In-plane magnetic field effect on hole cyclotron mass and g z factor in high-mobility SiGe/Ge/SiGe structures
I.L. Drichko, V.A. Malysh, I.Yu. Smirnov, L.E. Golub, S.A. Tarasenko, A.V. Suslov, O.A. Mironov, M. Kummer, H. von Känel
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov In-plane magnetic field effect on hole cyclotron mass and g z factor in high-mobilitySiGe/Ge/SiGe structures I.L. Drichko, V.A. Malysh, I.Yu. Smirnov, L.E. Golub, S.A. Tarasenko, A.V. Suslov, O.A. Mironov,
3, 4
M. Kummer, and H. von K¨anel A. F. Ioffe Physico-Technical Institute of Russian Academy of Sciences, 194021 St.Petersburg, Russia National High Magnetic Field Laboratory, Tallahassee, FL 32310, USA Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom International Laboratory of High Magnetic Fields and Low Temperatures, 53-421 Wroclaw, Poland Laboratorium f ¨ u r Festk¨orperphysik ETH Z¨urich, CH-8093 Z¨urich Switzerland The high-frequency (ac) conductivity of a high quality modulation doped GeSi/Ge/GeSi singlequantum well structure with hole density p =6 × cm − was measured by the surface acousticwave (SAW) technique at frequencies of 30 and 85 MHz and magnetic fields B of up to 18 T in thetemperature range of 0.3 – 5.8 K. The acoustic effects were also measured as a function of the tiltangle of the magnetic field with respect to the normal of the two-dimensional channel at T =0.3 K.It is shown, that at the minima of the conductivity oscillations, holes are localized on the Fermilevel, and that there is a temperature domain in which the high-frequency conductivity in the bulkof the quantum well is of the activation nature. The analysis of the temperature dependence of theconductivity at odd filling factors enables us to determine the effective g z factor. It is shown thatthe in-plane component of the magnetic field leads to an increase of the cyclotron mass and to areduction of the g z factor. We developed a microscopic theory of these effects for the heavy-holestates of the complex valence band in quantum wells which describes well the experimental findings. PACS numbers: 73.63.Hs, 73.50.Rb
I. INTRODUCTION
Modulation doped SiGe/Ge/SiGe structures with atwo-dimensional (2D) hole gas are attractive systemsfor both fundamental and applied studies since they arecompatible with silicon-based technology and, at thesame time, have a record high hole mobility among allgroup IV semiconductors. As compared to silicon-basedmetal-oxide-semiconductor field-effect-transistor struc-tures, they are also characterized by a strong spin-orbitcoupling and a large and strongly anisotropic g -factor,which is of interest for the study of spin-related phe-nomena. However, the details of the band structureand the quantum transport in p -SiGe/Ge/SiGe systemshave not yet been sufficiently explored. It is known, thatdue to the lattice constant mismatch the 2D hole chan-nel is located in strained Ge so that, the ground sub-band is formed by heavy-hole (hh) states while the light-hole (lh) subband is split off by a hundred meV. Thesplitting should suppress the hh-lh mixing and lead to astrong anisotropy of the hh g -factor tensor with vanish-ingly small in-plane component g k . One can therefore ex-pect that the transport properties of the hole channel inSiGe/Ge/SiGe are determined by the normal componentof the magnetic field only, as they are in a strained p -type channel of A III B V semiconductors. Such a behav-ior has been observed in p -Ge/SiGe multilayer structuresby studying the resistivity oscillations in tilted magneticfields up to a tilt angle of 60 ◦ . What concerns us aboutthe absolute value of the hh out-of-plane g -factor, | g z | , isthat the data available in literature vary from 20.4 forbulk Ge, to 16.5 for strained Ge, to 14.2 and 5.8 for p -Ge/GeSi multilayers, and to 1 for a single quantum well. These drastically different values may be the re-sult of many-body