Vanishing Zeeman energy in a two-dimensional hole gas

P. Del Vecchio,1, ∗ M. Lodari,2, ∗ A. Sammak,3 G. Scappucci,2, † and O. Moutanabbir1, ‡

1Department of Engineering Physics, Ecole Polytechnique de Montreal,Montreal, C.P. 6079, Succ. Centre-Ville, Montreal, Quebec, Canada H3C 3A7

2QuTech and Kavli Institute of Nanoscience, TU Delft,P.O. Box 5046, 2600 GA Delft, The Netherlands

3QuTech and Netherlands Organisation for Applied Scientific Research (TNO),Stieltjesweg 1, 2628 CK Delft, The Netherlands

A clear signature of Zeeman split states crossing is observed in Landau fan diagram of strainedgermanium two-dimensional hole gas. The underlying mechanisms are discussed based on a pertur-bative model yielding a closed formula for the critical magnetic fields. These fields depend stronglyon the energy difference between the top-most and the neighboring valence bands and are sensitiveto the quantum well thickness, strain, and spin-orbit-interaction. The latter is a necessary featurefor the crossing to occur. This framework enables a straightforward quantification of the hole-stateparameters from simple measurements, thus paving the way for its use in design and modelling ofhole-based quantum devices.

I. INTRODUCTION

The inherently large and tunable spin-orbit interaction(SOI) energies of holes and their reduced hyperfine cou-pling with nuclear spins are behind the surging interest inhole spin qubits with fast all-electrical control.1–5 Holescan also host superconducting pairing correlations, a keyingredient for the emergence of Majorana zero modes6–10

for topological quantum computing. Because of its at-tractive properties,1,11–22 strained Ge low-dimensionalsystem has been proposed as an effective building blockto develop these emerging quantum devices. Interest-ingly, the simplicity of this system makes it a textbookmodel to uncover and elucidate subtle hole spin-relatedphenomena leading, for instance, to the recent observa-tion of pure cubic Rashba spin-orbit coupling.23

Measuring Zeeman splitting (ZS) of hole states underan external magnetic field has been central in probinghole spin properties, as it is directly related to the holeg-factor, which is itself strongly influenced by the un-derlying SOI, strain, symmetry, and confinement.24,25 InIII-V semiconductors,24,26–38 hole spin splitting dependsnonlinearly on the out-of-plane magnetic field strengthB, causing Landau level crossings/anti-crossings28,39 andZeeman crossings/anti-crossings.12,40,41 The nonlinearityis usually modeled by a quadratic-in-field contribution toZS,24 which owes its existence to valence band mixing.Depending on the sign of the splitting, Zeeman energycan even vanish at some finite critical field, Bc. Theoret-ical studies attribute these nonlinearities to the mixingof heavy-hole (HH) and light-hole (LH) bands at finiteenergy.26 Alongside with valence band mixing, Rashbaand Dresselhaus spin-orbit coupling were also shown toinfluence the crossing field, due to the lattice inversionasymmetry and the confining potential.

Detailed mechanisms of ZS of hole states are yet to beunravelled and understood and furthermore, ZS treat-ments for zinc-blende or diamond crystals that explicitlyconsider strain and SOI strength remain conspicuouslymissing in literature. Note that in early calculations41

of Landau levels in Ge/SiGe quantum well (QW) to in-terpret cyclotron resonance experiments in Ref. 42, thecrossing of spin split states within the first HH sub-band was present and the corresponding field positionwas found to be sensitive to the strength of spin-orbitcoupling. In that work, the authors insisted on the im-portance of including explicitly the split-off hole band,which was required to achieve a good agreement with ex-periments. Crucially, studies that included both strainand SOI were diagonalizing numerically the full k · pmatrix.29,41 However, this mathematical rigor comes atthe expense of identifying the physics governing the non-linearities in ZS.

