Electron spin-phonon interaction symmetries and tunable spin relaxation in silicon and germanium
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J a n Electron spin-phonon interaction symmetries and tunable spin relaxation in siliconand germanium
Jian-Ming Tang ( 湯 健 銘 ) and Brian T. Collins Department of Physics, University of New Hampshire, Durham, NH 03824-3520, USA
Michael E. Flatt´e
Optical Science and Technology Center and Department of Physics and Astronomy,University of Iowa, Iowa City, IA 52242-1479, USA
Compared with direct-gap semiconductors, the valley degeneracy of silicon and germanium opensup new channels for spin relaxation that counteract the spin degeneracy of the inversion-symmetricsystem. Here the symmetries of the electron-phonon interaction for silicon and germanium are iden-tified and the resulting spin lifetimes are calculated. Room-temperature spin lifetimes of electronsin silicon are found to be comparable to those in gallium arsenide, however, the spin lifetimes insilicon or germanium can be tuned by reducing the valley degeneracy through strain or quantumconfinement. The tunable range is limited to slightly over an order of magnitude by intravalleyprocesses.
PACS numbers: 71.70.Fk, 72.25.Dc, 72.25.Rb, 72.10.Di
I. INTRODUCTION
The favorable material properties of silicon have per-mitted it to dominate the microelectronics industry forover half a century, however a new genre of spintronicsemiconductor devices, in which spins of electroniccarriers are manipulated instead of a charge current, re-quires long spin transport lengths and coherence times.Although spin injection into nonmagnetic semiconduc-tors was demonstrated over a decade ago, the recentsuccess at injecting spin-polarized current into silicon suggests incorporation of semiconductor spintronic de-vice concepts into hybrid silicon device architectures. Po-larized spins relax in semiconductors because the spin-orbit interaction entangles orbital and spin degrees offreedom, and thus ordinary scattering from defects orlattice vibrations leads to a loss of spin coherence. Inmaterials without inversion asymmetry the entanglementof spin and orbit manifests as an effective momentum-dependent (internal) magnetic field, causing spin pre-cession and D’yakonov-Perel’ spin relaxation. Withininversion-symmetric materials, such as silicon, the in-ternal magnetic field vanishes, but scattering betweenstates with spin-orbit entangled wave functions leads toElliott-Yafet spin relaxation. The spin coherence timesin silicon are long at low temperature, and the spin-orbit interaction and lattice symmetry reduces spin re-laxation rates relative to optically accessible (direct-gap)semiconductors.
The silicon band structure, how-ever, has multiple valleys that permits low-energy scat-tering of electrons by large momenta, which allows theElliott-Yafet process to be more effective. Numericalcalculations that include these effects have been success-ful at explaining the spin lifetime in silicon as a functionof temperature.
Tuning the spin lifetime in inversion-asymmetric semiconductors with a single direct gap havelargely focused on the electric-field-induced Rashba spin- orbit interaction that shortens the spin lifetime; un-addressed is the potential for new methods of tuning thespin lifetime associated with the valley degeneracy of thesemiconductor.Here we trace that origin of the intrinsic spin lifetime insilicon to the spin flips associated with large momentumtransfer events, and disentangle the intervalley contri-bution to spin relaxation from the intravalley contribu-tion. As the different processes involve different momen-tum transfers, and for the intrinsic spin-relaxation ratethe source of that momentum transfer will be electron-phonon scattering, the various contributions will be as-sociated with specific regions of the phonon dispersioncurves. Due to the high symmetry of the crystal lat-tice, several processes that might have been expected tocontribute will be forbidden by symmetry. We providea full symmetry analysis of the various contributions tothe spin-relaxation rate from phonon-mediated scatter-ing. The separation of spin-relaxation mechanisms bymomentum transfer also permits a direct calculation ofthe tuning of spin lifetime possible by splitting the en-ergies of the electron valleys and thus suppressing someof the intervalley scattering effects. Reducing the val-ley degeneracy in silicon, through applied strain or thegrowth of pseudomorphic SiGe quantum wells, will re-duce the effect of the spin-orbit interaction on electronscattering, lengthening the spin coherence time and spintransport length. We find an effective tuning range ofapproximately one order of magnitude.
