Theory of Hole-Spin Qubits in Strained Germanium Quantum Dots
L. A. Terrazos, E. Marcellina, Zhanning Wang, S. N. Coppersmith, Mark Friesen, A. R. Hamilton, Xuedong Hu, Belita Koiller, A. L. Saraiva, Dimitrie Culcer, Rodrigo B. Capaz
TTheory of Hole-Spin Qubits in Strained Germanium Quantum Dots
L. A. Terrazos, E. Marcellina, S. N. Coppersmith,
3, 2
Mark Friesen, A. R. Hamilton, Xuedong Hu, Belita Koiller, A. L. Saraiva, Dimitrie Culcer, and Rodrigo B. Capaz Centro de Educa¸c˜ao e Sa´ude, Universidade Federal de Campina Grande, Cuit´e, PB 58175-000, Brazil School of Physics, University of New South Wales, Sydney 2052, Australia Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA Department of Physics, University at Buffalo, SUNY, Buffalo, New York 14260-1500, USA Instituto de F´ısica, Universidade Federal do Rio de Janeiro, CP 68528, 21941-972 RJ, Brazil (Dated: February 6, 2020)We theoretically investigate the properties of holes in a Si x Ge − x /Ge/ Si x Ge − x quantum well in aperpendicular magnetic field that make them advantageous as qubits, including a large ( >
100 meV)intrinsic splitting between the light and heavy hole bands, a very light ( ∼ m ) in-plane effectivemass, consistent with higher mobilities and tunnel rates, and larger dot sizes that could ameliorateconstraints on device fabrication. Compared to electrons in quantum dots, hole qubits do notsuffer from the presence of nearby quantum levels (e.g., valley states) that can compete with spinsas qubits. The strong spin-orbit coupling in Ge quantum wells may be harnessed to implementelectric-dipole spin resonance, leading to gate times of several nanoseconds for single-qubit rotations.The microscopic mechanism of this spin-orbit coupling is discussed, along with its implications forquantum gates based on electric-dipole spin resonance, stressing the importance of coupling termsthat arise from the underlying cubic crystal field. Our results provide a theoretical foundation forrecent experimental advances in Ge hole-spin qubits. I. INTRODUCTION
Hole spin qubits in strained germanium possess favor-able properties for quantum computing, including (1) theabsence of valley degeneracy, which would otherwise com-pete with the spin degree of freedom for qubits formed inthe conduction band of Si or Ge [1], (2) the high naturalabundance of spin-0 nuclear isotopes in Ge, which maybe further purified, (3) the formation of hole states in p -type atomic orbitals whose wavefunction nodes occurat nuclear sites, suppressing unwanted hyperfine inter-actions [2, 3], and (4) the very light in-plane effectivemass [4–7], allowing for larger dots and relaxing con-straints on device fabrication. The light mass also im-proves carrier mobilities, which can exceed 10 cm /V sfor 2D Ge hole gases [4]. Leveraging these strengths,rapid progress has been made in implementing high-fidelity one and two-qubit gate operations [5, 8–16].Several of the most important advantages for qubits,such as the lifting of level degeneracy at the valence-band edge, the light effective mass, and access to Rashbaspin-orbit coupling (SOC), which enables fast gate oper-ations, are not available in the bulk. Rather, they emergein SiGe/Ge/SiGe quantum wells due to confinement orstrain. In fact, some properties (e.g., the light mass) arecompletely unexpected from the bulk band structure.In this work, we provide a theoretical foundation forthe emergent physics of Ge quantum wells, and explana-tions for recent experimental observations, through de-tailed ab initio band-structure calculations. We gain fur-ther insight into the origins of qubit-friendly materialsproperties by performing k · p calculations. We place spe-cial emphasis on understanding the Rashba coupling, andthe unexpected matrix element that connects states withorbital quantum numbers that differ by one. Taken to- gether, these ingredients enable electrically driven spinflips via electric-dipole spin resonance (EDSR), with fast,single-qubit gate times of order 0.2 GHz. In contrast withother recent work [15], we propose here to exploit thelarge out-of-plane value of the Land´e g -factor, so thatrelatively small external magnetic fields are needed forgate operation, making the qubit more compatible withsuperconducting gate structures, such as microwave res-onators. A large g -factor also helps to define the qubitwith respect to thermal broadening.The paper is organized as follows. In Sec. II, we pro-vide technical details of the theoretical methods used inthis work. We describe the model system, including theheterostructure and top gates (Fig. 1). We summarizethe ab initio simulations of the quantum-well portion ofthe device and our k · p Hamiltonian. We describe ourtheoretical approach for modeling EDSR in two steps.We first outline a model for electron confinement in thevertical direction (perpendicular to the plane of the quan-tum well) and the lateral confinement of a quantum dot,and use this to obtain the effective Rashba spin-orbitHamiltonian for our geometry. We then use this to de-termine the EDSR Rabi frequency when applying an in-plane ac electric field. In Sec. III, we describe the mainresults of our calculations, including the band-structuredetails obtained by ab initio methods (Fig. 2), the cor-responding in-plane and out-of-plane effective masses asa function of Ge concentration and strain (Fig. 3), andthe energy splittings between the valence bands (Fig. 4).We then apply k · p methods to help clarify the originsof energy-level splitting, and the lifting of degeneracy,by artificially separating the effects of strain and SOC(Fig. 5). Finally, we use our EDSR analysis to estimatethe expected Rabi frequency for a realistic range of de-vice parameters (Figs. 6 and 7). In Sec. IV, we discuss a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b FIG. 1. Cartoon depiction of a typical heterostructure andgate stack of a strained-Ge quantum well used to form hole-spin qubits in quantum dots. Here, a 20 nm strained-Ge quan-tum well is grown epitaxially on a strain-relaxed Si . Ge . alloy. For this arrangement, the strain in the Ge layer is ε ≈ − z is defined as the growth direction. our results and conclude by reviewing the predominantdecoherence mechanisms for Ge hole qubits. II. METHODS