To overcome these limitations and elucidate the un-derlying mechanisms of ZS, herein we uncover the clearsignature of ZS crossings in a Ge high-mobility two-dimensional hole gas (2DHG). We also derive a the-oretical framework describing the crossing of Zeemansplit states that includes explicitly the SOI strength andstrain. A closed formula for the crossing fields is obtainedand validated by experiment. In addition to establishingthe key parameters in Zeeman crossings, this analysis alsoprovides a toolkit for a direct quantification from sim-ple magnetotransport measurements of important physi-cal quantities including HH out-of-plane g-factor, HH-LHsplitting, and cubic Rashba spin-orbit coefficient.

II. EXPERIMENTAL DETAILS

The investigated 2DHG consists of a Ge/SiGe het-erostructure including a strain-relaxed Si0.2Ge0.8 buffersetting the overall lattice parameter, a compressively-strained Ge QW, and a Si0.2Ge0.8 barrier separating theQW from a sacrificial Si cap layer. The growth was car-ried out in an Epsilon 2000 (ASMI) reduced pressurechemical vapor deposition reactor on a 100 mm n-typeSi(001) substrate. The growth sequence starts with thedeposition of a Si0.2Ge0.8 virtual substrate. This virtualsubstrate is obtained by growing a 1.6µm strain-relaxed

arX

iv:2

006.

0010

2v2

[co

nd-m

at.m

es-h

all]

19

Oct

202

0

2

-18

-16

-14

-12

-10

-8

-6

-4

E (m

eV)

9876543210B (T)

(b) 6420Δρxx/ρ0

2

ν = 1

34

sGe

Ti/Au

Al2O3

Si0.2Ge0.8

(a)

0.2 mm

FIG. 1. (a) Optical micrograph of a Hall-bar shaped Ge/SiGeheterostructure field effect transistor and cross section of thegate stack and active regions of the strained Ge/SiGe het-erostructure below the red cut. The strained Ge (sGe) quan-tum well is 16 nm thick and the Si0.2Ge0.8 barrier on top is17 nm thick. (b) Landau level fan diagram reporting the mag-netoresistance ∆ρxx/ρ0 = (ρxx − ρ0)/ρ0 as a function of out-of-plane magnetic field B and energy E. Labels of fillingfactors ν = 1 – 4 are shown.

Ge buffer layer, a 1.6µm reverse-graded Si1−xGex layerwith final Ge composition x = 0.8, and a 500 nm strain-relaxed Si0.2Ge0.8 buffer layer. A 16 nm compressively-strained Ge quantum well is then grown on top of theSi0.2Ge0.8 virtual substrate, followed by a strain-relaxed17 nm-thick Si0.2Ge0.8 barrier. An in-plane compressivestrain ε‖ = −0.63% is found in the QW via X-ray diffrac-

tion measurements.12 A thin (< 2 nm) sacrificial Si capcompletes the heterostructure. This cap is readily oxi-dized upon exposure to the cleanroom environment afterunloading the Ge/SiGe heterostructure from the growthreactor.

Hall-bar field effect transistors (H-FETs) are fabri-cated and operated with a negatively biased gate to ac-cumulate a 2D hole gas into the QW and tune the carrierdensity. Fig. 1a shows an optical micrograph of the H-FET and a cross-section schematic of the active layersand the gate stack. A 170 nm deep trench mesa is dry-etched around the Hall-bar shaped H-FET in order toisolate the bonding pads from the device. The sample isdipped in HF to remove the native oxide prior to a 60 nmPt layer deposition via e-beam evaporation. Ohmic con-

tacts are obtained by diffusion of Pt into the quantumwell occurring during the atomic layer deposition of a30 nm Al2O3 dielectric layer at a temperature of 300 ◦C.Finally, a 10/200 nm-thick Ti/Au gate layer is deposited.An optimized Si0.2Ge0.8 barrier thickness of 17 nm waschosen, which is thin enough to allow for a large satura-tion carrier density12 (up to 7.5× 1011 cm−2), while pro-viding sufficient separation to reduce scattering of carri-ers in the QW from remote impurities,40 leading to largehole maximum mobility (2.6× 105 cm−2). Large densityrange and high mobility are key ingredients to observeLandau level fan diagrams in magnetotransport with theclarity required to reveal subtle spin-related features.