II. ELECTRON-PHONON SCATTERING
The intrinsic spin-relaxation time is determined byelectron-phonon scattering. For one-phonon absorption(+) and emission ( − ) processes, the scattering probabil-
50 100 T (K)10 -9 -8 -7 -6 -5 T ( s )
50 100 50010 m ob ilit y ( c m / V s )
730 K700 K210 K140 K630 K500 K210 K (a) (b)
FIG. 1. (Color online) Electron spin-relaxation time (a) andmobility (b) in bulk silicon shown by black solid lines. In-dividual scattering processes are intravalley acoustic process(dashed line), intravalley optical (Γ +25 ) process (plus symbol),intervalley ∆ processes (closed symbols), and intervalley Σprocesses (open symbols). ity from k to k ′ is M ± σ ′ σ ( k ′ , k ) = (cid:12)(cid:12)(cid:12)D ψ k ′ σ ′ , n q ∓ (cid:12)(cid:12)(cid:12) ˆ H ep ± (cid:12)(cid:12)(cid:12) ψ k σ , n q E(cid:12)(cid:12)(cid:12) , (1)where ˆ H ep ± is the time-independent part of the electron-phonon interaction Hamiltonian corresponding to ab-sorption or emission, q = k ′ − k , σ labels the spinstate, and n q is the phonon occupation number. Toevaluate M ± σ ′ σ ( k ′ , k ) for various types of scattering pro-cesses in the material, we use a first-nearest-neighbor sp tight-binding model (TBM) with on-site spin-orbitinteractions to obtain the wave functions, ψ k σ ( r ) = 1 √ N X j,a,l,s c als e i k · R ja φ al ( r − R ja ) χ s , (2)where N is the number of unit cells, j labels unit cells, a labels the two basis atoms within a unit cell, l labels theatomic orbital bases, χ is a two-component spinor, s isthe spin index, and R ja is the position vector of atoms.We choose the spin quantization axis to be aligned withthe z axis and determine the coefficients c als by maxi-mizing the expectation value of the spin operator h ˆ S z i .In the spherical band approximation, we express thematrix elements in terms of the deformation potentialsup to the first order in δ q = ( k ′ − k f ) − ( k − k i ), M ± σ ′ σ ≈ ¯ h ρV ω (cid:0) D ,σ ′ σ + D ,σ ′ σ | δ q | (cid:1) (cid:0) n q + ∓ (cid:1) , (3)where ρ is the density, V is the crystal volume, ω isthe phonon frequency, and k i and k f are the wavenumbers of the initial and final valley minima, respec-tively. For Si, there are six valleys on the ∆ axes, e.g., k = (2 π/a )(0 . , ,
50 100 T (K)10 -10 -9 -8 -7 -6 T ( s )
50 100 50010 m ob ilit y ( c m / V s )
430 K390 K340 K120 K (a) (b)
FIG. 2. (Color online) Electron spin-relaxation time (a) andmobility (b) in bulk germanium shown by black solid lines.The dashed line shows the intravalley acoustic phonon con-tribution and the plus symbol shows the intravalley opticalphonon contribution. The open symbols show the intervalley X processes for three effective phonon energies. at the L points, e.g., ( π/a )(1 , , D . We have assumed only the following typesof matrix elements are nonzero, h φ as ( R ) | ∂H∂ R | φ ap ( R ) i , h φ a ′ l ′ ( R ′ ) | ∂H∂ R | φ al ( R ) i , and h φ a ′ l ′ ( R ′ ) | ∂H∂ R ′ | φ al ( R ) i , andall have the same magnitude. Since the atomic potentialis not explicitly known in the TBM, we can only deter-mine the relative strengths of the processes. The overallmagnitude is later determined by fixing the mobility to be1450 cm /Vs in Si and 3800 cm /Vs in Ge. Calculateddeformation potentials for different phonon processes aresummarized in Table I. The relative strengths of differentprocesses are consistent with the semiempirical values inthe literature.