We consider a typical, electrically gated double-dot de-vice such as the one schematically depicted in Fig. 1. Theessential features include a SiGe/Ge/SiGe heterostruc-ture, an optional capping layer, and a set of patterned,nanometer-scale metal gates that are isolated from theheterostructure by oxide layer(s). When sandwiched be-tween strain-relaxed, Ge-rich SiGe alloys, the compres-sively strained Ge forms a type-I quantum well that cantrap either electrons or holes [17], although we focus ex-clusively on holes here. Note that the details of the gateand oxide layers are unimportant for the following dis-cussion.For the heterostructure, we specifically consideran accumulation-mode gating scheme [18–20] with nodopants. The SiGe barriers should be high enoughto form a quantum well. For example, a strain-relaxed Si . Ge . barrier gives a valence-band offset of ∼
170 meV [17], which is ample for trapping holes. Theheterostructure growth typically begins on a Si wafer.[Here, we assume a Si (001) substrate.] Due to latticemismatch, the growth of SiGe alloy on Si induces disloca-tions that are harmful for device operation. Such effectscan be mitigated by increasing the Ge content gradually,over several µ m, until the desired alloy is achieved; un-der ideal conditions, the resulting SiGe growth surface isstrain-relaxed and dislocation-free [21]. Alternative ap-proaches, such as reverse grading, are also common [4, 5].A pure Ge quantum well is then grown on the SiGe vir-tual substrate. The width of the well should be less thanthe critical thickness for forming additional dislocations; FIG. 2. Electronic band structures for (a) relaxed vs. (b)uniaxially-strained Ge, obtained using DFT. To the left ofeach plot we show the corresponding real and reciprocal spacecrystal structures (lower and upper diagrams, respectively),with lattice constants ( a and c ) and symmetry points (Γ, X,Z, and L), as indicated. [Note that the tetragonal deforma-tion is exaggerated in (c), for clarity.] (c),(d) Blown-up bandstructures corresponding to (a) and (b). Here, we focus onthe [100] ( x ) and [001] ( z ) axes because of their relevance forquantum dot formation, and we note that [100] and [010] areequivalent. In (a) and (c), cubic symmetry also makes the Xand Z points equivalent and enforces a degeneracy betweenthe top two hole bands at the Γ point. The lowest or “split-off” band is completely detached from the others. Away fromthe singular Γ point, the hole bands are all doubly-degenerate.In (b) and (d), the x - z degeneracy is lifted and only the x - y degeneracy remains. The resulting band structure is highlyanisotropic. however, this is not typically a problem for Ge-rich alloys.For example, the critical Ge thickness of a Si . Ge . barrier is ∼
30 nm [22]. Finally, we note that Ge formsunstable oxides [23] (similar problems also occur for SiGealloys [24]); it may therefore be beneficial to include a sil-icon capping layer, with a carefully chosen thickness [25].
A. Density Functional Theory
Realistic, quantitative predictions for the band struc-ture of strained Ge are key for assessing the viability ofhole-spin qubits. In this work, we compute the bandstructure using self-consistent, ab initio density func-tional theory (DFT), including spin-orbit interactions.The calculations are performed using the full-potentiallinearized augmented plane wave method (FP-LAPW),as implemented in the Wien2k package [26]. Using theaugmented plane wave plus local orbital (APW+lo) basisset [27–29], the wavefunctions are expanded in sphericalharmonics inside non-overlapping atomic spheres, with“muffin-tin” radii R MT , and in plane waves for the restof the unit cell (the interstitial region). In the presentcalculations we adopt R MT =0.95 ˚A for Ge, and use 405 k points in the irreducible wedge of the Brillouin zone.For the spherical-harmonic expansion, the maximum or-bital angular momentum is taken to be l max =10, whilethe plane-wave expansion in the interstitial region is ex-tended to k max =9.0 /R MT =9.47 ˚A − , and the charge den-sity is Fourier expanded up to G max =12 Ry. (These sim-ulation parameters were all checked and found to yieldnumerical convergence.) Electron-electron interactionsare described using the modified Becke-Johnson exchangepotential + local density approximation (LDA) correla-tions [30, 31], which is known to yield accurate calcula-tions of band gaps in semiconductors.The primitive Bravais lattice used in our simulationsis body-centered tetragonal with a two-atom basis con-sistent with the diamond structure. Details of the realand reciprocal lattice structure are depicted in the insetsof Figs. 2(a) and 2(b). For unstrained Ge, the tetrago-nal lattice parameters are given by a = b =4.0008˚A in theplane of the quantum well, and c = √ a =5.6580˚A in thegrowth direction. For a Si x Ge − x alloy with concentra-tion x , the lattice constant a ( x ) is modified, and if thequantum well is grown pseudomorphically, the same lat- tice constant is also imposed on the strained Ge. Wedefine the compressive strain as ε ( x ) = [ a ( x ) − a (0)] /a (0) < . (1)For the SiGe alloy, Vegard’s law gives ε ( x ) = − . x ,while Poisson’s ratio for germanium gives ν = 0 .
