J. At. Mol. Sci. doi: 1.48/jams.3111.431a Vol. 4, No., pp. 138-146 May 13 The properties of the polaron in semiconductor uantum dots induced by influence of Rashba spin-orbit interaction Xin-Jun Ma and Jing-Lin Xiao College of Physics and Electromechaincs, Inner Monglia University for Nationalities, Tongliao 843, China Received 11 March 1; Accepted (in revised version) 3 April 1 Published Online 8 March 13 Abstract. The properties of the effective mass of polaron in semiconductor uantum dots by influence of Rashba spin-orbit (SO) interaction are studied. The relations of the strength of confinement ω, the interaction energy and the effective mass of the polaron in the electron-lo phonon strong coupling region in a parabolic uantum dot on the vibration freuency is derived by using improved liner combination operator method. Numerical calculations for RbCl crystal are performed and the results show that the Rashba SO interaction makes the ground state energy and the effective mass of polaron split into two branches; the ground splitting energy and the effective mass will increase with the vibration freuency increasing. Whereas the interaction energy is sharply increased until the confinement strength reaches a certain value, then it will sharply decrease. PACS: 71.38.-k, 73..Mf Key words: semiconductor uantum dot, polaron, Rashba spin-orbit interaction, effective mass 1 Introduction In recent years there has attracted great attention in spin physics in semiconductors. Most of it is focused on spin-related optical and transport properties of low-dimensional semiconductor structures [1-3]. In particular, the spin-orbit (SO) interaction has attracted a lot of interest as it plays an important role in the field of semiconductor spintronics. SO interaction can arise in uantum dots (QDs) by various mechanisms related to electron confinement and symmetry breaking and are generally introduced in the Hamiltonian via the Rashba [4] and Dresselhaus terms [5]. Dresselhaus term is obtained from the electric field produced by the bulk inversion asymmetry of the material and Rashba term is Corresponding author. Email address: xiaojlin@16.com (J. -L. Xiao) http://www.global-sci.org/jams 138 c 13 Global-Science Press
X. -J. Ma and J. -L. Xiao / J. At. Mol. Sci. 4 (13 ) 138-146 139 generated due to the structural asymmetry of the heterostructure. Rashba splitting has been observed in many experiments and it constitutes the basis of the proposed electronic nanostructures. The strength of these interactions not only depends on the characteristics of the material but can be controlled by an external electric field. In the literature, most of the theoretical studies about the solution of the SO effects in QDs are carried out by means of perturbative schemes or numerical simulations [6, 7]. Analytical solution of the problem has recently been treated by employing various techniues [8-11]. For zero-magnetic field and a hard-wall confining potential the exact analytical results have been obtained by Boulgakov and Sadreev[1]. Tsitsishvili et al present an analytic solution to Rashba coupling in a uantum dots [13]. Tapash Chakraborty report on a theoretical approach developed to investigate the influence of the Bychkov- Rashba interaction on a few interacting electrons confined in a uantum dot [14]. There have been much work about the influence of the Rashba SO interaction on the electron system, the study of the effect of the Rashba SO interaction on the polaron, however, is uite rare so far. In this paper, we find the Rashba SO will induce the splitting of the ground state energy and the effective mass of the polaron. Theoretical Model We consider a uasi-two-dimensional uantum dot normal to the z axis. Therefore, we confine ourselves to considering only the motion of the electron in the x-y plane. The Hamiltonian of the electron-phonon system is given by H= P m +V(ρ)+H ph+h ph e +H SO, (1) where the first term denotes the kinetic energy of the electron and the second term represents the confining potential in a single QD that is V(ρ)= 1 mω ρ, () where m is the bare band mass of the electron and ω is the confinement strength of the uantum dot. The Hamiltonian of the phonons H ph is given by H ph = hω L b + b, (3) where b + (b ) is the creation (annihilation) operator of a bulk LO phonon with wave vector 1,=( //, z ). The electron-phonon interaction term H ph e is expressed as H ph e = [V exp(i r)b +h.c.], (4)
