Intense terahertz laser fields on a two-dimensional hole gas with Rashba spin-orbit coupling
aa r X i v : . [ c ond - m a t . m t r l - s c i ] S e p Intense terahertz laser fields on a two-dimensional hole gas with Rashba spin-orbitcoupling
Y. Zhou
Hefei National Laboratory for Physical Sciences at Microscale and Department of Physics,University of Science and Technology of China, Hefei, Anhui, 230026, China (Dated: November 30, 2018)We investigate the influence on the density of states and the density of spin polarization for atwo-dimensional hole gas with Rashba spin-orbit coupling under intense terahertz laser fields. ViaFloquet theorem, we solve the time-dependent Schr¨odinger equation and calculate these densities.It is shown that a terahertz magnetic moment can be induced for low hole concentration. Differentfrom the electron case, the induced magnetic moment is quite anisotropic due to the anisotropicspin-orbit coupling. Both the amplitude and the direction of the magnetic moment depend on thedirection of the terahertz field. We further point out that for high hole concentration, the magneticmoment becomes very small due to the interference caused by the momentum dependence of thespin-orbit coupling. This effect also appears in two-dimensional electron systems.
PACS numbers: 71.70.Ej, 78.67.De, 73.21.Fg, 78.90.-t
Optical properties of semiconductors are sensitive toexternal conditions. Almost fifty years ago Franz andKeldysh pointed out that under static electric fields theabsorption coefficient becomes finite below the band gap,and the above-gap absorption spectrum shows oscilla-tions. In the late 1990s, Jauho and Johnsen studied theoptical properties of semiconductors under strong ac fieldand developed dynamic Franz-Keldysh effect (DFKE),which presents the blueshift of the main absorption edgeand the fine structure near the band gap. This ef-fect is particularly obvious for semiconductors under in-tense terahertz (THz) field, thereby leading to extensivetheoretical and experimental interests on THz electro-optics. Very recently Cheng and Wu brought thespin degrees of freedom into the study of the THz fieldinduced effects. They studied a two-dimensional electrongas (2DEG) with the Rahhba spin-orbit coupling (SOC)under intense terahertz field. It is shown that the THzfield can efficiently modify the density of states (DOS) ofthe 2DEG and excite a magnetic moment oscillating atTHz frequency. Later, Jiang et al. studied similar effectsof quantum dots , and further discussed the spin dis-sipation under THz driving fields. However, all theseworks concentrate on electron systems. Up to now thereis no study on the spin properties of hole systems underintense THz field. In this letter we study the effect of in-tense terahertz laser fields on a two-dimensional hole gas(2DHG) with Rashba SOC and show that this systemhas some new properties different from the previouslystudied electron system.We consider a p -type GaAs (100) quantum well (QW).The growth direction is denoted as the z axis. A uni-form THz radiation field (RF) E THz ( t ) = E cos(Ω t ) =( E x , E y ,