27 = − [ c ( x ) − c (0)] / [ a ( x ) − a (0)] [32]. Combining these formu-las yields an analytical expression for c ( x ).The main results of our DFT calculations are reportedin Sec. III A. To simplify the calculations, we do notspecifically consider a quantum well geometry. Instead,we adopt a range of strain parameters consistent with astrained Ge quantum well sandwiched between strain-relaxed Si x Ge − x for the range x ∈ (0 , . ε ∈ ( − ,
0) percent.
B. k · p Theory for Strained Germanium k · p theory provides insight into energy-splitting mech-anisms associated with SOC and strain for the top-most valence bands. We consider the 6 × | j, m j (cid:105) ∈ {| , (cid:105) , | , (cid:105) , | , − (cid:105) , | , − (cid:105) , | , (cid:105) , | , − (cid:105)} ,the LKPB Hamiltonian is given by H LKPB = P + Q − S R − S/ √ √ R − S ∗ P − Q R −√ Q (cid:112) / SR ∗ P − Q S (cid:112) / S ∗ √ Q R ∗ S ∗ P + Q −√ R ∗ − S ∗ / √ − S ∗ / √ −√ Q ∗ (cid:112) / S −√ R P + ∆ 0 √ R ∗ (cid:112) / S ∗ √ Q ∗ − S/ √ P + ∆ , (2)where P = P k + P ε , Q = Q k + Q ε ,R = R k + R ε , S = S k + S ε . (3)Here, the k subscripts denote the Luttinger-Kohn Hamiltonian matrix elements, defined as [35] P k = (cid:126) m γ ( k x + k y + k z ) , Q k = − (cid:126) m γ (2 k z − k x − k y ) ,R k = √ (cid:126) m [ − γ ( k x − k y ) + 2 iγ k x k y ] , S k = √ (cid:126) m γ ( k x − ik y ) k z , (4)while the ε subscripts denote the Bir-Pikus strain matrix elements, defined as [33] P ε = − a v ( ε xx + ε yy + ε zz ) , Q ε = − b v ε xx + ε yy − ε zz ) ,R ε = √ b v ( ε xx − ε yy ) − idε xy , S ε = − d v ( ε xz − iε yz ) , (5)where m is the bare electron mass, γ =13.38, γ =4.24, and γ =5.69 are Luttinger parameters [35], a v =2 . b v = − .
16 eV, are d v = − .
06 eV are deformation poten-tial parameters [36], and { ε ij } are strain tensor com-ponents. Experiments in bulk, relaxed Ge [17] give∆=0.296 eV as the energy splitting of the split-off bandbelow the light and heavy-hole bands at k =0; this valueis taken as an input parameter in the present work. Notethat ε xx (= ε yy ) is equivalent to ε ( x ), defined in Sec. II A.The Pikus-Bir expressions in Eq. (5), are generic. In thiswork, we focus on the special case of uniaxial strain, forwhich ε zz = − C /C ) ε xx and ε xy = ε yz = ε zx =0, lead-ing to R ε = S ε =0, and we adopt C =129.2 GPa and C =47.9 GPa as the elastic stiffness constants for thestrain-stress tensor [32]. The resulting strain splits thefourfold-degenerate valence band edge into two twofold-degenerate levels. The main results of these calculationsare presented in Sec. III B. C. Calculating the Rashba Spin-Orbit Coupling
There are two prerequisites for observing Rashba SOCin a quantum well: a broken structural symmetry and anintrinsic SOC. The broken symmetry is provided here byan asymmetric confinement potential of the form V z ( z ) = (cid:26) eF z z ( | z | < d/ ∞ (otherwise) , (6)where F z is the average electric field across the quantumwell, and the well width, d =20 nm, is held fixed for allour calculations. (Note that d is not an important pa-rameter in this calculation, since the electric field drawsthe hole wavefunction to the top of the quantum well, sothat it does not interact strongly with the bottom of thewell.) The total Hamiltonian for the vertical confinementof holes is then given by H z = H LKPB ( k , ˆ k z ) + V z ( z ) , (7)where ˆ k z ≡ − i ∂∂z .To evaluate Eq. (7), we first truncate H LKPB to thefour-dimensional, j = 3 / | m j | ∈ { / , / } [37]: ϕ i ( z ) = sin [ πd ( z + d )] exp [ − b i ( zd + )] π (cid:114) d exp( − bi ) sinh( bi )2 π bi +2 b i ( | z | < d/ . (8)Here, i is the band index and { b i } are the correspondingvariational parameters. We determine b i by minimizingthe eigenvalues of H z in the limit of k = 0, in whichcase the Hamiltonian is already diagonal and the bandsdecouple.A large intrinsic SOC occurs naturally for holes in Gedue to the coupling between the different valence bands.We compute the effective Hamiltonian for holes in thetopmost ( | m j | =3/2) band by applying a Schrieffer-Wolfftransformation to Eq. (7), using the states shown in Eq. (8), to eliminate the coupling to the second band [38],obtaining H + H R , where H = (cid:126) ( k x + k y ) / m x is thekinetic energy in the effective mass approximation. Theeffective Rashba Hamiltonian can be expressed as thesum of two terms, H R = iα R ( k σ − − k − σ + )+ iα R ( k + k − k + σ + − k − k + k − σ − ) , (9)where σ ± = σ x ± iσ y are Pauli spin matrices, and the cou-pling constants α R and α R are derived in Ref. [38].Here, α R arises from the spherically symmetric com-ponent of the Luttinger-Kohn Hamiltonian, while α R arises from the cubic-symmetric component. D. Calculating the ESDR Rabi Frequency