Various electron-phonon scattering rates are computedas follows. For the intravalley acoustic process, a linearphonon dispersion, ω = c | q | is used and the scatteringrate including both absorption and emission is1 τ A = √ D A m / πρc ¯ h k B T D √ E E T , (4)where m = ( m L m T ) / is the averaged effective mass,and c = 3 / (1 /c L + 2 /c T ) is the averaged speed ofsound. We use ρ = 2329 kg/m , m L = 0 . m e , m T = 0 . m e , c L = 8500 m/s, and c T = 5900 m/s forSi, and ρ = 5323 kg/m , m L = 1 . m e , m T = 0 . m e , c L = 4900 m/s, and c T = 3500 m/s for Ge. A ther-mal averaging of the initial electron energy is carried outwith the Boltzmann distribution, appropriate for non-degenerate systems, h g ( E ) i T ≡ √ πT / Z ∞ g ( E ) E / e − E/T dE. (5)In this regime the spin lifetime is independent of the car-rier density; for degenerate carrier densities the spin life-
TABLE I. Deformation potentials. The unit for D ’s iseV/˚A, and for D ’s is eV. For intravalley acoustic processes, D A = 3 . D A, ↑↓ = 0 . D A = 3 . D A, ↑↓ = 0 .
032 eV for Ge. For spin-flip processes, the super-script xy indicates that both the initial and final valleys are inthe x - y plane, and the superscript z indicates that the initialand final valleys are separated along the z direction. T ω isthe effective phonon frequency in Kelvin. The deformationpotentials listed here do not include the valley degeneracy offinal states. The parentheses indicate that the D term isnegative.Phonon T ω (K) D D D xy , ↑↓ D xy , ↑↓ D z , ↑↓ D z , ↑↓ Si Γ +25
730 0 0.34 0 0.14 0 0.19∆ ′ + ∆
700 4.5 (3.2) 0 0.031 0 0.01∆
210 0 0.04 0 0.028 0 0.04∆
140 0 2.7 0 0.009 0 0.0015Σ + Σ
630 3.2 (2.3) 0.03 0.11 0.044 0.17Σ + Σ
500 3 (1.1) 0.018 0.075 0.026 0.097Σ + Σ
210 0.0083 2.2 0.0083 0.041 0.0059 0.058Ge Γ +25
430 3.5 4.6 0 0 0 0 X
390 0.24 1 0.24 0.16 0 0.17 X
340 3.8 3.4 0.08 0.054 0.12 0.089 X
120 0 2.7 0 0.11 0 0.1 time will be shorter than for nondegenerate carrier den-sities as the distribution function spreads out to a larger k range for which the spin mixing upon scattering willbe larger. For the intravalley optical process and eachintervalley process, we use an effective phonon frequency ω , listed in Table I as temperature T ω . The zeroth-order and the first-order scattering rates are1 τ (0) ω = D m / √ πρω ¯ h h n ω D p (0)+ E T + ( n ω + 1) D p (0) − E T i , (6)1 τ (1) ω = √ D m / πρω ¯ h h n ω D p (1)+ E T + ( n ω + 1) D p (1) − E T i , (7)where p (0) ± = p E ± ¯ hω θ ( E ± ¯ hω ) , (8) p (1) ± = (2 E ± ¯ hω ) p E ± ¯ hω θ ( E ± ¯ hω ) . (9)The final electron spin-relaxation times including all scat-tering processes for Si and Ge are plotted, respectively,in Figs. 1 and 2 as a function of temperature. III. DISCUSSION
Our results can be qualitatively understood from theselection rules, derived from symmetry, that apply toscattering processes between the valley minima, i.e., D ’s. The selection rules without spin-orbit couplingwere discussed by Lax and Hopfield and are listed inTable II. When electron spin is included, the irreducible TABLE II. Selection rules with the inclusion of time-reversalsymmetry without spin. For electron representations, ∆ t is∆ at k transformed to a valley on a perpendicular axis to k , and L t is L transformed from one L valley to a dif-ferent valley. The phonon representation ∆ (2 k ) becomes∆ ′ (2 k − (4 π/a )(1 , , ( k ) ⊗ ∆ ( k ) = Γ +1 ⊕ Γ +12 ∆ ( k ) ⊗ ∆ ( − k ) = ∆ (2 k )∆ ( k ) ⊗ ∆ t = Σ Ge L +1 ⊗ L +1 = Γ +1 ⊕ Γ +25 L +1 ⊗ L +1 t = X TABLE III. Selection rules with the inclusion of time-reversalsymmetry with spin. The spin-flip part