In the magnetotransport studies, the longitudinal andtransversal (ρxx and ρxy) component of the 2DHG re-sistivity tensor were measured via a standard four-probelow-frequency lock-in technique. The measurements arerecorded at a temperature of T = 260 mK, measured atthe cold finger of a 3He dilution refrigerator. A source-drain voltage bias Vsd = 0.1 mV is applied at a frequencyof 7.7 Hz. The magnetoresistance characterization of thedevice is performed by sweeping the voltage gate Vg andstepping B with a resolution of 15 mV and 25 mT, re-spectively. The energy E is obtained using the relationE = pπh2/m∗, where we obtain the carrier density pby Hall effect measurements at low B and we use theeffective mass m∗ measured as a function of density insimilar heterostructures.40 The ρxx vs. energy profiles inthe upper panels of Figs. 3(a)-3(d) have been smoothedfor clarity by using a Matlab routine based on Savitzky-Golay filtering method.

III. MAGNETOTRANSPORT STUDIES OFSTRAINED GE 2DHG

The fan diagram in Fig. 1b shows the normalized mag-netoresistance oscillation amplitude ∆ρxx/ρ0 = (ρxx −ρ0)/ρ0 as a function of energy and out-of-plane externalmagnetic field B aligned along the growth direction zand perpendicular to the 2DHG plane, where ρ0 is theρxx value at B = 0. The Zeeman split energy gap, corre-sponding to odd integer filling factors ν, deviates from itslinear dependence on B, vanishes when the magnetic fieldreaches a critical value Bc, and then reopens at higherB values. We clearly observe the associated crossing ofZeeman split states for odd integers ν = 3, 5, 7, and 9.Partial signatures of Zeeman crossings occurring at sim-ilar magnetic fields were observed in earlier studies,12,40

albeit the fan diagram measurements were limited in den-sity range12 or affected by thermal broadening.40 Theseobservations point to an underlying mechanism that isindependent of the QW position with respect to the sur-face gate.

3

IV. THEORETICAL FRAMEWORK FOR HOLEDISPERSION IN STRAINED GE 2DHG

To identify the mechanisms behind the non-linearitiesin ZS and the parameters affecting the crossing field, wedeveloped a perturbative model to describe the hole dis-persion as a function of the out-of-plane magnetic field.The model assumes an abrupt and infinite band offset be-tween the QW and its barriers and is based on a 6-bandk · p Hamiltonian for HH, LH and split-off (SO) bands.The total Hamiltonian H for the hole dispersion is writ-ten as43 : H = Hk +Hε +HSO +HB + V where Hk is afunction of the wavevector operator k = (kx, ky, kz), Hε

is the Bir-Pikus Hamiltonian and depends on the straintensor components εij , HSO is the spin-orbit term pro-portional to the spin-orbit energy ∆ and HB includes theinteraction of the free electron spin with the magneticfield. V is the infinite well potential for a square well ofwidth L. We consider QWs grown along [001] directionand subjected to biaxial bi-isotropic strain. Thus, εij = 0if i 6= j, εxx = εyy ≡ ε‖ and εzz = −D001ε‖, where D001

is the Poisson ratio and ε‖ is the in-plane lattice strain.We first rewrite the total Hamiltonian H in two terms :

H = H0(ε‖; kz) + H ′(n,B; kz), where the integer n ≥ 1labels the spin-split Landau pairs such that ν = 2n−1 atcrossings. The eigenstates of H0 consist of pure HH sub-bands of energy EHH

l and two superpositions of LH andSO holes of energy Eηl . Here, η = {+,−} is a generic la-bel to distinguish the two orthogonal LH-SO states andl ≥ 1 is the subband index. The perturbation H ′ in-troduces the magnetic field and is eliminated to secondorder by a Schrieffer-Wolff transformation, resulting inan effective Hamiltonian for the 2-fold HH subband. Re-markably, the resulting effective 2×2 Hamiltonian for theHH subband does not couple spin-up (+) and spin-down(−) projections. The HH dispersion as a function of B isthus simply the diagonal entries of the effective matrix.We have

E(2)+,l,n(B) = EHH

l + 3n(n+ 1) (κ− Fl)µBB

2

B∗l− [(2n− 1) (γ1 + γ2) + 3κ− 6nFl]µBB,

(1)