There are also several prerequisites for implementingEDSR: an artificial [39] or intrinsic SOC, a large, staticmagnetic field to define the quantization axis, and an in-plane ac electric field. Here, we assume Rashba SOC, asdescribed in the previous section. Contrary to other pro-posals that we have seen, we also assume an out-of-plane B -field whose orientation takes advantage of the largeout-of-plane g -factor [40], g z , to reduce the constraintson the field magnitude. The effective qubit Hamiltonianfor EDSR is then given by H q = H ( k → − i ∇ − e A / (cid:126) ) + H R ( k → − i ∇ − e A / (cid:126) )+ V d ( x, y ) + ( g z / µ B B z σ z + eE ac x cos( ωt ) σ x , (10)where A = ( B z / − y, x, V d ( x, y ) = 12 m x ω ( x + y ) , (11)where (cid:126) ω is the energy splitting between the orbital lev-els when B z = 0. If we now assume that B z >
0, but set E ac = 0, the eigenstates of H q are defined as Fock-Darwinorbitals [41, 42], for which the ground state ( n = 0) isgiven by φ ( x, y ) = 1 a √ π exp (cid:2) − ( x + y ) / a (cid:3) , (12)and the first excited states ( n = 1) are given by φ ± ( x, y ) = 2 a √ π ( x ± iy ) exp (cid:2) − ( x + y ) / a (cid:3) . (13)For an out-of-plane magnetic field, we note that the dotis confined both electrostatically and magnetically, withan effective radius of a = (cid:112) (cid:126) / | eB | / (1 / ω /ω c ) / ,where ω c = | eB | /m x is the cyclotron frequency.For hole-spin qubits, the logical (spin) states areformed exclusively within the ground-state orbital, φ . strain , ε Γ - X e ff e c t i v e m a ss , m x / m Γ - Z e ff e c t i v e m a ss , m z / m top VB2 nd VBsplit-off band0.0 x in Si x Ge -x FIG. 3. Effective masses for the top three valence bands, inunits of the free-electron rest mass m , obtained using DFT.Due to uniaxial strain along [001], the masses in different di-rections, m ∗ x and m ∗ z , are inequivalent. While m ∗ z is foundto vary smoothly with substrate composition, m ∗ x changesabruptly near x ≈
0, indicating an inversion of the band char-acter: the top band becomes lighter than the second band, asconsistent with the right-hand side of Fig. 2(d).
However, the EDSR spin-flip mechanism involves virtualtransitions to φ ± via a second-order process that com-bines ac driving and SOC. The driving term in Eq. (10), eE ac x cos( ωt ), is applied through one of the nearby topgates [43], generating an orbital transition with ∆ n = ±
1. Initial proposals for hole-based EDSR [2] there-fore required Dresselhaus SOC, which can generate such∆ n = ± α R term of the Rashba coupling,Eq. (9), is cubic in k , as consistent with ∆ n = ±
3, andtherefore does not support EDSR. An important conclu-sion of the present work is that the α R term, whichis not typically considered in such calculations, providesthe required ∆ n = ± f R , we evalu-ate the full Hamiltonian, Eq. (10), using the Fock-Darwinbasis states, and perform a Schrieffer-Wolff transforma-tion to eliminate the coupling to the excited states. Forresonant driving, with ω = (cid:112) ω + ω c /
4, we obtain hf R = − eE ac α R a (cid:104)(cid:16) + (cid:17) − (cid:16) + (cid:17)(cid:105) − e E ac α R B z (cid:126) (cid:104)(cid:16) + (cid:17) + (cid:16) + (cid:17)(cid:105) , (14)where ∆ ≡ − (cid:126) ω − (cid:126) ω c , ∆ ≡ − (cid:126) ω − (cid:126) ω c − gµ B B z , ∆ ≡ − (cid:126) ω + (cid:126) ω c + gµ B B z , ∆ ≡ − (cid:126) ω + (cid:126) ω c . (15)This result is explicitly proportional to E ac α R . More-over, f R is found to be linear in B z , as readily verified ene r g y d i ff e r en c e s ( e V ) x in Si x Ge x top VB – split offtop VB – 2 nd band- 0.4% - 0.8%strain , ε FIG. 4. Energy differences between the hole bands at the Γpoint as a function of the Si concentration in the substrate, x , obtained using DFT. Upward-pointing blue triangles cor-respond to the splitting between the top of the valence bandand the split-off band, while downward-pointing teal trianglesshow the splitting between the first and second bands. by expanding Eq. (14) in powers of (small) B z . Basedon these results, and those of the previous section, weestimate the Rabi frequencies that could be expected inrealistic quantum dot devices, as described in Sec. III D. III. RESULTS
In this section we present the main results of each ofthe previous calculations.