effects and spectrum nonparabolicity which are also responsible for the observed dependenceof the in-plane effective mass on the hole density. In this paper, we report a comprehensive study ofthe high-frequency conductivity of the high-mobility 2Dhole gas embedded in a SiGi/Ge/SiGe structure in tiltedmagnetic fields. The measurements are carried out bymeans of the contactless surface acoustic wave tech-nique. It probes the “bulk” electric properties of thetwo-dimensional system and provides information on the2D hole energy spectrum unaffected by chiral edges whichplay a key role in conventional four-probe measurementsof the quantum Hall effect. The experimental data allowus to determine the hole cyclotron mass from the tem-perature dependence of Shubnikov-de Haas (SdH) oscil-lations and the effective g z factor for our samples. Wefind that both the cyclotron mass and the g z factor canbe tuned by applying an in-plane magnetic field. Rais-ing the in-plane field component B k leads to an increaseof the cyclotron mass and decrease of the g z factor. Wehave developed a microscopic theory of these effects forthe complex valence band of germanium and have shownthat the theory describes well the experimental data. II. SAMPLE AND METHOD
The experiments were carried out on a p -typeSiGe/Ge/SiGe heterostructure with a single hole chan-nel (sample K6016). The layer structure of the sam-ple is illustrated in Fig. 1a. The two-dimensional holegas has a density of p ≈ × cm − and mobility µ ≈ × cm /(V · s) at 4.2 K. The structure was grownby low-energy plasma-enhanced chemical vapor deposi-tion (LEPECVD) on a Si(001) substrate by making useof the large dynamic range of growth rates. The buffer,graded at the rate of about 10% /µ m to the final Ge con-tent of 70%, and the 4- µ m-thick Ge . Si . layer weregrown at the high rate of 5 ÷
10 nm/s gradually loweringthe substrate temperature T s from 720 o C to 450 o C. Theactive part, consisting of a 20-nm-thick pure Ge layersandwiched between cladding layers with a Ge content ofabout 60%, and the Si cap were grown at the low rate ofabout 0.3 nm/s at T s = 450 o C. Modulation doping wasachieved by introducing dilute diborane pulses into thecladding layer at a distance of about 30 nm above thechannel. In the structure grown on a relaxed Si . Ge . buffer layer, the Ge channel is compressively strained inthe interface plane due to the lattice mismatch of about1 . ≈
100 meV for our sample.The properties of the 2D hole gas are studied by acontactless acoustoelectric method. The technique wasfirst employed in Ref. 18 for GaAs/Al x Ga − x As het-erostructures. The experimental setup is illustrated inFig. 1(b). A surface acoustic wave (SAW) is excitedon the surface of a piezoelectric LiNbO platelet by aninter-digital transducer. The SAW propagating along thelithium niobate surface induces a high-frequency elec-tric field which penetrates into the hole channel locatedin the SiGe/Ge/SiGe structure slightly pressed to thepiezoelectric platelet by means of springs. The field pro-duces an ac electrical current in the channel. As theresult of the interaction of the SAW electric field withholes, the wave attenuates and its velocity is modified,governed by the high-frequency conductivity σ ac . This“sandwich-like” experimental configuration enables con-tactless acoustoelectric experiments on non-piezoelectric2D systems, such as SiGe/Ge/SiGe. The measurementswere done at SAW frequencies of 30 and 85 MHz, inexternal magnetic fields B of up to 18 T, and in the tem-perature range of 0.3 – 5.8 K. The samples were mountedon a one-axis rotator, which enabled us to change the an-gle Θ between the quantum well (QW) normal and themagnetic field. III. EXPERIMENTAL RESULTS