of ∆ (∆ ′ in Table I)in Si and of Γ +25 in Ge are also forbidden by time reversal.Si ∆ ⊗ ∆ = Γ ± ⊕ Γ ± ⊕ Γ − ⊕ Γ − ∆ ⊗ ∆ = ∆ ∆ ⊗ ∆ t = 2Σ ⊕ Σ ⊕ Σ Ge L +6 ⊗ L +6 = Γ +1 ⊕ Γ +25 L +6 ⊗ L +6 t = 2 X ⊕ X ⊕ X representation at the conduction-band minima in Si (Ge)becomes ∆ ( L +6 ) instead of ∆ ( L +1 ). We analyzed theselection rules including time-reversal symmetry usingthe same subgroup technique developed in Refs. 26 and28 and list the results in Table III. The allowed phononrepresentations at q (right-hand side of the equations)are obtained from the characters at k and k ′ (the tworepresentations on the left-hand side of the equations).Details of our calculations of the selection rules are pre- -0.2 -0.1 0 0.1 0.2 ∆ E (eV) -9 -8 -7 -6 T ( s ) -0.2 -0.1 0 0.1 0.2 10 -10 -9 -8 -7 (a) (b) Si Ge
FIG. 3. (Color online) Spin-relaxation time at T = 300 K asa function of valley energy shift, ∆ E , shown by black solidlines. The energy shift is the energy offset of four (three)valleys relative to the remaining two (one) in Si (Ge). Thesymbols in panel (a) have the same meaning as in Fig. 1 andin panel (b) are the same as in Fig. 2. sented in the Appendix.In Si the spin-orbit interaction mixes spin up and downby about 1% in wave function and gives a spin-flip prob-ability about 10 − . It can be seen from Fig. 1 that theΣ phonons (also known as the f processes) dominatethe mobility and the spin flip near room temperature.For mobility, ∆ phonons (also known as the g processes)also contribute substantially, consistent with the selec-tion rules of ∆ ′ and Σ in Table II. For spin flip, the Σ and Σ phonons become allowed with nonzero spin-orbitinteraction in addition to Σ (Table III). The intravalleyoptical (Γ +25 ) phonons and the ∆ phonons remain for-bidden for spin flip by time reversal and, therefore, arenot as effective as the Σ phonons. Although the cou-pling strength for the low-energy Σ phonon is weaker,this is somewhat compensated by the temperature de-pendence of the phonon distribution and it ends up thatall Σ phonons contribute approximately the same to spinflip near room temperature.In Ge the spin-flip probability due to spin-orbit inter-action is about one order of magnitude larger, but thenumber of possible phonon processes is reduced. So thespin-relaxation time is only about one order of magni-tude smaller than in Si as shown in Fig. 2. For mobil-ity, the intervalley X and the intravalley optical (Γ +25 )phonons are more important due to the selection rulesin Table II. For spin flip, the high-energy ( X ) phononis allowed with finite spin-orbit interaction, but the low-energy ( X ) phonon is still forbidden by time reversal.Although the coupling to Γ +25 phonons is as strong as thecoupling to X phonons, the spin-flip part is forbiddenby time reversal and ineffective (Table III).Now that the structure and symmetry of the spin-relaxation mechanisms has been clarified, the analysis ofthe effect of strain is straightforward. Strain (or quan-tum confinement) breaks valley degeneracy and can elim-inate multivalley scattering processes. In Si, a [001]strain can change the lowest-energy valley degeneracyfrom six to two (located on the same axis). In Ge, a[111] strain can yield just a single nondegenerate valleyat the conduction-band edge. We have performed a sim-ple estimate that only takes into account the valley en-ergy shift, ∆ E , by modifying ± ¯ hω → ± ¯ hω − ∆ E andthe initial electron distribution in each valley. As shownin Fig. 3, the spin lifetime averaged over all valleys canbe lengthened substantially; 1% strain gives about 0 . .