A. DFT Estimates
DFT results are plotted in the main panels of Figs. 2(a)and 2(b), where we compare Ge band structures for thecases of x = 0 (unstrained Ge) and x =0.25 (strained Ge).In the first case, the cubic symmetry ensures that the en-ergy dispersion is identical for wavevectors in the plane ofthe quantum well ( k x , k y ) and the growth direction ( k z ).(Here, the subscript x refers to the [100] axis, not the al-loy composition.) In the second case, the X and Z pointsare inequivalent; both are shown in the figure. Focus-ing on holes, Figs. 2(c) and 2(d) show blown-up views ofthe top of the valence band. Since the quantum dot wavefunctions are constructed mainly from Bloch states at thevery top of the band, the essential physics is captured inthe band curvature at the Γ point, which is proportionalto the inverse effective mass. In the case of strain, we ob-serve anisotropic behavior in the x (in-plane) and z (out-of-plane) directions. Figures 2(c) and 2(d) also highlightthe large energy splitting between light and heavy holebands under strain, which is crucial for defining the qubitstates.Figure 3 provides a more detailed picture of the in-plane ( m ∗ x ) and out-of-plane ( m ∗ z ) effective masses, ob-tained for strains in the range ε ∈ ( − ,
0) percent. Thecorresponding values of x in the Si x Ge − x barrier alloyare also shown. We note that the in-plane mass of the toptwo bands changes abruptly near x =0. Remarkably, m ∗ x O h no strain & no spin-orbit Γ no SOC Γ + Γ no strainincreasing SOC increasing strain increasing strain Δ = 0 ε = 0 Δ = 0 .296 eV ε = 0 Δ = 0 .296 eV ε = −1% Δ = 0 ε = 0 Δ = 0 ε = −1% Δ = ε = −1% D j =1/2 Γ + Γ Γ + Γ + Γ + Γ + Γ O h D D increasing SOC ene r g i e s a t k = ( e V ) -0.2 j =3/2 Γ | ⟩| ⟩ | ⟩ | ⟩| ⟩| ⟩ | ⟩| ⟩ | ⟩ | ⟩ | ⟩| ⟩| ⟩| ⟩ | ⟩| ⟩ | ⟩| ⟩ | ⟩ | ⟩ | ⟩| ⟩ | ⟩ | ⟩ | ⟩| ⟩ p x spinupspin down | ⟩ | ⟩| ⟩| ⟩ +3/2 p y p z +1/2 −1/2 −3/2 +1/2 −1/2 -0.40.2 FIG. 5. Energy levels calculated at the Γ point, using the k · p method, at zero magnetic field, which allows us to artificiallydecouple the effects of SOC (represented by the split-off band gap ∆ of bulk Ge) and strain ( ε ). The five panels show resultswhen these two parameters are independently varied between zero and their final values, corresponding to a strained quantumwell with x =0 .
25. Level degeneracies are indicated by color: black for sixfold, blue for fourfold, and red for twofold. The pointsymmetry groups and corresponding irreducible representations for the hole states are indicated in each case. The center panelrepresents the case with no SOC and no strain, in which the p x , p y , and p z orbitals and both spin states are degenerate. Movingto the right, the strain is increased without including SOC, yielding a fourfold degenerate band spanned by p x and p y , and atwofold degenerate p z band. Including SOC, the p orbitals hybridize, creating states with different combinations of orbitals(represented now as tori) and spins, resulting in three doublets. Moving from the center panel to the left, including SOC butno strain yields a split-off, doubly-degenerate j =1 / j =3 / m j ) and by the inclination of the green vectors in relation to the verticaldirection. Including strain, the bands hybridize slightly such that j is no longer a good quantum number. Here, the fullystrained spectrum is identical to the far right-hand side of the figure. becomes lightest for the top band, over the experimen-tal regime of interest ( x (cid:38) m ∗ z , the top band remains heaviest for all x consid-ered here, and is a smooth function of the strain. Theseresults are in reasonable agreement with several recentexperiments [4–6], and they agree very well with Ref. [7],in which band nonparabolicity is explicitly accounted for.We note that although m ∗ x appears to change abruptly inFig. 3(a), smooth behavior is recovered when consideringthe narrower range of x =0-0.01.Figure 4 shows the corresponding results for the energydispersion of the valence-band edges. In the limit x → . µ eV,which compares well with the experimentally measuredvalue of 0.296 µ eV [17]. For x >