E(2)−,l,n(B) = EHH

l + 3(n− 2)(n− 1) (κ− Fl)µBB

2

B∗l− [(2n− 1) (γ1 + γ2)− 3κ− 6(n− 1)Fl]µBB,

(2)

with

B∗l =κ− Fl

µB (γ2 + γ3)2

[∑η=±

(lηl +

√2sηl)2

EHHl − Eηl

]−1(3a)

≈ κ− FlµB (γ2 + γ3)

2

(EHHl − ELH

l

)(3b)

and

Fl =32α0γ

23

L2

∞∑j=1j 6=l

[1− (−1)l+j

]l2j2

(l2 − j2)2

∑η=±

(lηj − s

ηj /√

2)2

EHHl − Eηj

(4a)

≈ 32α0γ23

L2

∞∑j=1j 6=l

1

EHHl − ELH

j

[1− (−1)l+j

]l2j2

(l2 − j2)2 . (4b)

Here, µB is the Bohr magneton, γi and κ are the Lut-tinger parameters, α0 = h2/(2m0) with m0 the free elec-tron mass and lηl and sηl are respectively the LH andSO contributions of the lth η subband. The character-istic field B∗l controls the crossing positions and is fill-ing factor-independent, while Fl indicates the couplingstrength between the HH subband and neighboring ηstates. As we focus on the HH ground subband (l = 1),l subscripts will be omitted for simplicity. The obtained

Zeeman splitting energy EZ ≡ E(2)−,n(B)−E(2)

+,n(B) of thenth spin-split Landau pair is :

EZ = 6(κ− F )µBB

[1− (2n− 1)

(B

B∗

)]. (5)

Solving for EZ = 0 results in a second order approxi-mation for the filling factor-dependent crossing field Bc :

B(2)c (n) =

B∗

2n− 1. (6)

The energy difference that separates the HH subbandedge from the energy at a crossing position can also befound from the second order equations. When n → ∞(or ν →∞) this energy difference is independent of n :

∆E =

[γ1 + γ2 −

3

4(κ+ 3F )

]µBB

∗. (7)

Equation (5) also yields the HH weak-field g-factor :

g∗ = 6(κ− F ). (8)

The approximations (3b) and (4b) hold only when SOIis large enough so that the SO band can be neglectedfrom the k · p framework. An explicit criterion for this is(Appendix B) :

∆� α0γ2

(πL

)2+

(−b)2

(1 +D001)|ε‖|, (9)

where b is a valence band deformation potential.In addition to the perturbation scheme, H is also nu-

merically diagonalized by projecting it into the positionbasis via the substitution kz → −i∂/∂z, in which the

4

12

10

8

6

4

2

0

Bc

(T)

1615141312111098765L (nm)

(a) (b)

-20

-15

-10

-5

0

E (

meV

)

109876543210B (T)

ν = 1

2

4

NumericalFinite ∆Infinite ∆

NumericalFinite ∆Infinite ∆

ԑ// = 0% - 0.3% - 0.6% - 0.9%

FIG. 2. (a) Fan diagram of the ground HH subband in a 16 nm Ge well subject to 0.6% compressive strain. Solid curves are thedispersion obtained from the numerical solution of H, while the dashed curves are obtained from the second order dispersionassuming finite or infinite SOI respectively. Circles indicate the Zeeman crossings. Filling factors ν are also indicated. (b)ν = 3 crossing field as a function of the well thickness at various strain values obtained from the numerical solution of H (solidcurves) and through Eq. (6) assuming finite or infinite SOI (dashed curves).

z-derivative is implemented by finite differences over thesimulation domain. A constant mesh grid size of 0.01 nmis used for every diagonalization. The Matlab eigs()routine is used to retrieve the desired subset of eigenval-ues. The Ge Luttinger parameters γ1,2,3 and deformationpotentials are taken from Ref. 44, while the parameter κis taken from Ref. 45. Explicit matrix representations ofH0 and H ′ are presented in Appendix A. See Appendix Bfor additional details on the eigenvalues and eigenvectorsof H0.