Figure 2 shows the dependencies of the SAW attenua-tion change ∆Γ ≡ Γ( B ) − Γ(0) and normalized change ofthe SAW velocity ∆ v/v (0) ≡ [ v ( B ) − v (0)] /v (0) on themagnetic field B applied along the QW normal z . The de-pendencies are presented for different temperatures. Allcurves contain pronounced oscillations, which are causedby the formation of Landau levels in the two-dimensionalhole gas and microscopically are similar to the oscilla-tions of the magnetoconductivity in the Shubnikov-deHaas and quantum Hall effects. We note that Γ(0) is
30 nm SiGe spacer
20 nm Ge channel
180 nm SiGe cladding Ge=0.6
Si<001> substrate a b
90 nm SiGe with 10 (cid:71) spikes of diboraneSiGe-graded buffer layer
SiGe buffer with 4 (cid:80) mconstant Ge=0.7
FIG. 1: (Color online) (a) Cross-section of the studied sampleand (b) sketch of the acoustic experimental setup. negligibly small compared to Γ( B ) due to very high elec-tric conductivity of the hole gas at zero magnetic field, σ (0) ≈ × − Ω − . (cid:3) FIG. 2: (Color online) Magnetic field dependences of ∆Γ and∆ v/v (0) measured in a field B k z up to 18 T at the SAWfrequency f = 30 MHz, and in the temperature ranges of (a)0.3–1.8 K and (b) 2–5.8 K. Arrows denote the positions ofinteger filling factors. From the experimentally measured values of the SAWabsorption and the relative change of the SAW veloc-ity, one can calculate the real σ and imaginary σ com-ponents of the high-frequency conductivity of the holechannel by using Eqs. (1) and (2) of Ref. 15. Below, wefocus on the real part of the ac conductivity only sinceit enables us to determine the parameters of the energyspectrum. The corresponding dependence of σ on themagnetic field for different temperatures is presented inFig. 3. The magnetic field dependence of the conductiv-ity contains the Shubnikov-de Haas oscillations evolvinginto the integer quantum Hall effect at strong fields. Thepositions of the even and odd filling factors ν correspond-ing to the orbital and spin splitting of the Landau levels,respectively, are shown by vertical arrows.Comparison of the temperature dependence of the con-ductivity at odd and even filling factors allows us to de- -8 -7 -6 -5 -4 -3
12 8 (cid:86) ( (cid:58) - ) B z (T) (cid:81) =234567 T e m pe r au r e I n c r ea s e (cid:3) FIG. 3: (Color online) Dependence of σ on the magneticfield at different temperatures for B k z and for a SAW fre-quency f = 30 MHz. The positions of integer filling factorsare marked by arrows. termine the absolute value of the g z factor. The proce-dure is the following. For odd filling factors, we founda temperature range (0.5-5.8 K) where the conductivityin the oscillation minima is of activation nature and de-scribed by the Arrhenius law σ odd1 ∝ exp (cid:18) − ∆ odd k B T (cid:19) . (1)Here ∆ odd is the activation energy and T is the tem-perature. Thus, the slope of the linear dependence ofln σ on 1 /T yields the activation energy. The corre-sponding Arrhenius plots for the filling factors ν =3, 5,and 7 together with linear fits are presented in Fig. 4a.The activation energy is given by ∆ odd = ∆ Z − Γ B ,where ∆ Z = | g z | µ B z is the Zeeman splitting, µ is theBohr magneton, and Γ B is the Landau level broadening.The latter also depends on the magnetic field, the cal-culation in the self-consistent Bohr approximation yieldsΓ B = C √ B z with C being the field-independent param-eter. For even filling factors, the activation conductivity isalso described by the Arrhenius law σ even1 ∝ exp (cid:18) − ∆ even k B T (cid:19) , (2)where ∆ even = ~ ω c − ∆ Z − Γ B , ~ ω c = ~ eB z / ( m c c ) is theenergy spacing between the orbital Landau levels and m c is the cyclotron mass. The cyclotron mass for ourGe/SiGe structure m c ≈ . m is known with high ac-curacy from analysis of the Shubnikov-de Haas oscilla-tions. The Arrhenius plots for the even filling factors ν =10, 12, and 14 together with linear fits are presentedin Fig. 4b. The best fit of the activation conductivity for odd andeven filling factors by Eqs. (1) and (2) yields | g z | =6 . ± . C = 0 . ± .