0, the band degener-acy is lifted by a significant amount, of order 100 meVfor typical quantum-well heterostructures. In contrastwith the effective mass, no abrupt change occurs for thevalence-band edges near x =0.To summarize the present results, DFT predicts a sud-den change in the in-plane mass of the top band as thestrain decreases from zero, with m x becoming very light.Moreover, the degeneracy of the top two bands is lifted,and the energy splitting between all the bands is en-hanced. These results are all consistent with recent ex-periments. B. k · p Analysis k · p theory allows us to explore the mechanisms thatcause the changes in the band structure and clarifytheir separate roles. In Fig. 5, we plot the edges ofthe top three valence bands, as a function of eitherstrain or SOC. By following the progression from a singlesixfold-degenerate band (center panel) to three twofold-degenerate bands (outer panels), we infer that the split-ting of the top two bands requires both strain and SOC.The calculations also show that the hybridization of thetopmost bands occurs at second order, via strain-inducedcoupling to the split-off band. Since this effect is weak,the total angular momentum in the top band, which de-fines the qubit, is still given by j =3 / m j = ± / m s = ± k · p calculations, we note that strain has beenintroduced perturbatively. Hence, although the energysplitting of the lowest valence band is exact when ε = 0,since it is taken as an input parameter, the calculatedenergies become increasingly inaccurate for higher strainvalues. For example, when ε = − >
50% larger than the k · p estimate. Likewise, the k · p energy splitting of 0.06 eVbetween the top two valence bands is approximately halfthe DFT estimate of 0.13 eV.To summarize, the k · p theory reproduces the generalfeatures of the band structure that was obtained more e l e c t r i c f i e l d , F z ( M V / m ) dot radius, a (nm) f R (GHz) FIG. 6. Color map of the EDSR Rabi frequency f R as afunction of both the vertical electric field F z and the effec-tive dot radius a , with magnetic field B z =0.06 T, quantumwell width d =20 nm, and microwave driving amplitude [44] E ac =0.1 MV/m. All materials parameters assume a Si con-centration of x =0.25 in the barrier alloy. rigorously using DFT. Although k · p methods are lessaccurate than DFT, they allow us to clarify that bothstrain and SOC are required to fully lift the band degen-eracy at k = 0. C. Quantum Well Corrections to the Energy
The energies plotted in Fig. 4 were obtained withoutincluding the quantum-well subband confinement ener-gies, which differ for different bands, and can be sizeable.Here we show that the subband contribution to the holeenergy does not compromise the energy splitting betweenthe top two valence bands or change the effective orderingbetween them.The subband energies differ for the top two valencebands due to their different effective masses. We canestimate these effects by assuming a triangular, verti-cal confinement potential, as in Eq. (6). Here, we as-sume an electric field value of F z ≈ ep/(cid:15) , which is thefield required to accumulate a 2D hole gas with density p = 4 × cm − , and we linearly interpolate the di-electric constant in the Si x Ge − x barrier layer, obtain-ing the relation (cid:15) ( x )=(16 . − . x ) (cid:15) , where (cid:15) is thevacuum permittivity. We further assume that the ver-tical extent of the wavefunction is less than the quan-tum well width, allowing us to ignore the bottom edgeof the well. The triangular potential has known solu-tions [45], yielding a confinement energy of 2 . E forthe first subband and 4 . E for the second subband,where E =( (cid:126) e p / m ∗ z (cid:15) ) / is a characteristic energyscale and m ∗ z depends on both the alloy composition andthe particular valence band. (Note that we do not con-sider band-nonparabolicity effects here, although theycan be significant due to the large energies involved.) Inthis way, when x =0 .
25, we obtain a total energy splitting(including both band and subband energies) of 140 meVfor the lowest-energy confined holes in the first and sec- f R ( G H z ) τ R ( n s ) z = 0.06 T E x = 0.1 MV/md = 20 nm ε xx = ε yy = -1% ε zz = 0.09%F z = 4.5 MV/m f R ( G H z ) a (nm) ! R ( n s ) a.b. a (nm) f R ( G H z ) τ R ( n s ) a.b. FIG. 7. Calculated values of (a) the EDSR Rabi frequency f R , and (b) the corresponding X π gate time ( τ R =1 /f R ), asa function of the effective dot radius a . Here, the simulationparameters are the same as in Fig. 6, with F z = 4 . ond valence bands, with the first band still having thelowest energy. In comparison, the energy splitting be-tween the first and second subbands within the top va-lence band, is 27.7 meV, which therefore represents thepredominant leakage channel for the qubit. We concludethat band and subband excitations of the qubit level aremuch larger than other relevant energy scales in this sys-tem, including the thermal energy of the hole reservoirs(5-15 µ eV), the inter-dot tunnel couplings ( ∼ µ eV),and exchange interactions ( ∼ µ eV). D. Rabi Frequency Estimates