0) cos(Ω t ) is applied in the x - y plane with the pe-riod T = 2 π/ Ω. The angle between the electric field and x axis is θ E . By using the Coulomb gauge, the vector andscalar potentials can be written as A ( t ) = − E sin(Ω t ) / Ωand φ ( t ) = 0, respectively. We assume the well width is small enough so that only the lowest subband is rel-evant. For this structure, the lowest subband is heavyhole (HH) like. By applying a suitable strain, it can belight hole (LH) like. The confinement is assumed to belarge enough so that the lowest HH and LH subbandsare well separated and we can consider the HHs and LHsseparately. The Hamiltonian can be written as ( ~ ≡ H λ ( K , t ) = K m ∗ λ + H λso ( K , t ) . (1)Here λ = LH, HH and K = k − e A ( t ), with k standingfor the electron momentum. m ∗ λ is the effective mass.For GaAs QW, H so is mainly due to the Rashba term, and can be written as H λso ( K ) = ( σ x Ω λx + σ y Ω λy ). Here, σ is the Pauli matrix. For HHs, we have Ω HHx ( K ) = 2 K y ( γ HHa K y + γ HHb K x ) , (2)Ω HHy ( K ) = 2 K x ( γ HHa K x + γ HHb K y ) , (3)with γ HHa = E z ( γ h h + γ h h ), γ HHb = E z ( γ h h − γ h h ). For LHs,Ω LHx ( K ) = 2 K y ( γ LHa K y + γ LHb K x + γ LHc h k z i ) , (4)Ω LHy ( K ) = − K x ( γ LHa K x + γ LHb K y + γ LHc h k z i ) , (5)with γ LHa = E z ( γ l l + γ l l ), γ LHb = E z ( γ l l − γ l l ), γ LHc = E z γ l l . It is noted from these equations that themagnitude of the Rashba term can be tuned by the exter-nal electric field E z applied on the sample. In Eqs.(2)-(5), γ h h , γ h h , γ l l , γ l l , γ l l are the Rashbacoefficients. They depend both on the property of ma-terial and QW well width.Similar to Refs. 9 and 10, by employing the Floquettheorem, the solution of the Schr¨odinger equation withtime-dependent Hamiltonian H λ ( K , t ) can be written asΦ λs ( k , t ) = e − i { ( E λ k + E λem ) t − b k · E [cos(Ω t ) − − γ sin(2Ω t ) } × e − q λs ( k ) t ∞ X n = −∞ φ λn,s ( k ) e in Ω t (6)Here s = ± represents the two helix spin branches; E λ k = k / m ∗ λ is the kinetic energy of HHs or LHs; E λem = e E / (4 m ∗ λ Ω ) is the energy induced by the RFdue to the DFKE; b = e/ ( m ∗ λ Ω ); γ = E em / (2Ω). φ λn,s ( k ) = ( φ λ,σn,s ( k )) ≡ (cid:0) φ λ, +1 n,s ( k ) φ λ, − n,s ( k ) (cid:1) in Eq. (6) are the ex-pansion coefficients of the Floquet states with σ = 1 ( − ↑ (-down ↓ ) in the laboratory coor-dinates (along the z axis). q s ( k ) is the correspondingeigenvalue and can be determined by[ n Ω − q λs ( k )] φ σn,s + (cid:8) [ D λ ( k ) ± iσD λ ( k )] + 2( e/ [ D λ ( k ) ± iσD λ ( k )] (cid:9) φ − σn,s + (cid:8) i ( e/ D λ ( k ) ± iσD λ ( k )] + 3 i ( e/ [ D λ ( k ) ± iσD λ ( k )] (cid:9) ( φ − σn +1 ,s − φ − σn − ,s ) − ( e/ [ D λ ( k ) ± iσD λ ( k )]( φ − σn +2 ,s + φ − σn − ,s ) − i ( e/ [ D λ ( k ) ± iσD λ ( k )]( φ − σn +3 ,s − φ − σn − ,s ) = 0 , (7)where D HH = γ HHa k y + γ HHb k x k y , D HH = γ HHa k x + γ HHb k y k x ,D LH = γ LHa k y + γ LHb k x k y + γ LHc h k z i k y , D LH = γ LHa k x + γ LHb k y k x + γ LHc h k z i k x ,D HH = 3 γ HHa k y E y + γ HHb ( k x E y + 2 k x k y E x ) , D HH = 3 γ HHa k x E x + γ HHb ( k y E x + 2 k y k x E y ) ,D LH = 3 γ LHa k y E y + γ LHb ( k x E y + 2 k x k y E x ) + γ LHc h k z i E y , D λ = 3 γ λa k x E x + γ λb (2 k y E x E y + k x E y ) ,D λ = 3 γ λa k y E y + γ λb (2 k x E y E x + k y E x ) , D LH = 3 γ LHa k x E x + γ LHb ( k y E x + 2 k y k x E y ) + γ LHc h k z i E x ,D λ = γ λa E y + γ λb E x E y , D λ = γ λa E x + γ λb E y E x . All eigenvalues can be written as q s,n = q s, + n Ω where q s, is the eigenvalue in the region ( − Ω / , Ω / q s,n and q s, are physically equivalent. Wealso find s = + branch and s = − branch satisfying therelations: φ σn, − = − σφ − σ, ∗− n, + , (8) q − ( k ) = − q + ( k ) . (9)With the help of Green function, we can calculate thedensity of states (DOS) ρ σ,σ and the density of spin po-larization(DOSP) ρ σ, − σ , ρ σ ,σ ( T, ω ) = 12 π Z ∞−∞ d k X s = ± ∞ X l ,l n,m = −∞ e i ( n − m )Ω T × R σ ,σ ( s ; n, m ; k ) J l ( − b k · E sin(Ω T )) × J l (2 γ cos(2Ω T )) δ ( ω − [ E k + E em − ( l + 2 l + n + m )Ω / q s ( k )]) , (10)in which J n ( x ) is the Bessel function of n th order, R σ ,σ ( s ; n, m ; k ) = ( η σ † φ n,s ( k ))( φ m,s † ( k ) η σ ) with η σ standing for the eigenfunction of σ z . It is seen from Eq.