06 meV T − / . The extracteddependence of the Zeeman splitting ∆ Z on the magneticfield is shown in the inset in Fig. 4a. Note, that the the-oretical estimation C = p e ~ / ( πm c cτ q ) (see Ref. 20)yields the very close value C ≈ .
69 meV T − / for thequantum relaxation time τ q ≈ The obtained value of the out-of-plane g -factor differs from that of the heavy holes in bulkGe, | g bulk | = | K| ≈ .
4. We attribute the difference torenormalization of the energy spectrum in Ge quantumwells due to size quantization, strain, and interaction ef-fects. The dispersion of heavy holes in quantum wellsis typically non-parabolic so that the in-plane effectivemass and g -factor depend on the Fermi energy. We alsonote that in the above analysis we neglected possible os-cillations of the hole g -factor in the magnetic field due toexchange interaction. While the exchange contributionto the g-factor may be important for 2D electron systems,experimental results and theoretical analysis reveal thatit is suppressed in GaAs-based hole systems. Theproblem of the exchange interaction in strained Ge-basedsystems requires further study and is out of the scope ofthis paper. b l n (cid:86) -1 ) (cid:81)(cid:32)(cid:26)(cid:81)(cid:32)(cid:24)(cid:81)(cid:32)(cid:22) a (cid:39) Z ( m e V ) B z (T) l n (cid:86) -1 ) (cid:81)(cid:32)(cid:20)(cid:23)(cid:81)(cid:32)(cid:20)(cid:21)(cid:81)(cid:32)(cid:20)(cid:19) (cid:3) FIG. 4: (Color online) Dependence of ln σ on 1 /T for (a) oddand (b) even filling factors. Lines are the result of linear fit-ting. Inset shows the obtained Zeeman splitting vs perpen-dicular magnetic field for odd filling factors. To study the g -factor anisotropy, the SAW absorptionand velocity change were measured in tilted magneticfields of magnitude B TOT . Figure 5 shows the obtaineddependence of the real part of the conductivity σ on thenormal component of magnetic field B z = B TOT cos Θfor various tilt angles Θ. One can see that the positionsof the conductivity minima are determined by the normalcomponent of the magnetic field, which is in agreementwith the 2D character of the hole states. Such behav-ior comes from the significant strain in p -SiGe/Ge/SiGestructures resulting in the splitting between the hh and lhsubbands of about 100 meV which exceeds the Fermi en-ergy of 14 meV. Therefore, the in-plane component of thehh g -factor vanishes and the Zeeman splitting is deter-mined by B z . Similar dependencies are also presentedin Fig. 6a where the conductivity oscillations correspond-ing to small filling factors ν = 2, 3, and 4 are shown asa function of the total magnetic field. With increasingthe tilt angle, the positions of the oscillation minima areshifted towards higher magnetic fields while the oscilla-tion amplitudes remain almost the same. Note, however,that study of conductivity at small filling factors requireshigh B z and therefore for ν =4 we were limited by 60 ◦ tiltangle. -9 -8 -7 -6 -5 -4 B (cid:52) (cid:81)(cid:32) (cid:52) (deg)