16 eV shift in Ge. Positive energyshift (∆ E >
0) corresponds to the configuration thatfour valleys out of six are shifted to higher energy in thecase of Si, and three out of four in the case of Ge. Neg-ative shift reverses the ordering. For positive shifts, theintervalley Σ processes in Si or the X processes in Gecan be completely eliminated. This tuning is eventuallylimited by the intervalley ∆ and the intravalley optical(Γ +25 ) processes in Si, and by the intravalley acoustic pro-cesses in Ge. For negative shifts, the elimination of theintervalley processes is only partial, so the tuning rangeis much smaller. IV. CONCLUSIONS
We have presented a thorough symmetry analysis ofthe electron spin-phonon interaction processes for siliconand germanium, finding a spin lifetime for nondegener-ate carriers at room temperature comparable to thosein III-V semiconductors when the scattering determin-ing the carrier mobility is dominated by phonons. How-ever, strain or quantum confinement can lift the valleydegeneracy, which lengthens the spin lifetime substan-tially (over an order of magnitude at room temperature).
ACKNOWLEDGEMENTS
This work was supported in part by an ONR MURIand an ARO MURI.
Appendix A: Selection Rules
In this Appendix we show the calculations of the char-acters of the product representations for obtaining theselection rules in Tables II and III. Each electron-phononscattering process involves three subgroups of the wavevectors, k , k ′ and q . Instead of using the intersectiongroup formed by the elements common to all three sub-groups and creating the corresponding character tables,the selection rules are derived using the existing charac-ter tables of the subgroups at these wave vectors. Thetwo multiplication rules, with and without time-reversalsymmetry, will be presented. We choose to computethe characters of the phonon representations at q fromthe products of the electron representations at k and k ′ .Once the characters at q are obtained, the decompositioninto irreducible representations is done in the usual wayusing the subgroup of the wave vector q . TABLE IV. Double group characters at ∆ = k (1 , , ∆ ∆ ′ ∆ ′ ∆ ∆ ∆ ( E |
0) 1 1 1 1 2 2 2( ¯ E |
0) 1 1 1 1 2 − − C | , ( ¯ C |
0) 1 1 1 1 − C | τ ) λ − λ − λ λ √ λ −√ λ
2( ¯ C | τ ) λ − λ − λ λ −√ λ √ λ iC | τ ) , i ¯ C | τ ) λ λ − λ − λ iC | , i ¯ C |
0) 1 − − λ = e ik a/ , ¯ E is 2 π rotation, N ∆ star ∆ ( C ) = 1,∆ = ∆ ⊗ D / , ∆ = ∆ ⊗ D / We first carry out calculations without time-reversalsymmetry. To obtain the product character at q for eachclass C using the characters at k and at k ′ , the usualcharacter multiplication rules are modified as follows. χ i ⊗ j q ( C ) = χ i k ′ ( C ) h χ j k ( C ) i ∗ N q star k ( C ) , (A1) TABLE V. Double group characters at L = ( π/a )(1 , , L ± L ± L ± L ± L ± L ± ( E |
0) 1 1 2 1 1 2( ¯ E |
0) 1 1 2 − − − C | τ ) 1 − i − i
03( ¯ C | τ ) 1 − − i i C |
0) 1 1 − − − C |