14 X. -J. Ma and J. -L. Xiao / J. At. Mol. Sci. 4 (13 ) 138-146 where r=(ρ,z). The Fourier coefficient for the interaction are described by V = i( hω LO e h )( ) 1/4 ( 4πα mω LO v )1/, (5a) α=( )( mω L ) 1/ ( 1 1 ), hω L h ǫ ǫ where ε and ε are the optical dielectric and the static dielectric constant. α is the electron-lo phonon coupling strength. v is the volume of the crystal. The contribution of the Rashba effect to the single-electron Hamiltonian can be expressed as (5b) H SO = ᾱ h (p xσ y p y σ x ) (6) where σ x, σ y, are the Pauli spin matrices and the Rashba parameter α R is linearly dependent on the expectation value of the electric field E. In semiconductor structures, it is determined by many factors such as the geometry [15]. α R = c E, there the coefficient c is inversely proportional to the energy gap and effective mass [16,17]. The Rashba interaction usually dominates in uantum dots obtain in a heterostructure [6,7,18]. We have chosen to include the Rashba term than the Dresselhaus term. Following Tokuda [19], one also introduce the improved linear combination of the creation operator a + j and annihilation operator a j to represent the momentum and position of the electron in the x-y plane ( m hλ ) 1 P j = (a j +a + j +P j ), (7a) ( h ) 1 ρ j = i (a j a + mλ j ), (7b) j= x,y (7c) where λ and P are variational parameters. Carrying out a unitary transformation ( ) U= exp (b + f b f ) * (8) where f ( f * ) is the variational parameter. The transformed Hamiltonian can be rewritten as H = H + P Z m The ground state wavefunction of the system is chosen as Ψ = ϕ(z) (aχ 1 (9) +bχ 1) b a (1) where ϕ(z) is the normalized electron wavefunction along the z direction and ϕ(z) ϕ(z) =δ(z) since ( ) the electrons ( are ) considered to be confined in an infinitesimally narrow layer. χ 1 = and χ 1 1 = label the up and down states of the z component of 1
X. -J. Ma and J. -L. Xiao / J. At. Mol. Sci. 4 (13 ) 138-146 141 the spin, a and b are coefficients. a is the ground state of polaron, and b is the zero phonon state, which satisfied b b = a j a =. To obtain the effective mass of strongcoupling polaron, the minimization of the energy should be performed by constraining the total momentum operator P T of the system parallel to the x-y plane. The total momentum of the system is parallel to the x-y plane operator can be represented as P T = P + b + b h (11) The minimization problem is now carried out by using the Lagrange multipliers, choosing an arbitrary constant multiplier. Now the ground state splitting energy of the whole system can be obtain as E ± = min F ± (u,p,λ, f ) (1) F ± (u,p,λ, f )= a b U 1 (H P u)u b a (13) F ± (u,p,λ, f ) may be called the variational parameter function. Using the variational method, we get the variational parameter f ( f ) and p. Substituting f ( f ) and p into F, we have F ± (u,λ)= hλ + hω λ mu α R h m± α R h α h λω 1/ L mu (1+ 1 π 3 u mλ hω ) (14) L Performing the variation of F ± (u,λ) with respect to λ, we obtain the expression of the vibration freuency of strong-coupling polaron in a parabolic uantum dot λ ωl α π λ 3 ω = (15) By solving euation (15) we obtain the vibration freuency of the strong-coupling polaron λ=λ (16) Substituting these parameters and the function in euation (14), the ground state splitting energy of the strong-coupling polaron can be expressed as E ± = hλ + hω λ mu α R h m± α R h α h λ ω 1/ L mu (1+ 1 π 3 The spin-splitting energy at zero magnetic fields can be expressed as u mλ hω ) (17) L E SO =± α R h mu α R h m (18)
14 X. -J. Ma and J. -L. Xiao / J. At. Mol. Sci. 4 (13 ) 138-146 For the momentum expectation value of the strong-coupling polaron in a parabolic uantum dot in the x-y plane, we find [ P = b a U 1 P T U a o b = m 1± α R hu + α ( ) 3] 3 λ u (19) π It is evident from the structure of this expression that u has the meaning of velocity which may be regarded as the average velocity of the polaron in the x-y plane, and the factor before u, namely m ±= m [ 1± α R hu + α 3 π ( λ ω L ) 3 ] ω L u () can be interpreted as the effective mass of the strong-coupling polaron in a parabolic uantum dot. Finally, the effective Hamiltonian of strong-coupling polaron in a parabolic uantum dot can be expressed as H e f f = H kin +H int (1) where H kin = P z m + P m () H int = hλ + hω 1 ( ) 1 λ α R α hω L λ π ω L h m (3) are the kinetic energy and the interaction energy of strong-coupling in a parabolic uantum dot. From E. () one can see that the effective mass m ± of strong-coupling in a parabolic uantum dot depends not only on the freuency of phonon ω L and vibration freuency λ but also on α R the strength of spin-orbit coupling. From E. (3) one can see that the interaction energy H int of strong-coupling in a parabolic uantum dot depends not only on the freuency of phonon ω L, the strength of confinement ω, the vibration freuency λ but also on α R the strength of spin-orbit coupling. 