72 71.2 65.5 62.6 57 54 51 48.5 45.5 42.5 36.5 33.5 30.5 21.5 0 82.6 80.7 79.7 79.6 78.7 78 77.3 77.2 76.8 75.6 74.1 73.7 73.3 73 72.4 (cid:86) ( (cid:58) - ) B z (T) T il t ang l e i n c r ea s e (cid:81)(cid:32) (cid:81)(cid:32) (cid:81)(cid:32) (cid:81)(cid:32) (cid:81)(cid:32) (cid:81)(cid:32) (cid:81)(cid:32) (cid:81)(cid:32) FIG. 5: (Color online) Dependence of σ on the normalcomponent B z of the magnetic field for different tilt anglesΘ=(0 ÷ o ; f =30 MHz, T =0.3 K. -9 -8 -7 -6 -5 (cid:52) (degree) 0 o o o o o B TOT (T) (cid:86) ( (cid:58) - ) (cid:81) =4 (cid:81) =3 (cid:81) =2 a -8 -7 -6 -5 -4 (cid:52) (degree) 0 o o o o o o o o o o (cid:86) ( (cid:58) - ) Bz (cid:3) (T) T il t ang l e i n c r ea s e b FIG. 6: (Color online) (a) Dependence of σ on the totalmagnetic field B TOT for different tilt angles; (b) σ vs B z forthe tilt angles Θ=0 o - 80 o ; f =30 MHz, and T =0.3 K. Arrowsdenote the positions of integer filling factors. Figure 6b shows the magnetoconductivity oscillationsas a function of B z at various angles Θ for large fillingfactors ν ≥
5, where a strong in-plane field B k can beapplied to the sample. Surprisingly, we observe an effectof B k on the oscillations: The conductivity in the min-ima increases with increasing tilt angle at large Θ and the oscillation amplitude decreases. We emphasize thatsuch a behavior is observed for both even and odd fillingfactors, see Fig. 6b. Therefore, it cannot be explained bychanges in relative positions of the Landau levels since,in that case, the amplitudes of conductivity oscillationscorresponding to even and odd ν would change in an-tiphase. We attribute the observed features to the effectof the in-plane magnetic field on the hole cyclotron mass, g z factor, and Landau level broadening in the complexvalence band of germanium.According to the Arrhenius law Eq. (1), the decrease ofthe oscillation amplitudes at odd filling factors indicatesa decrease of the activation energy ∆ odd . We suggest thatthe decrease is caused by the reduction of the absolutevalue of the hole g -factor | g z | and increase of the Landaulevel broadening in the in-plane magnetic field. To secondorder in B k , the dependence of g z and C on the magneticfield is given by g z ( B k ) = g z (0) + α s B k , (3) C ( B k ) = C (0) + βB k , where g z (0) and C (0) are the g -factor and the parame-ter of the Landau level broadening at zero in-plane field,which are calculated above, α s and β are parameters.Below, see part IV, we present the microscopic theory ofthe Zeeman splitting of heavy-hole states in QWs andshow that exactly such a dependence of the g z -factor onthe in-plane field follows from the theory. Equations (1)and (3) yieldln σ odd1 ( B k ) = ln σ odd1 (0) + α s µ B z + β √ B z k B T B k , (4)where we take into account that g z < σ odd1 (0) is the conductivity in perpendicularmagnetic field for given odd filling factor (an in-planefield independent term). To determine the field correc-tions to both the g z factor and Landau level broaden-ing we plot in Fig. 7 the dependence of ln σ ( B k ) on B k for ν = 5 and 7 in the range of activation behav-ior of conductivity. The dependencies are linear, as ex-pected, and yield α (exp) s ≈ . × − T − , β (exp) ≈ × − meV T − / . For the magnetic field B z = 5 .