0) 1 1 − − i | τ ) Z ± χ ( Z ) ± χ ( Z ) Z can be any of the six classes shown above. L ± = L ± ⊗ D / = L ± ⊗ D / where i and j are the irreducible representations, and N q star k ( C ) is the number of wave vectors that are un-changed by the class C , out of a set of nonequivalent k points. This set of nonequivalent k points is generatedby the subgroup of q and is called “ q star of k .” Theproduct character is simply zero if C is not a commonclass of all three subgroups. TABLE VI. Group characters at Γ = (0 , , ± Γ ± Γ ± Γ ± Γ ± N Γ star ∆ N Γ star L ( E |
0) 1 1 2 3 3 6 43( C |
0) 1 1 2 − − C | τ ) 1 − − C | τ ) 1 − − C |
0) 1 1 − i | τ ) ± ± ± ± ± iC | τ ) ± ± ± ∓ ∓ iC | ± ∓ ± ∓ iC | ± ∓ ∓ ± iC | τ ) ± ± ∓ k (1 , , Σ Σ Σ N Σ star ∆ ( E |
0) 1 1 1 1 2( C | τ ) λ λ − λ − λ iC | τ ) λ − λ − λ λ iC |
0) 1 − − λ = e ik a/ Listed here are the character tables for the doublegroups representing the symmetry of the electron statesat ∆ for Si (Table IV) and at L for Ge (Table V), and thetables for the groups at ∆, Γ, Σ, and X for phonons (Ta-bles IV and VI–VIII). The numbers of invariant q -star- k points, N q star k ( C ), are listed in the tables for phonons.Note that the group at X has 14 irreducible representa-tions, 14 classes, and 32 elements. Only the 4 physicallyadmissible representations and 8 classes are listed in the TABLE VIII. Group characters at X = (2 π/a )(1 , , X X X X N X star L ( E |
0) 2 2 2 2 4( C |
0) 2 2 − − C | τ ) , ( C ′ | τ + t xy ) 0 0 − iC |
0) 2 − E | t xy ) − − − − C | t xy ) − − C | τ + t xy ) , ( C ′ | τ ) 0 0 2 − iC | t xy ) − t xy = ( a/ , , table. The other 6 classes are not relevant because theyhave zero character for the 4 physical representations.Further simplification is possible because one can workwith just 4 classes; the other 4 classes, correspondingto an additional lattice translation, t xy = ( a/ , , τ = ( a/ , , λ ) due to this transla-tion in some characters should always cancel out in thefinal character product ( k ′ = k + q ) and not affect theselection rules. We will ignore these explicit phase fac-tors in our character product tables. However, care needsto be taken with the product classes of L and L t whencomparing to the classes at X . The product of two groupelements with τ can result in a lattice translation, t xy ,which is explicitly included in the subgroup of X , but notin the subgroup of L . All physical representations at X are odd under t xy . The characters at L = ( π/a )(1 , , t xy , but are even at L t = ( π/a )( − , , C | τ ) at L and at L t isactually ( C | τ + t xy ) at X . This is why the product of L and L t contains X , but not X . Time reversal can add additional constraints to theselection rules if there exists a group element that con-nects the time-reversed process ( − k ′ → − k ) to the orig-inal process ( k → k ′ ). That is, we need an element Q from the subgroup of q that interchanges k and − k ′ . Toincorporate the time reversal symmetry, the charactermultiplication rule is modified to the symmetric or anti-symmetric combination of the characters of the originalprocess and the process connected via Q , χ i ⊗ j q ± ( C ) = 12 h χ i ⊗ j q ( C ) ± χ j k ( C ) N q star k ( QC ) i , (A2)where C is the class of the square of elements in C and Q is the element that interchanges k and − k ′ . The pos-itive (symmetric) sign is used in the cases