Let us now test the accuracy of the perturbative modelcompared to the dispersion given by solving numericallyH. We take Ge as the QW material with width L andstrain ε‖ as free parameters. Since Ge has a rather high

spin-orbit energy ∆ = 260 meV,46 it is worthwhile to lookalso at the behavior of the model with approximations(3b) and (4b). We also focus on relaxed or compressivelystrained wells, which always result in a HH-like valenceband edge. The calculated fan diagram of the ground HHsubband is displayed in Fig. 2a for a 16 nm-thick well withε‖ = −0.6%, similar to the system analyzed in Fig. 1. As-suming finite ∆, the model reproduces perfectly well thenumerical fan diagram up to ∼ 2 T, which implies that6(κ − F ) is a very accurate approximation for the HHg-factor at low fields. As the magnetic field increases,quadratic terms in B become more important and thedispersions eventually cross. The dispersion of a statewith spin-up projection in a given spin-split Landau pairalways has a bigger curvature than the spin-down one,which can be straightforwardly inferred from the coeffi-cients n(n + 1) and (n − 2)(n − 1) in (1) and (2). Forthat reason, a Zeeman crossing cannot occur, at least tosecond order, if the spin-up state lies closer to the bandgap than the spin-down one. Crossing fields are indicated

in Fig. 2a for filling factors ν = 3 and ν = 5. The nu-merical solution of H gives a crossing field Bc = 7.27 Tfor ν = 3, whereas the second order formula (Eq. (6))

gives B(2)c = 5.04 T. Here the second order approxima-

tion underestimates Bc as it diverges from the numericaldispersion before the crossing. When assuming ∆→∞,however, the dispersion diverges less dramatically thanits finite SOI counterpart and instead overestimates thecrossing field. Assuming an infinite SOI for this particu-lar system turns out to be a good approximation, becausethe right-hand side of (9) equals 21.2 meV, which is muchsmaller than spin-orbit gap in Ge.

Fig. 2b depicts the behavior of the crossing field asa function of the well thickness and strain, with andwithout the assumption of an infinite SOI. The crossing

field Bc is well approximated by B(2)c for a well thickness

> 10 nm with reduced strain levels, as in our experiments.For narrower and highly strained wells, third or higherperturbative terms become more important. These couldbe included in the model, but at the cost of extremelycumbersome equations, even with infinite SOI. On the

other hand, for ∆ → ∞, B(2)c misses completely the in-

crease of the crossing field for thin wells, which highlightsthe explicit role of the SOI strength. This is consistentwith criterion (9) : thin wells increase the right-hand sidein (9) as 1/L2, thus requiring ∆ to be even larger for thiscriterion to be satisfied.

V. DISCUSSION

From the present model, we see that Zeeman cross-ings still occur under the assumption of an infinite QW

5

(e)

(a) (b)

(c) (d)

3001500ρxx (Ω/sq.)

ExperimentTheory B*/(2n-1)

400

200

0ρ xx

(Ω/s

q.)

9

8

ν

-18-16-14-12-10E (meV)

Bc = 2.90 T

400

200

0ρ xx

(Ω/s

q.)

7

6

ν

-18-16-14-12-10E (meV)

Bc = 3.65 T

30

20

10

0ρ xx

(Ω/s

q.)

3

2

ν

-18-16-14-12-10E (meV)

Bc = 8.36 T400

200

0ρ xx

(Ω/s

q.)

5

4

ν

-18-16-14-12-10E (meV)

Bc = 4.95 T

10

8

6

4

2

0

Bc

(T)

10987654321n

FIG. 3. Experiment vs. Theory. (a)-(d) ρxx as a functionof filing factor ν and energy E around the crossings of Zeemansplit states. The upper parts of each panel shows a cross-section at odd filling factors ν = 3, 5, 7, 9. (e) Experimentalcrossing fields (dots) for ν = 3, 5, 7, 9, 11, 13, 15, 17 fitted usingEq. (6) (solid line). The fitting parameter B∗ = 25.258 T.