In Sec. II D, particularly in Eq. (14), we obtained gen-eral results for the EDSR Rabi frequency f R as a functionof system parameters. In Fig. 6, we now plot the depen-dence of the Rabi frequency on the dot radius a and thevertical electric field F z . In Fig. 7, we further show aline-cut through this data, and a corresponding plot of τ R = 1 /f R , representing the gate time for an X π gateoperation. Generally, we find that larger dots yield fastergate operations due to their smaller orbital energies. (Wenote that, for sufficiently large a , the perturbative meth-ods used here become inaccurate.) To take an example,for a typical vertical field of 4.8 MV/m and effective dotradii in the range of 30-60 nm, Rabi frequencies can beof order 0.2 GHz, corresponding to a 5 ns gate time foran X π gate. Such fast gates are very promising for high-fidelity quantum gate operations. IV. DISCUSSION AND CONCLUSIONS
Recent experimental work has already demonstratedthat holes in germanium are promising as qubits. In thiswork, we have explored how confinement and strain arecritical for achieving such strong performance, particu-larly in the context of EDSR-based gate operations. Wehave also demonstrated that operating the qubits in anout-of-plane magnetic field may be advantageous becauseof the highly anisotropic g -factor.To conclude, we comment on the expected decoher-ence mechanisms affecting Ge hole spins. As mentionedin the introduction, hyperfine interactions are suppressedfor hole spins due to the p -orbital character of the va-lence band [2, 3], and the low natural abundance ( < ACKNOWLEDGEMENTS
We are grateful to D. J. Paul, G. Scappucci, D. E.Savage, O. P. Sushkov, D. S. Miserev and M. A. Eriks-son for illuminating discussions and helpful information.Work in Brazil was performed as part of the INCT-IQand supported by Centro Nacional de Processamentode Alto Desempenho em So Paulo (CENAPAD-SP,project UNICAMP/FINEP-MCT), CNPq (304869/2014-7, 308251/2017-2, and 309861/2015-2), and FAPERJ (E-26/202.915/2015 and 202.991/2017). Work in Australiawas supported by the Australian Research Council Cen-tre of Excellence in Future Low-Energy Electronics Tech-nologies (project CE170100039). Work in the U.S.A.was supported in part by ARO (W911NF-17-1-0274 andW911NF-17-1-0257), NSF (OISE-1132804), and the Van-nevar Bush Faculty Fellowship program sponsored by theBasic Research Office of the Assistant Secretary of De-fense for Research and Engineering and funded by the Of-fice of Naval Research through grant N00014-15-1-0029.The views and conclusions contained in this documentare those of the authors and should not be interpretedas representing the official policies, either expressed orimplied, of the Army Research Office (ARO), or the U.S.Government. The U.S. Government is authorized to re-produce and distribute reprints for Government purposesnotwithstanding any copyright notation herein. [1] M. Friesen, M. A. Eriksson, and S. N. Coppersmith,Appl. Phys. Lett. , 202106 (2006).[2] D. V. Bulaev and D. Loss, Phys. Rev. Lett. , 097202(2007).[3] K. De Greve, P. L. McMahon, D. Press, T. D. Ladd,D. Bisping, C. Schneider, M. Kamp, L. Worschech,S. H¨ofling, A. Forchel, and Y. Y., Nature Phys. , 872(2011).[4] A. Dobbie, M. Myronov, R. J. H. Morris, A. H. A. Has-san, M. J. Prest, V. A. Shah, E. H. C. Parker, T. E.Whall, and D. R. Leadley, Appl. Phys. Lett. , 172108(2012).[5] N. W. Hendrickx, D. P. Franke, A. Sammak, M. Kouwen-hoven, D. Sabbagh, L. Yeoh, R. Li, M. L. V. Tagliaferri,M. Virgilio, G. Capellini, G. Scappucci, and M. Veld-horst, Nature Commun. , 2835 (2018).[6] C. Morrison and M. Myronov, Phys. Status Solidi A ,2809 (2016).[7] M. Lodari, A. Tosato, D. Sabbagh, M. A. Schubert,G. Capellini, A. Sammak, M. Veldhorst, and G. Scap-pucci, Phys. Rev. B , 041304(R) (2019).[8] S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, andL. P. Kouwenhoven, Nature , 1084 (2010).