without spinand the negative (antisymmetric) sign is used with spin.Again, all the relevant characters for the product repre-sentations are listed in Tables IX–XIII. The element Q can be identified as the first entry in the column of QC when C is the identity class. For ∆ phonons in Si and Γphonons in Ge, Q is simply the identity element. Uponcloser examination, we also found that the spin-flip partof the ∆ phonon process in Si and of the Γ +25 phononprocess in Ge are forbidden because the final state is ex-actly the time reverse of the initial state (e.g., | α i = | k ↑i and | ˆ T α i = | − k ↓i ), h ˆ T α | ˆ H ep | α i = h ˆ T α | ˆ T ˆ H † ep ˆ T − | ˆ T α i , (A3) where ˆ T is the time-reversal operator, ˆ T = − ˆ1 in thepresence of half-integer spin, and ˆ H ep = ˆ T ˆ H † ep ˆ T − isthe time-reversal invariant electron-phonon interactionHamiltonian. S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.Daughton, S. von Moln´ar, M. L. Roukes, A. Y. Chtchelka-nova, and D. M. Treger, Science , 1488 (2001). Semiconductor Spintronics and Quantum Computation ,edited by D. D. Awschalom, N. Samarth, and D. Loss(Springer-Verlag, Berlin, 2002). D. D. Awschalom and M. E. Flatt´e, Nature Physics , 153(2007). R. Fiederling, M. Keim, G. Reuscher, W. Ossau,G. Schmidt, A. Waag, and L. W. Molenkamp, Nature ,787 (1999). Y. Ohno, D. K. Young, B. Beschoten, F. Matsukura,H. Ohno, and D. D. Awschalom, Nature , 790 (1999). A. T. Hanbicki, B. T. Jonker, G. Itskos, G. Kioseoglou,and A. Petrou, Appl. Phys. Lett. , 1240 (2002). J. Strand, B. D. Schultz, A. F. Isakovic, C. J. Palmstrøm,and P. A. Crowell, Phys. Rev. Lett. , 036602 (2003). C. Adelmann, X. Lou, J. Strand, C. J. Palmstrøm, andP. A. Crowell, Phys. Rev. B , 121301 (2005). I. Appelbaum, B. Huang, and D. J. Monsma, Nature ,295 (May 2007). B. T. Jonker, G. Kioseoglou, A. T. Hanbicki, C. H. Li, andP. E. Thompson, Natute Physics , 542 (2007). S. P. Dash, S. Sharma, R. S. Patel, M. P. de Jong, andR. Jansen, Nature , 491 (2009). L. Grenet, M. Jamet, P. No´e, V. Calvo, J. Hartmann, L. E.Nistor, B. Rodmacq, S. Auffret, P. Warin, and Y. Samson,Appl. Phys. Lett. , 032502 (2009). J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett. ,4313 (1998). T. F. Boggess, J. T. Olesberg, C. Yu, M. E. Flatt´e, and W. H. Lau, Appl. Phys. Lett. , 1333 (2000). B. Beschoten, E. Johnston-Halperin, D. K. Young, M. Pog-gio, J. E. Grimaldi, S. Keller, S. P. DenBaars, U. K.Mishra, E. L. Hu, and D. D. Awschalom, Phys. Rev. B , 121202 (2001). Optical Orientation , edited by F. Meier and B. P. Za-kharchenya (North-Holland, Amsterdam, 1984). J. L. Cheng, M. W. Wu, and J. Fabian, Phys. Rev. Lett. , 016601 (2010). P. Li and H. Dery, Phys. Rev. Lett. , 107203 (2011). W. H. Lau and M. E. Flatt´e, J. Appl. Phys. , 8682(2002). N. S. Averkiev, L. E. Golub, A. S. Gurevich, V. P.Evtikhiev, V. P. Kochereshko, A. V. Platonov, A. S. Shkol-nik, and Y. P. Efimov, Phys. Rev. B , 033305 (2006). D. J. Chadi, Phys. Rev. B , 790 (1977). D. K. Ferry, Phys. Rev. B , 1605 (1976). O. Madelung,
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TABLE IX. Characters of the product representations at Γ.