(no barrier effects), an infinite band gap (6-band k · p),and even an infinite spin-orbit gap (4-band k · p for HHand LH). Consequently, LH-HH mixing plays a crucialrole in the crossing of spin-split states. Our assumptionsalso imply that structure inversion asymmetry (SIA) hasno role in the observed crossing in ZS energy. SIA is in-deed suppressed in infinite wells without external electricfields. Thus, Rashba SOI does not have a dominant effecton the value of Bc. The role of SOI and strain is, however,more evident in Eqs. (6) and (3). SOI and strain affect

B(2)c mostly through the energy splitting EHH − Eη and

the parameter F . Compressive strain typically increasesEHH − Eη, which explains the increase of Bc at highercompressive strain. SOI also increases EHH−Eη, mainlythrough the spin-orbit energy ∆ for η = + or through theout-of-plane effective mass for η = −. At ∆ = 0 and any

strain, the HH subbands share the same spectrum as theη = + or η = − states. Eq. (3) then gives B∗ = 0 henceno Zeeman crossing occurs. SOI lifts this degeneracy be-tween HH and η states and thus allows the existence ofZeeman crossings.

The experimental observation of Zeeman crossings arefurther highlighted by plotting portions of the fan dia-gram from Fig. 1b as a function of energy and filling fac-tor (Fig. 3a-d). The upper part of each panel shows theρxx as a function of the energy E at odd-integer valuesof filling factors from ν = 3 to 9. Fingerprints of Zee-man crossing are observed for filing factors up to ν = 17.In addition to describing the crossings in Zeeman splitstates, the theoretical framework described above also al-lows a straightforward evaluation of several parameters.First, we fit the crossing fields extracted from Fig. 3a-d(ν = 3, 5, ..., 17) with Eq. (6) using B∗ as the sole fittingparameter. This yields B∗ = 25.258 T and the cross-ing fields obtained from Eq. (6) match the experimentalvalues with a relative error < 4% for ν = 3, 5, 7, 9, 11and < 10% for ν = 13, 15, 17 (Fig. 3e). Zeeman cross-ings also approach a fixed energy value as ν increases,as demonstrated in Eq. (7). From Fig. 1(b), we have∆E ≈ 17 meV. Knowing B∗ and ∆E gives the value ofF , leading to HH effective mass and weak-field g-factor.A rearrangement of Eq. (7) gives

F =4

9

(γ1 + γ2 −

∆E

µBB∗

)− κ

3≈ 1.52. (10)

From Eqs. (8) and (10), we extract g∗ = 11.35, whichis close to 12.9 obtained by solving H numerically. Anexpression for the subband-edge HH in-plane effectivemass m∗ involving the parameter F can also be de-rived by inserting Eq. (5) from Ref. 47 into Eq. (8) :

m∗/m0 = (γ1 + γ2 − 3F )−1 ≈ 0.077. This value is also

close to those reported in the literature at similar holedensity.40,48 A close relation exists between the crossingfields, the HH g-factor and the HH-η splitting (Eqs. (3)and (3b)). Knowing two of these quantities is enough toobtain the third. For the system described in Fig.1, thecriterion (9) is also satisfied, thus the HH-LH splitting isfound directly from Eq. (3b) :

EHH − ELH =6 (γ2 + γ3)

2µBB

∗

g∗≈ 76.0 meV. (11)

A numerical solution of H yields a HH-LH splitting of62.8 meV. This value does not change significantly whenan effective out-of-plane electric field is introduced in H.This is expected from square QWs whose HH-LH split-ting is dominated by strain and quantum confinement.23

For that reason, we assume that the HH-LH splittingdoes not change with hole concentration, or applied gatevoltages. From the HH-LH splitting energy (Eq. (11)),one can finally estimate the cubic Rashba coefficient α3 :

6

α3 =eα2

0γ3

12 (γ2 + γ3)3

(g∗

µBB∗

)2

≈ 4.25× 105 e A4, (12)

where e is the elementary charge. α3 appears inthe cubic Rashba SOI Hamiltonian of HH states23 :H3 = β3i(k