(10) that these densities are periodic functions of T withperiod T . The DOSP is nonzero only when both theRF and the SOC are present. Furthermore, the induced magnetic moment can be written as M ( T ) = (cid:16) M x ( T ) , M y ( T ) , M z ( T ) (cid:17) = 2 gµ B n ↑ + n ↓ Z E F ( T ) −∞ dω (cid:16) Re ρ ↑ , ↓ , − Im ρ ↑ , ↓ ,
12 ( ρ ↑ , ↑ − ρ ↓ , ↓ ) (cid:17) , (11)where the Fermi energy E F ( T ) is determined by n σ = R E F ( T ) −∞ ρ σ,σ ( ω, T ) dω where n σ represents the hole con-centration. Eq. (11) has been simplified by using thefact that ρ σ ,σ = ρ ∗ σ ,σ . It is evident that E F ( T ) and M ( T ) both oscillate with the period T . Due to time re-versal symmetry, the DOSP is an odd function of thetime ρ σ, − σ ( T, ω ) = − ρ σ, − σ ( − T, ω ), and therefore theDOSP averaged over time reduces to zero. Besides, theDOS is an even function ρ σ,σ ( T, ω ) = ρ σ,σ ( − T, ω ) and ρ ↑ , ↑ ( T, ω ) = ρ ↓ , ↓ ( T, ω ), thus the RF in the x - y planecannot induce magnetic moment along the z axis. Thesecharacters are similar to those of a 2DEG. We numerically solve the eigen-equation Eq. (7) andcalculate the DOS and the DOSP through Eq. (10).One can further obtain the magnetic moment by us-ing Eq. (11). In the calculation we choose a = 10 nm, E z = 30 kV/cm. The material parameters of GaAs areas follows: γ = 6 . γ = 2 . γ = 2 .
9, ∆ = 0 . g LH = 1 . g HH = 3 . m ∗ LH = 0 . m and m ∗ HH = 0 . m . In order to ensure the validity ofthe model which we adopt, we must keep the highestsideband of HH well separated from the lowest sidebandof LH. Moreover, the HH and LH bands can be splittedby 50 meV by adjusting the applied strain. Accordingto these, we choose E = 0 . . -5 0 5 10 15 20 25 30-1.5 -1 -0.5 0 0.5 1 1.5 D O S ( m e V - m - ) ω (meV) FIG. 1: Time-averaged DOS of HHs under THz field with E = 0 . . -0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 -0.002-0.001 0 0.001 0.002 0.003-1.5 -1 -0.5 0 0.5 1 1.5 D O S P ( m e V - m - ) (a)E F (meV) Γ ( m - ) Re ρ ↑ , ↓ Im ρ ↑ , ↓ Γ -0.08-0.06-0.04-0.02 0 0.02 0.04 0.06-1.5 -1 -0.5 0 0.5 1 1.5-0.003-0.002-0.001 0 0.001 0.002 D O S P ( m e V - m - ) ω (meV) (b) Γ ( m - ) Re ρ ↑ , ↓ Im ρ ↑ , ↓ Γ FIG. 2: DOSP of HHs for (a) θ E = 0 and (b) θ E = π/ T = T / E = 0 . . E F .Note the scale of E F and Γ are on the upper and right frameof the figure. -8-6-4-2 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1T (T ) M ( T ) ( - g µ B ) θ E = π /4 θ E = π /6 θ E =0 FIG. 3: Magnetic moment M of HHs versus time for θ E = 0, θ E = π/ θ E = π/ E = 0 . . n ↑ = n ↓ = 0 . × cm − and E F is about 0.35 meV. In Fig. 1 we compare the time-averaged DOS with andwithout the THz field. Due to DFKE, the main absorp-tion edge has a blueshift and the DOS becomes finitebelow the band gap. The DOSP at T = T / θ E = 0 and (b) θ E = π/
