( s ) as a function ofspin orientation s , as well as of the conduction electrondensity and confinement strength for the representativesquare and triangular wells.Moreover, the consequences of various stress configu-rations have been worked out in details. Importantly,this newly discovered spin relaxation mechanism com-bined with the Si 2DEG setup provides interesting possi-bilities to tune spin lifetime as well as its dependence onspin orientation (or applied magnetic field direction) sub-stantially by on-chip gate voltages and possibly by localstress. Such a tunability of spin relaxation in MOSFET-type Si devices could have potential spintronic applica-bility.Also crucially, we provide experimental ways (elabo-rated in Sec. IV) to verify our spin relaxation mechanismand distinguish it from the DP spin relaxation effect fromthe generalized Rashba/Dresselhaus field in Si 2DEGs, byexploiting their different dependence on impurity densi-ties and types, on the interface symmetry properties, andon 2DEG plane, spin and stress orientations.4Regarding a general expansion of this model, we pointout that for 2DEG near the interface with considerableamount of disorder, a variation of our impurity-driven in-tervalley spin-flip process may become quantitatively im-portant in determining the spin relaxation rate. As men-tioned in the introduction, DP spin relaxation mechanismalone leads to much longer spin lifetime for low-mobility2DEG than observed experimentally [18]. However,spin lifetime is apparently shorter in 2DEG near typi-cal Si/SiO interfaces, indicating impurity-driven Elliott-Yafet spin relaxation. While our spin-flip matrix ele-ments [Eq. (2)] apply specifically to substitutional im-purities in Si with their given symmetry, it is a basicrule that lower-symmetry disorder inherits the allowedtransition matrix elements. Thus the key idea of zeroth-order intervalley spin-flip scattering [22] robustly holdsfor irregular defects, with additional scattering channelspotentially open depending on the specific defects. It istherefore possible that interface impurities (even whenthey are completely nonmagnetic as our theory entirelyrestricts itself to– any magnetic interface impurities willof course very strongly affect spin relaxation near thesurface through direct magnetic spin-flip scattering) areplaying a strong role in determining the 2D spin relax-ation time in disordered Si/SiO MOSFETs by partic-ipating in the Yafet process identified and analyzed inthe current work. Obviously, figuring this out remainsan open and important future experimental challenge inSi spintronics.This work is supported by LPS-MPO-CMTC.
Appendix A: physics of Intervalley coupling in Siand symmetry analysis
To be self-contained, we provide the essential physicalpicture of intervalley coupling in bulk Si and the relevantsymmetry analysis and selection rules for Sec. II.Bulk Si has the crystal structure [Fig. 10(a) in ab-sence of the impurity substitution] consisting of two setsof interpenetrating face center cubic lattices, and a spacesymmetry group O h . Its lowest conduction band in thewavevector space has its bottoms not at the center ofthe Brillouin zone but along the cubic axes directions.Crystal symmetry determines that 6 energy valleys residecylindrically along ± x, ± y and ± z axes. This well-knownmultivalley picture of Si supplies relevant information forthe electron states involved in this work. The transitionsbetween these electron states residing near the bottomof the conduction valleys can obviously be classified intothree groups [see Fig. 10(b)]: (I) within the same valley(“intravalley”), (II) between two opposite valleys (“inter-valley g process”), and (III) between two non-oppositevalleys (“intervalley f process”).The particular scattering potential we deal with comesfrom the impurity which replaces one of the Si atoms.This impurity immediately invalidates the translationalsymmetry of the Si crystal, and as a result the symme- try of the Hamiltonian system falls into a point grouparound the impurity [Fig. 10(a)]. This point group hasthe same symmetry operations as a tetrahedron molecule: C rotation about x, y or z axis, C rotation about bodydiagonals, σ reflection about the face diagonal planes,and S ( C followed by reflection) about x, y or z axis,and is called the T d group. K W X U z y x g f T d T d (a) (b) FIG. 10: (a) The Si crystal lattice, with one of its Si atomsreplaced by an impurity denoted as “X”. As a result, thesymmetry of the whole Hamiltonian system is reduced to thatof the T d point group. (b) The Brillouin zone of the Si crystal.The yellow ellipsoids mark the low-energy surface of the 6conduction valleys. Two representative examples are markedfor intervalley g and f processes. To utilize the symmetry property of this system for se-lection rules, we work with symmetrized electrons statesby linearly combining 6 different valleys rather thanstates in each individual valley as one is used to. The6 combinations are as follows [41] ψ A = 1 √ , , , , , ψ E I = 12 (1 , , − , − , , , (A2) ψ E II = 12 √ , , , , − , − ψ T I = 1 √ , − , , , , , (A4) ψ T II = 1 √ , , , − , , , (A5) ψ T III = 1 √ , , , , , − x , − x , + y , − y , + z or − z axis, respectively. Each new state is given a nameat the subscript of ψ , following the well-established nam-ing system (see the T d group character table in [68] or[22]). The selection rules immediately follow, since onlythe same-symmetry states can couple while different-symmetry states are not mixed by the scatterer potentialwhich transforms as the identity in this group. Once weget the scattering matrix elements that do not vanish,we can easily make linear combinations between them totransfer back to the familiar intravalley and intervalley g and f processes [22].5Thus far, we have not considered spin degrees of free-dom or SOC. To include spin, we can expand the basis tobe the product space of 6 valleys and 2 spins. It turns outtwo ¯ F states emerge from this valley-spin coupling. Tobe concrete, the multiplication expressions are as follows.The pure spin transforms as ¯ E , and then we have A × ¯ E = ¯ E , (A7) E × ¯ E = ¯ F , (A8) T × ¯ E = ¯ E + ¯ F . (A9)We may follow a similar procedure as the spinless caseto obtain spin-dependent scattering selection rules [22]. Only states with the same symmetry can be coupled.Among all the 5 nonvanishing couplings [each of the 4states in Eqs. (A7)-(A9) coupling to itself, as well asthe inter-coupling of the two ¯ F states from Eqs. (A8)and (A9)], we find that there are spin-flip terms in twoof them: the difference between ¯ E and ¯ F self-couplingmatrix elements from Eqs. (A9), and the inter-couplingmatrix element between two different ¯ F . That leads tothe two terms in Eq. (2), respectively. After transform-ing back to the intravalley, intervalley g and f processes,we find [22] both terms contribute to the f -process spinflip. [1] I. ˇZuti´c, J. Fabian, and S. Das Sarma, Spintronics: Fun-damentals and applications, Rev. Mod. Phys. , 323(2004).[2] R. Jansen, Silicon spintronics, Nat. Mater. , 400(2012).[3] V. Sverdlov and S. Selberherr, Silicon spintronics:Progress and challenges, Phys. 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