04 Tcorresponding to ν = 5 the effect of B k on the Zeemansplitting is more than twice as large as the effect of B k on the level broadening.The oscillations corresponding to even filling factorsare also damped out with increasing the tilt angle Θ, seeFig. 6b. This indicates a decrease of the activation en-ergy ∆ even which can be attributed to the increase of thecyclotron mass m c and the Landau level broadening bythe in-plane component of the magnetic field. Similarbehavior was observed for n -type 2D systems in a num-ber of papers and ascribed to the increase of m c . To second order in B k , the effect is phenomenologicallydescribed by m m c ( B k ) = m m c (0) − α c B k , (5) l n (cid:86) B ||2 (T ) (cid:81) =7 (cid:81) =5 (cid:3) FIG. 7: Dependence of ln σ on B k for the filling factors ν =5and 7. where m is the free electron mass and α c is a parameter.From the Arrhenius law at even filling factors, Eq. (2),we obtainln σ even1 ( B k ) = ln σ even1 (0)+ (2 α c − α s ) µ B z + β √ B z k B T B k , (6)where σ even1 (0) is the conductivity in perpendicular mag-netic field for given even filling factor (the term indepen-dent of B k ).Figure 8 shows the dependence of ln σ ( B k ) on B k mea-sured at different B z corresponding to the filling factors ν = 10, 12, and 14. In accordance with Eq. (6), the de-pendencies are linear. Fitting the experimental data byEq. (6) with the parameters α s and β obtained aboveyields α (exp) c ≈ × − T − . Comparing 2 α (exp) c with α (exp) s we conclude that the dominant contribution to thevariation of ∆ even is due to change of the cyclotron mass. (cid:81) =14 (cid:81) =12 l n (cid:86) B II2 (T ) (cid:81) =10 (cid:3) FIG. 8: (Color online) Dependence of ln σ on B k at theconductivity minima corresponding to the even filling factors ν =10, 12, and 14. Below we calculate the parameters α s and α c deter-mining the corrections to the spin splitting and cyclotron mass, respectively, for the heavy-hole subband in QWsand compare them with the values obtained from ourexperiment. IV. THEORY
We describe the effect of the in-plane magnetic fieldon the cyclotron mass and g z factor in the frameworkof the Luttinger model. In the axial approximation, theeffective Hamiltonian of holes in Ge quantum wells in anexternal magnetic field has the form H = H + U ( z ) + H Z + V , (7)where H is the Luttinger Hamiltonian for zero in-planemomentum, H = 12 m (cid:20)(cid:18) γ + 52 γ (cid:19) I − γJ z (cid:21) p z , (8) γ and γ are the Luttinger parameters, J i ( i = x, y, z )are the 4 × / I isthe identity matrix, p z = − i ~ ∂/∂z , U ( z ) is the diagonalmatrix of confinement potentials which are different forthe heavy-hole and light-hole subbands due to strain, H Z is the Zeeman Hamiltonian, H Z = − K µ ( J · B ) , (9) K is the parameter of the Zeeman splitting of hole statesat the Γ point of the Brillouin zone in bulk material, V isthe contribution to the Luttinger Hamiltonian account-ing for the in-plane momentum and magnetic field, V = 12 m (cid:18) γ + 52 γ (cid:19) I (cid:0) P x + P y (cid:1) (10) − γm (cid:0) J x P x + J y P y + 2 { J x J y }{ P x P y } (cid:1) , P = − i ~ ∇ − ( e/c ) A , e > A is thevector potential of the magnetic field B , and the bracesdenote the symmetrized product { CD } = ( CD + DC ) / ∝ B or ∝ B , do notseem to give a substantial contribution to the cyclotronmass and g z factor in moderate in-plane magnetic fields.We calculate the hole energy spectrum in the sym-metric heterostructure subjected to the magnetic field intwo steps. First, we solve the Schr¨odinger equation forthe case of H Z = V = 0 and find the envelope func-tions ϕ hn ( z ) and ϕ lm ( z ) and energies E hn and E lm ofthe heavy-hole and light-hole states, respectively, where n, m = 1 , , . . . are the subband indices. Each state istwo-fold degenerate. Then we use perturbation theoryand calculate the in-plane effective mass for the h B is oriented in the ( yz ) plane and choose the vectorpotential in the form A = ( zB y , xB z , B y and emerge in the second, third, andfourth orders of the perturbation theory. The knowledgeof the in-plane effective masses and the Zeeman splittingenables us to obtain the quasi-classical structure of theLandau levels.The calculation shows that the energy