3 Numerical results and discussion To show more obviously the influence of α R the strength of spin-orbit coupling, the vibration freuency λ and the influence of the confinement strength ω on the properties of strong coupling in a parabolic uantum dot, taking in a uantum dot of the RbCl crystal (the data for a RbCl crystal []: ε = 4.58, ε =., hω L =.317 mev, α=3.81, ω L = 3.39 1 13 S 1 ) for example, we perform a numerical evaluation. Fig. 1 shows the relationships between polaron ground state energy E b =(α R =, ground state splitting energy E ± of the strong coupling in a parabolic uantum dot of the
X. -J. Ma and J. -L. Xiao / J. At. Mol. Sci. 4 (13 ) 138-146 143 7 6 Ground state energy (mev) 5 4 3 1.6.8 3. 3. 3.4 /1 14 Hz Figure 1: The polaron ground state energy E b =(α R = ) and ground state splitting energy E ± calculated by using E. (17). Energy as a function of vibration freuency λ in the RbCl crystal. They are expressed by the black line, dash line respectively (α= mev nm, u= 1 4 m/s). RbCl crystal with the vibration freuency λ. We assume that α R = mev nm u= 1 4 m/s. From the figure, one can see that polaron ground state energy E b and ground state splitting energy E ± will increase strongly with increasing the vibration freuency λ.the Rashba SO interaction originating from the lack of inversion symmetry which causes a local electric field perpendicular to the plane of heterostructure splits the ground state energy of the polaron into two branches, each of them is not stand for the splitting energy of polaron spin-up (spin-down) energy, but the result of common affect of polaron spin-up and spin-down.when the material selected, the increase α R can be attributed to the increase in E by the negative gate voltage because it can not significantly charge the energy gap and effective mass, therefore, the coefficient can be treated as a constant. In another words, by applying a voltage to a gate the effective electric field in the conducting channel and thus the Rashba coupling parameter can be controlled. The Rashba parameter takes values in the range of 1 11 1 1 ev m [1]. When the splitting parameter α R is very small, the spin splitting takes less than 1% of the polaron ground state energy. When the splitting parameter α R is very large such as mev nmthe spin splitting energy is about E SO =6.5 mev. It takes about 6% of the polaron ground state energy. With the present of the phonons, the total energy of the partial is decreased, so the state of the polaron is much stable than electron s. Therefore, the splitting state of the polaron is much stable. Furthermore E state is the most stable in the splitting state of the polaron. So we can see that Rashba effect can not be neglected. Fig. shows the relationship between the confinement strength ω and the vibration freuency λ in RbCl crystal. We assume that α R = mev nm. From the figure, one can see that polaron the confinement strength ω will increase strongly with increasing the vibration freuency λ. This is because of the exist of confining potential ω limits the moving of electrons increase with increasing of the confining potential that is the
144 X. -J. Ma and J. -L. Xiao / J. At. Mol. Sci. 4 (13 ) 138-146 1 8 Hz 6 4 4 6 8 1 /1 14 Hz Figure : The confinement strength ω as a function of the vibration freuency λ in RbCl crystal which calculated by using E. (15). decrease of ρ, the enhancement of the energy electron thermal motion which the medium is phonon and the interaction of electron-phonon enhance with diminishing the extent of particle s moving, cause the increase of the vibration freuency, so it shows the novel uantum size effect. Fig. 3 shows the relationship between the vibration freuency λ and the effective mass m ± /m in RbCl crystal. We assume that α R= mev nm. From the figure, one can see that polaron the effective mass m ±/m will increase strongly with increasing the vibration freuency λ. As we all know polaron is the interaction between electron and phoron, 9 8 7 6 m /m 5 4 3 1...4.6.8 1. Hz Figure 3: The effective mass m ± /m calculated by using E. (). Effective mass as a function of the vibration freuency λ in RbCl crystal.