[9] Y. Hu, F. Kuemmeth, C. M. Lieber, and C. M. Marcus,Nature Nano. , 47 (2012).[10] R. Li, F. E. Hudson, A. S. Dzurak, and A. R. Hamilton,Nano. Lett. , 7314 (2015).[11] S. D. Liles, R. Li, C. H. Yang, F. E. Hudson, M. Veld-horst, A. S. Dzurak, and A. R. Hamilton, Nature Com-mun. , 3255 (2018). [12] L. Vukuˇsi´c, J. Kukuˇcka, H. Watzinger, J. M. Milem,F. Sch¨affler, and G. Katsaros, Nano. Lett. , 7141(2018).[13] H. Watzinger, J. Kukuˇcka, L. Vukuˇsi´c, F. Gao, T. Wang,F. Sch¨affler, J.-J. Zhang, and G. Katsaros, Nature Com-mun. , 3902 (2018).[14] W. J. Hardy, C. T. Harris, Y.-H. Su, Y. Chuang,J. Moussa, L. N. Maurer, J.-Y. Li, T.-M. Lu, and D. R.Luhman, Nanotechn. , 215202 (2019).[15] N. W. Hendrickx, D. P. Franke, A. Sammak, G. Scap-pucci, and M. Veldhorst, Nature , 487 (2020).[16] A. Sammak, D. Sabbagh, N. W. Hendrickx, M. Lodari,B. Paquelet Wuetz, A. Tosato, L. Yeoh, M. Bollani,M. Virgilio, M. A. Schubert, P. Zaumseil, G. Capellini,M. Veldhorst, and G. Scappucci, Advanced FunctionalMaterials , 1807613 (2019).[17] F. Sch¨affler, Semicond. Sci. Techn. , 1515 (1997).[18] S. J. Angus, A. J. Ferguson, A. S. Dzurak, and R. G.Clark, Nano Lett. , 2051 (2007).[19] K. Eng, T. D. Ladd, A. Smith, M. G. Borselli, A. A. Kise-lev, B. H. Fong, K. S. Holabird, T. M. Hazard, B. Huang,P. W. Deelman, I. Milosavljevic, R. Schmitz, R. S. Ross,M. F. Gyure, and A. T. Hunter, Sci. Adv. , e1500214(2015).[20] D. M. Zajac, T. M. Hazard, X. Mi, K. Wang, and J. R.Petta, Appl. Phys. Lett. , 223507 (2015).[21] D. L. Harame, S. J. Koester, G. Freeman, P. Cottrel,K. Rim, G. Dehlinger, D. Ahlgren, J. S. Dunn, D. Green-berg, A. Joseph, F. Anderson, J.-S. Rieh, S. A. S. T.Onge, D. Coolbaugh, V. Ramachandran, J. D. Cressler, and S. Subbanna, Applied Surface Science , 9 (2004).[22] D. J. Paul, Laser Photon. Rev. , 610 (2010).[23] Y.-H. Su, Y. Chuang, C.-Y. Liu, J.-Y. Li, and T.-M. Lu,Phys. Rev. Materials , 044601 (2017).[24] F. K. LeGoues, R. Rosenberg, T. Nguyen, F. Himpsel,and B. S. Meyerson, J. Appl. Phys. , 1724 (1989).[25] D. Laroche, S.-H. Huang, Y. Chuang, J.-Y. Li, C. W. Liu,and T. M. Lu, Appl. Phys. Lett. , 233504 (2016).[26] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, andJ. Luitz, WIEN2k: An Augmented Plane Wave plus LocalOrbitals Program for Calculating Crystal Properties (T.U. Vienna, Austria, Vienna, 2001).[27] G. K. H. Madsen, P. Blaha, K. Schwarz, E. Sj¨ostedt, andL. Nordstr¨om, Phys. Rev. B , 195134 (2001).[28] O. K. Andersen, Phys. Rev. B , 3060 (1975).[29] E. Sj¨ostedt, L. Nordstr¨om, and D. J. Singh, Solid StateCommun. , 15 (2000).[30] F. Tran and P. Blaha, Phys. Rev. Lett. , 226401(2009).[31] A. D. Becke and E. R. Johnson, J. Chem. Phys. ,221101 (2006).[32] J. Wortman and R. Evans, J. Appl. Phys. , 153 (1965).[33] G. L. Bir and G. E. Pikus, Symmetry and Strain InducedEffects in Semiconductors (Wiley, New York, 1974).[34] C. G. Van de Walle, Phys. Rev. B , 1871 (1989).[35] J. M. Luttinger, Phys. Rev. , 1030 (1956).[36] M. V. Fischetti and S. E. Laux, J. Appl. Phys. , 2234(1996). [37] G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki,Phys. Rev. B , 3241 (1983).[38] E. Marcellina, A. R. Hamilton, R. Winkler, and D. Cul-cer, Phys. Rev. B , 075305 (2017).[39] Y. Tokura, W. G. van der Wiel, T. Obata, andS. Tarucha, Phys. Rev. Lett. , 047202 (2006).[40] R. Winkler, Spin-Orbit Coupling Effects in Two- Di-mensional Electron and Hole systems (Springer, Berlin,2003).[41] V. Fock, Zeitschrift f¨ur Physik , 446 (1928).[42] C. G. Darwin, Mathematical Proceedings of the Cam-bridge Philosophical Society , 86 (1931).[43] E. Kawakami, P. Scarlino, D. R. Ward, F. R. Braakman,D. E. Savage, M. G. Lagally, M. Friesen, S. N. Copper-smith, M. A. Eriksson, and L. M. K. Vandersypen, Na-ture Nano. , 666 (2014).[44] J. Salfi, J. A. Mol, D. Culcer, and S. Rogge, Phys. Rev.Lett. , 246801 (2016).[45] J. H. Davies, The Physics of Low-Dimensional Semicon-ductors (Cambridge University Press, Cambridge, 1998).[46] Y. Hu, H. O. H. Churchill, D. J. Reilly, J. Xiang, C. M.Lieber, and C. M. Marcus, Nature Nano. , 622 (2007).[47] O. E. Dial, M. D. Shulman, S. P. Harvey, H. Bluhm,V. Umansky, and A. Yacoby, Phys. Rev. Lett. ,146804 (2013).[48] T. Hayashi, T. Fujisawa, H.-D. Cheong, Y. H. Jeong, andY. Hirayama, Phys. Rev. Lett. , 226804 (2003).[49] X. Hu, Phys. Rev. B , 165322 (2011).[50] J. K. Gamble, M. Friesen, S. N. Coppersmith, and X. Hu,Phys. Rev. B86