spectrum in thesubband h E ( N ) h , ± = ~ eB z ( N + 1 / c m c ( B y ) ± µ g z ( B y ) B z , (11)where m c ( B y ) = p m x ( B y ) m y ( B y ) is the cyclotron mass, m x ( B y ) and m y ( B y ) are the in-plane effective masses inthe directions perpendicular and parallel to the in-plane component of the magnetic field, respectively, and N isan integer number. The in-plane component of the g -factor tensor vanishes at B = 0 in the uniaxial approx-imation and therefore gives only higher order correc-tions ( ∝ B ) to the Zeeman splitting for tilted magneticfields.The g z factor determining the spin splitting of Landaulevels is given by Eq. (3) where the value in the perpen-dicular field g z (0) has the form g z (0) = − K + 12 γ m X n | p zh ,ln | E ln − E h , (12) p zhn,lm = h hn | p z | lm i are the matrix elements of the mo-mentum operator, and the coefficient α s is given by α s = 12 γ e m c X n,m,k " z h ,ln E ln − E h + γ + γ m ( z ) h ,h | p zh ,ln | ( E ln − E h ) − γ − γ m p zh ,ln z ln,lm { p z z } lm,h + p zh ,ln ( z ) ln,lm p zlm,h ( E ln − E h )( E lm − E h ) − γ + γm ( z ) h ,hn p zhn,lm p zlm,h + z h ,hn (cid:16) { p z z } hn,lm p zlm,h − p zhn,lm { p z z } lm,h (cid:17) ( E hn − E h )( E lm − E h )+ 3 γ m p zh ,ln { p z z } ln,hm { p z z } hm,lk p zlk,h − { p z z } h ,ln p zln,hm p zhm,lk { p z z } lk,h + 2 p zh ,ln p zln,hm { p z z } hm,lk { p z z } lk,h ( E ln − E h )( E hm − E h )( E lk − E h ) − γ m | p zh ,ln | |{ p z z } h ,lm | + | p zh ,lm | |{ p z z } h ,ln | ( E ln − E h ) ( E lm − E h ) . (13)Here we take into account that the matrix elements z µn,µ ′ m = h µn | z | µ ′ m i and z µn,µ ′ m = h µn | z | µ ′ m i arereal while the matrix elements p zµn,µ ′ m are purely imagi-nary ( µ, µ ′ = h, l ). The major contribution to α s comestypically from the terms containing only one light-holeenergy E ln in the denominator.Figure 9 shows the dependence of α s responsible forthe g z factor renormalization on the Luttinger parame-ters calculated after Eq. (13) for strain-free rectangularQWs with infinitely high barriers. In this model, thedependence of α s on the QW width a is simplified to α s ∝ a . The curves in Fig. 9 are plotted for a = 200 ˚A.It follows from the calculation that the correction to the g z factor caused by the in-plane magnetic field dependson the material parameters.The in-plane masses are given by m m x,y ( B y ) = m m k − ( α c ± δ ) B y , (14) where m k is the in-plane mass at zero magnetic field,1 m k = γ + γm − γ m X n | p zh ,ln | E ln − E h . (15)The parameter α c , which determines renormalization ofthe cyclotron mass in the in-plane field, and the param-eter δ describing the mass anisotropy have the form α c = α s / ξ + ξ , δ = ξ + ξ , where α s is given by Eq. (13), ξ = e m c X m,n ( ( γ + γ ) z h ,hn E hn − E h − γ m p zh ,ln { p z z } ln,h p zh ,lm { p z z } lm,h ( E ln − E h ) ( E lm − E h )+ 6 γ ( γ + γ ) m z h ,h p zh ,ln { p z z } ln,h ( E ln − E h ) − z h ,hn (cid:16) { p z z } hn,lm p zlm,h + p zhn,lm { p z z } lm,h (cid:17) ( E hn − E h )( E lm − E h ) ) , (16) ξ = 6 γ (cid:18) em c (cid:19) X n,m,k " |{ p z z } h ,ln | ( E ln − E h ) + 6 γm { p z z } h ,ln p zln,hm p zhm,lk { p z z } lk,h ( E ln − E h )( E hm − E h )( E lk − E h ) , (17) ξ = 6 γ e m c X m,n " I h ,ln ( z ) ln,h E ln − E h − γ − γm p zh ,ln z ln,lm { p z z } lm,h ( E ln − E h )( E lm − E h ) + 6 γ m p zh ,ln { p z z } ln,h p zh ,lm { p z z } lm,h ( E ln − E h )( E hm − E h )( E lk − E h ) , (18) -4 -3 -2 -1 =5 =10 =15 =20 s ( T - ) FIG. 9: (Color online) Dependence of the coefficient α s on theLuttinger parameters in strain-free QW of width a = 200 ˚A. and I h ,ln = h h | ln i are the overlap integrals of the enve-lope functions. We note that the dominant contributionto the cyclotron mass renormalization comes from thecoupling of the h h ξ , Eq. (16), and issimilar to that in electron systems. However, in con-trast to the conduction band where the effective massis modified only for the direction perpendicular to themagnetic field, in hole systems both components of theeffective mass tensor are renormalized.Equations (13) and (16) enable one to calculate therenormalizations of the cyclotron mass and Zeeman split-ting in hole systems caused by the in-plane magneticfield.