4. It is seen from Fig. 2(a)that only the imaginary part of DOSP is finite. FromEq. (11), we can find that the induced magnetic mo-ment is along the y axis. This is similar to the 2DEGcase with Rashba SOC. In Fig. 2(b), we can see thatRe ρ ↑ , ↓ = − Im ρ ↑ , ↓ . Thus the induced magnetic momentis along the (1 , , / √ This is due to the anisotropy of the SOCHamiltionian.In Fig. 3, the magnetic moment of M is plotted asfunction of time for θ E = 0, θ E = π/ θ E = π/ E = 0 . . θ E . The magnetic moment isthe smallest for θ E = 0 and the largest for θ E = π/ E F , whereΓ = R E F −∞ Im ρ ↑ , ↓ dω . It is noted that Γ is very small forlarge enough E F , hence M becomes negligible when theconcentration of the hole gas is high. This can be under-stood as follows: By interchanging the order of integral,one has Z ∞−∞ ρ ↑ , ↓ dω = Z ∞−∞ d k X l J l ( − b k · E sin(Ω T )) × X s,m,n R ↑ , ↓ ( s ; n, m ; k ) e i ( n − m )Ω T × X l J l (2 γ cos(2Ω T )) . By virtue of Eqs. (8) and (9), one gets R ↑ , ↓ ( s ; n, m ; k ) = − R ↑ , ↓ ( − s ; − m, − n ; k ). Thus the terms of s = + branchcompensate those of s = − branch, and the integral ofthe DOSP over the whole range ( −∞ , ∞ ) is zero. On theother hand, the DOSP decays to very small value withincreasing ω due to the interference caused by the mo-mentum dependence of the SOC. Hence the contributionto the magnetic moment at large ω is negligible. Accord-ingly, M becomes very small when E F is large, i.e., thehole concentration is high. Our calculation shows thatthis is also true for 2DEG with Rashba SOC.In conclusion, we study the effects of the intense THzfield on 2DHG with Rashba SOC. We calculate the DOSand DOSP. We also show that the a THz magnetic mo-ment can be excited for low hole concentration. It isnoted that the direction of the THz field has a stronginfluence on the angle between the induced magnetic mo-ment and the THz field, as well as on the amplitude of the magnetic moment, which is quite different from2DEG with Rashba SOC case. We also point out thatthe magnetic moment becomes very small if the hole con-centration is high enough, due to the interference causedby the momentum dependence of the SOC. This effectalso appears in 2DEG.The author would like to thank M. W. Wu for propos-ing the topic as well as the directions during the in-vestigation. This work was supported by the NationalNatural Science Foundation of China under Grant No.10574120, the National Basic Research Program of Chinaunder Grant No. 2006CB922005, the Knowledge Innova-tion Project of Chinese Academy of Sciences and SRFDP.The author would also like to thank J. H. Jiang for help-ful discussions and I. C. da Cunha Lima for proof readingof this manuscript. W. Franz, Z. Naturforsch. Teil A , 481 (1958). L. V. Keldysh, Sov. Phys. JETP , 788 (1958). A. P. Jauho and K. Johnsen, Phys. Rev. Lett. , 4576(1996). K. Johnsen and A. P. Jauho, Phys. Rev. B , 8860 (1998). K. B. Nordstrom, K. Johnsen, S. J. Allen, A. -P. Jauho,B. Birnir, J. Kono, and T. Noda, Phys. Rev. Lett. , 457(1998). J. Cerne, K. Kono, T. Inoshita, M. Sundaram, and A. C.Gossard, Appl. Phys. Lett. , 3543 (1997). J. Kono, M. Y. Su, T. Inoshita, T. Noda, M. S. Sherwin,S. J. Allen, Jr., and H. Sakaki, Phys. Rev. Lett. , 1758(1997). C. Phillips, M. Y. Su, M. S. Sherwin, J. Ko, and L. Col-dren, Appl. Phys. Lett. , 2728 (1999). J. L. Cheng and M. W. Wu, Appl. Phys. Lett. , 032107(2005). J. H. Jiang, M. Q. Weng, and M. W. Wu, J. Appl. Phys. , 063709 (2006). J. H. Jiang and M. W. Wu, Phys. Rev. B , 035307(2007). Y. A. Bychkov and E. Rashba, Sov. Phys. JETP Lett. ,78 (1984). R. Winkler,