3−σ+ − k3+σ−), where k± = kx ± iky and

σ± = (σx± iσy)/2 with σx,y the Pauli spin matrices, andβ3 = α3Ez, with Ez = ep/ε the effective out-of-planeelectric field in the accumulation mode 2DHG,25 p thehole density and ε the Ge dielectric constant. The ob-tained α3 is almost twice as large as the one obtained forGe QW in Ref. 23, which had a bigger HH-LH splitting of110 meV. As mentioned above, we expect α3 to be inde-pendent of the gate voltage or hole concentration, sinceit depends mostly on the HH-LH splitting. The Zeemancrossings appear at a density p ∼ 6.1× 1011 cm−2, corre-

sponding to Ez ≈ 6.8×10−4 V A−1

(by taking ε = 16.2ε0

for Ge), which yields β3 ≈ 290 eV A3. Note that α3

or β3 are hitherto hard to measure in these high mo-bility systems with established methodologies : weakanti-localization measurements are impractical due to thesmall characteristic transport field BL associated withµm-scale mean free paths49,50 ; Shubnikov-de Haas os-cillations lack sufficient spectral resolution before onsetof ZS to resolve the beatings associated with spin-splitsubbands.17

VI. CONCLUSION

In summary, Zeeman energy crossing of HH states isobserved in a Ge 2DHG under out-of-plane magneticfields and discussed within a perturbative model describ-ing the hole dispersion. Only second order perturbationin the magnetic field is necessary to describe the crossingin which SOI emerges as an essential feature. However,our analysis indicates that SIA has no effective role. Ad-ditionally, this analysis also provides a straightforward

framework to evaluate several physical parameters defin-ing the hole states from simple magnetotransport mea-surements. Crucially, the detailed knowledge of param-eters such as the effective g-factor, the in-plane effectivemass, and the cubic Rashba coefficient of the underly-ing material platform will provide the necessary input tofurther advance design and modelling of hole spin qubitsand other hole-based quantum devices.

Acknowledgment

O. M. acknowledges support from NSERC Canada(Discovery, SPG, and CRD Grants), Canada ResearchChairs, Canada Foundation for Innovation, Mitacs,PRIMA Quebec, Defence Canada (Innovation for De-fence Excellence and Security, IDEaS), and NRC Canada(New Beginnings Initiative). G. S. and M. L. acknowl-edge financial support from The Netherlands Organiza-tion for Scientific Research (NWO).

Data availability

Datasets supporting the findings of this studyare available at 10.4121/uuid:c64b0509-2247-4d51-adc0-90e361b928a4

Appendix A: Hamiltonian matrix representation

The matrix representation of H is presented in thefollowing |j,m〉 angular momentum basis25 :

{∣∣∣∣32 , 3

2

⟩,

∣∣∣∣32 , 1

2

⟩,

∣∣∣∣32 ,−1

2

⟩,

∣∣∣∣32 ,−3

2

⟩,

∣∣∣∣12 , 1

2

⟩,

∣∣∣∣12 ,−1

2

⟩}The magnetic field-free Hamiltonian H0 is

H0 = −α0

(γ1 − 2γ2) k2z 0 0 0 0 0

(γ1 + 2γ2) k2z 0 0 −√

8γ2k2z 0

(γ1 + 2γ2) k2z 0 0√

8γ2k2z

† (γ1 − 2γ2) k2z 0 0γ1k

2z + ∆/α0 0

γ1k2z + ∆/α0

+ 16×6(2−D001)avε‖ +

1 0 0 0 0 0

−1 0 0√

2 0

−1 0 0 −√

2† 1 0 0

0 00

(1 +D001)bε‖,

7

where av and b are deformation potentials. Its eigen-states and eigenvalues are described in Appendix B. Forperpendicular-to-plane magnetic fields it is convenient towrite kx and ky in terms of the ladder operator a :

kx =1√2λ

(a+ a†

)ky =

i√2λ

(a− a†

),

where the magnetic length λ =√h/eB, e being the

elementary charge. Also, [a, a†] = 1, a |N〉 =√N |N − 1〉

and a†a |N〉 = N |N〉, where N is an integer. In the axialapproximation the vector

|N − 1〉 |l〉3/2,3/2|N〉 |l〉3/2,1/2

|N + 1〉 |l〉3/2,−1/2|N + 2〉 |l〉3/2,−3/2

|N〉 |l〉1/2,1/2|N + 1〉 |l〉1/2,−1/2

is an eigenstate of H, where 〈z | l〉j,m is the spatial en-

velope function of the hole component with angular mo-mentum |j,m〉 and subband index l. This ansatz allowsto write H as a function of the quantum numbers N andto eliminate the ladder operators a.51 The perturbationH ′ takes the form (with the electron g-factor g0 = 2) :