C QC N ( QC ) C χ ∆ k ( C ) χ ∆ ⊗ ∆ Γ χ ∆ ⊗ ∆ Γ+ C χ ∆ k ( C ) χ ∆ ⊗ ∆ Γ χ ∆ ⊗ ∆ Γ − ( E |
0) ( i | τ ) 0 ( E |
0) 1 6 3 ( E |
0) 2 24 12( C |
0) ( iC | τ ) 4 ( E |
0) 1 2 3 ( ¯ E | − C | τ ) ( iC |
0) 0 ( C |
0) 1 2 1 ( C | , ( ¯ C |
0) 0 4 2( C | τ ) ( iC |
0) 2 ( E |
0) 1 0 1 ( ¯ E | − i | τ ) ( E |
0) 6 ( E |
0) 1 0 3 ( E |
0) 2 0 − iC | τ ) ( C |
0) 2 ( E |
0) 1 4 3 ( ¯ E | − iC |
0) ( C | τ ) 2 ( C |
0) 1 0 1 ( C | , ( ¯ C |
0) 0 0 0( iC |
0) ( C | τ ) 0 ( E |
0) 1 2 1 ( ¯ E | − ⊗ ∆ = Γ +1 ⊕ Γ +12 ⊕ Γ − and ∆ ⊗ ∆ = Γ ± ⊕ Γ ± ⊕ ± ⊕ Γ ± without time-reversal symmetryTABLE X. Characters of the product representations at ∆ = 2 k . C C χ ∆ k ( C ) χ ∆ ⊗ ∆ ∆ χ ∆ ⊗ ∆ ∆+ C χ ∆ k ( C ) χ ∆ ⊗ ∆ ∆ χ ∆ ⊗ ∆ ∆ − ( E |
0) ( E |
0) 1 1 1 ( E |
0) 2 4 1( C |
0) ( E |
0) 1 1 1 ( ¯ E | − C | τ ) ( C |
0) 1 1 1 ( C | , ( ¯ C |
0) 0 2 1( iC | τ ) ( E |
0) 1 1 1 ( ¯ E | − iC |
0) ( E |
0) 1 1 1 ( ¯ E | − Q is the identity element. ∆ ⊗ ∆ = ∆ ⊕ ∆ ′ ⊕ ∆ without time-reversal symmetryTABLE XI. Characters of the product representations at Σ = k (1 , , C QC N ( QC ) C χ ∆ k ( C ) χ ∆ ⊗ ∆ t Σ χ ∆ ⊗ ∆ t Σ+ C χ ∆ k ( C ) χ ∆ ⊗ ∆ t Σ χ ∆ ⊗ ∆ t Σ − ( E |
0) ( C | τ ) 0 ( E |
0) 1 2 1 ( E |
0) 2 8 4( C | τ ) ( E |
0) 2 ( E |
0) 1 0 1 ( ¯ E | − iC | τ ) ( iC |
0) 0 ( E |
0) 1 2 1 ( ¯ E | − iC |
0) ( iC | τ ) 2 ( E |
0) 1 0 1 ( ¯ E | − ⊗ ∆ t = Σ ⊕ Σ and ∆ ⊗ ∆ t = 2Σ ⊕ ⊕ ⊕ without time-reversal symmetryTABLE XII. Characters of the product representations at Γ. C C χ L +1 L ( C ) χ L +1 ⊗ L +1 Γ χ L +1 ⊗ L +1 Γ+ C χ L +6 L ( C ) χ L +6 ⊗ L +6 Γ χ L +6 ⊗ L +6 Γ − ( E |
0) ( E |
0) 1 4 4 ( E |
0) 2 16 4( C | τ ) ( E |
0) 1 2 2 ( ¯ E | − C |
0) ( C |
0) 1 1 1 ( ¯ C | − i | τ ) ( E |
0) 1 4 4 ( E |
0) 2 16 4( iC |
0) ( E |
0) 1 2 2 ( ¯ E | − iC | τ ) ( C |
0) 1 1 1 ( ¯ C | − Q is the identity element. Only classes that have non-trivial characters are shown. L +6 ⊗ L +6 = Γ +1 ⊕ Γ +2 ⊕ Γ +12 ⊕ +15 ⊕ +25 without time-reversal symmetryTABLE XIII. Characters of the product representations at X . C QC N ( QC ) C χ L +1 L ( C ) χ L +1 ⊗ L +1 t X χ L +1 ⊗ L +1 t X + C χ L +6 L ( C ) χ L +6 ⊗ L +6 t X χ L +6 ⊗ L +6 t X − ( E |
0) ( C |
0) 0 ( E |
0) 1 4 2 ( E |
0) 2 16 8( C |
0) ( E |
0) 4 ( E |
0) 1 0 2 ( ¯ E | − C | τ ) , ( C ′ | τ + t xy ) ( C ′ | τ ) , ( C | τ + t xy ) 2 ( E |
0) 1 − E | − iC |
0) ( iC |
0) 2 ( E |
0) 1 2 2 ( ¯ E | − L ⊗ L t = X ⊕ X and L +6 ⊗ L +6 t = 2 X ⊕ X ⊕ X ⊕ X4