V. DISCUSSION AND SUMMARY
The experimental results discussed above demonstratethat, in SiGe/Ge/SiGe quantum wells, the in-plane com-ponent of the magnetic field leads to a decrease of theeffective g z factor and to an increase in the hole cyclotronmass. The effects are described by Eqs. (3) and (5), re-spectively, which also follow from the microscopic theory,with the fitting parameters α (exp) s ≈ . × − T − and α (exp) c ≈ × − T − .Figure 10 shows the theoretical dependence of the co-efficients α c and α s describing the effective mass and g z factor renormalization, respectively, on the QW width.The curves are calculated for a rectangular Ge quantumwell with the infinitely high barriers, taking into accountthe strain-induced splitting of the hh and lh subbands(solid curves) and neglecting the strain (dashed curves).Both α c and α s increase with the QW width. The coef-ficient α c is almost independent of strain since its domi-nant part is determined by the structure of the hh sub-bands only. In contrast, the Zeeman splitting renormal-ization occurs due to the mixing of the hh and lh sub-bands and α s is therefore sensitive to the hh-lh splittinginduced by strain. The dependence of α s on strain ismore pronounced for wide QWs where the hh-lh splittingdue to size quantization is comparable to or smaller thanthe strain-induced splitting. The absolute values of α c and α s extracted from the experiment correspond to thecalculated values for the QW width of about 100 ˚A ac-cording to Fig. 10. The nominal QW width in the studiedsample is 200 ˚A, but the effective length of the hole con-finement might be considerably smaller due to the build-in electric field produced by ionized dopants incorporatedin the barrier above the QW (see Sec. II). This electricfield pushes the holes to the upper interface thereby re-ducing the effective QW width.To summarize, we have performed contactless acous-toelectric measurements of the high-frequency conductiv-ity of the two-dimensional hole gas in a p -SiGe/Ge/SiGestructure subjected to a strong magnetic field. It hasbeen shown that in certain temperature domains and in-teger filling factors the conductivity is of the activationnature. The analysis of the activated conductivity atodd and even filling factors in perpendicular magneticfield enabled us to determine | g z | ≈ .
7. By applyinga tilted magnetic field we observed that at fixed normalcomponent of the field, the conductivity in oscillationminima increases with increasing Θ ( Θ > ◦ ) at botheven and odd filling factors for ν ≥
5. Such a behavioris attributed to the decrease of the cyclotron frequencyand Zeeman splitting of holes and the broadening of theLandau levels by the in-plane component of the mag- netic field. We have developed a microscopic theory ofthe heavy-hole cyclotron mass and g z factor renormaliza-tion in the framework of the Luttinger Hamiltonian. Thistheory describes the experimental data and predicts therenormalization to be more pronounced in wide quantumwells. FIG. 10: Dependence of (a) α c and (b) α s on the QW widthfor a strained (∆ = 100 meV, solid curves) and strain-free(∆ = 0, dashed curves) Ge quantum well. The curves arecalculated for the Luttinger parameters γ = 13 and γ = 5. Acknowledgments
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