H ′ = −α0

λ2

(2N − 1)γ+ + 3κ −2λ

√6Nγ3kz −

√3N(N + 1)γ

(2N + 1)γ− + κ 0(2N + 3)γ− − κ

† . . .

0 2λ√

3Nγ3kz√

6N(N + 1)γ

−√

3(N + 1)(N + 2)γ√

2[(2N + 1)γ2 + κ+ 1] −6λ√N + 1γ3kz

2λ√

6(N + 2)γ3kz −6λ√N + 1γ3kz −

√2[(2N + 3)γ2 − κ− 1]

(2N + 5)γ+ − 3κ −√

6(N + 1)(N + 2)γ 2λ√

3(N + 2)γ3kz(2N + 1)γ1 + 2κ+ 1 0

(2N + 3)γ1 − 2κ− 1

where N ≥ −2, γ± = γ1 ± γ2 and γ = γ2 + γ3. Notethat α0/λ

2 = µBB.

Appendix B: Eigenvalues and eigenvectors of H0

If B = 0 (and kx = ky = 0) the subbands are eitherpure HH states or pure spin-1/2 states (LH-SO superpo-sition). The subband eigenstates are

|HH, σ, l〉 =

∣∣∣∣32 , 3σ

2

⟩|l〉 (B1)

|η, σ, l〉 =

(lηl

∣∣∣∣32 , σ2⟩

+ σsηl

∣∣∣∣12 , σ2⟩)|l〉 , (B2)

where σ = {+,−} is the pseudo-spin index (spin-up-down respectively). We have

lηl =ξ− + δ−η√

(ξ− + δ−η)2

+ 8ξ2−

, sηl =−√

8ξ−√(ξ− + δ−η)

2+ 8ξ2−

,

(B3)

with

δ± =∆

2±√

(ξ− + ∆/2)2

+ 8ξ2−, (B4)

ξ± = −α0

[(1± 1

2

)γ1 + γ2

](lπ

L

)2

+

[(1± 1

2

)(2−D001)av − (1 +D001)b/2

]ε‖.

(B5)

The spatial part is

〈z | l〉 =

√2

Lsin

(lπz

L

). (B6)

The energy spectrum for the HH and η-states is

EHHl = −α0 (γ1 − 2γ2)

(lπ

L

)2

+ [(2−D001)av + (1 +D001)b] ε‖

(B7)

Eηl = ξ+ − δη. (B8)

8

Infinite SOI regime is reached when ∆� |ξ−|. Undercompressive strain this expands to

∆� α0γ2

(lπ

L

)2

+(−b)

2(1 +D001)|ε‖|. (B9)

Assuming ∆ → ∞ is a good approximation only if ∆satisfies criterion (B9). The square root in δ± can thenbe eliminated by a Taylor expansion, and the followingresults immediately follow :

δ− = −ξ−, δ+ = ∆ + ξ−,

l−l = 1, s−l = 0,

l+l = 0, s+l = 1.

Consequently,

|−, σ, l〉 → |LH, σ, l〉 =

∣∣∣∣32 , σ2⟩|l〉 (B10)

|+, σ, l〉 → |SO, σ, l〉 =

∣∣∣∣12 , σ2⟩|l〉 (B11)

E−l → ELHl = −α0 (γ1 + 2γ2)

(lπ

L

)2

+ [(2−D001)av − (1 +D001)b] ε‖

(B12)

E+l → ESO

l = −α0γ1

(lπ

L

)2

−∆ + (2−D001)avε‖,

(B13)

corresponding to a pure LH and pure SO spectrum.

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