X. -J. Ma and J. -L. Xiao / J. At. Mol. Sci. 4 (13 ) 138-146 145-6 -5 H int /mev -4-3 - -1..5 1. 1.5..5 3. 3.5 /1 14 /Hz Figure 4: The interaction energy H int calculated by using E. (3). Interaction energy as a function of the vibration freuency λ in RbCl crystal. which cause the effective mass of the electron changed. The Rashba SO interaction makes the density in the material of in different spin state, that is to say, spin state splits on their energies. This energy s split is the result of the split of the polaron s effective mass. Fig. 4 shows the relationship between the interaction energy H int and the vibration freuency λ in RbCl crystal. From the figure, we can see that the interaction energy increases rapidly to the maximum H intmax = 58.6 mev at λ =.8 1 14 Hz and decrease rapidly with increasing the λ. We obtain H int = hλ 3 a h wl π λ1/ α R m, after inserting E. (15) to E. (3). In the point of mathematical view, the increasing speed of λ h is bigger than λ 1, but the coefficient of h is smaller than 3 a h wl π, so it appear the maximum. Inserting λ =.8 1 14 Hz (at the maximum point) into E. (15), ω is obtained. h r =( mw l ) 1 is the radius of polaron and l =( h mω ) 1 is the effective confinement length of the QD. We can find that r = l = 1.3 nm. It is said that, when the polaron s radius is eual to the effective confinement length of the QD, the interaction energy is the maximum. From that, we can draw the conclusion that interaction energy has sympathetic vibration character with the change of the confinement strength ω, so felicitous confinement strength ω is favor of stabilization of the polaron in semiconductor uantum dots. 4 Conclusion In this paper, we have adopted improved liner combination operator method to investigate the ground state of a polaron which in the electron-lo phonon strong coupling region in a parabolic uantum dot by considering the influences of the Rashba SO interaction. The numerical calculation is performed on RbCl crystal. We find that the spin-
146 X. -J. Ma and J. -L. Xiao / J. At. Mol. Sci. 4 (13 ) 138-146 splitting energy and effective mass of polaron induced by Rashba SO interaction is sensitive to α R the strength of spin-orbit coupling. The total polaronic correction is negative and decreases the ground state energy. The vibration freuency and the effective mass will increase with the confinement strength increasing. Whereas the interaction energy is sharply increased until the confinement strength reaches a certain value, then it will sharply decrease, so felicitous confinement strength ω is favor of stabilization of the polaron in semiconductor uantum dots. From this paper, we can see that the Rashba effect plays a fundamental role in the understanding of the properties of the spin-dependent ground state energy of the polaron in semiconductor uantum dots. Acknowledgments The authors would like to acknowledge the project support by National Natural Science Foundation of China under Grant No. 13474 References [1] F. Meier and B. P. Zakharchenya, Optical Orientation (North-Holland, Amsterdam,1984). [] S. S. Li, K. Chang, J. B. Xia, and K. Hirose, Phys. Rev. B. 68(3)4536. [3] S. S. Li, J. B. Xia, J. L. Liu, F. H. Yang, Z. C. Niu, S. L. Feng, and H. Z. Zheng, J. Appl. Phys. 9 (1) 6151. [4] E. I. Rashba, Fiz. Tverd. Tela (Leningrad) (196) 14. [5] G. Dresselhaus, Phys. Rev. 1 (1955) 58. [6] M. Governale, Phys. Rev. Lett. 89 () 68. [7] S. Bandyopadhyay, Phys. Rev. B 61 () 13813. [8] M. Valìn-Rodrìguez, A. Puente, and L. Serra, Eur. Phys. J. B 34 (3) 359. [9] E. Tsitsishvili, G. Lozano, and A. O. Gogolin, Phys. Rev. B 7 (4) 115316. [1] D. Frustaglia and K. Richer, Phys. Rev. B 69 (4) 3531. [11] R. de Sousa and D. S. Sarma, Phys. Rev. B 68 (3) 15533. [1] E. N. Bulgakov and A. F. Sadreev, JETP Lett. 73 (1) 55. [13] E. Tsitsishvili, G. Lozano, and A. O. Gogolin, Phys. Rev. B 7 (4) 115316. [14] T. Chakraborty, P. Pietiläinen, Phys. Rev. B 71 (5) 11335. [15] N. Junsaku, A. Tatsushi, and T. Hideaki, Phys. Rev. Lett. 78 (1997) 1335. [16] G. Lommer, F. Malcher, and U. Rossler, Phys. Rev. Lett. 6 (1998) 78. [17] Q. F. Sun, J. Wang, and H. Guo, Phys. Rev. B 71 (5) 16531. [18] O. Voskoboynikov, C. P. Lee, and O.Tretyak, Phys. Rev. B 63 (1) 16536. [19] N. Tokuda, Solid State Commun. 35 (198) 15. [] E. Kartheuser, Dielectric Properties of Polar Crystals. Polarons in Ionic Crystals and Polar Semiconductors (Amsterdam, North-Holland, 197). p. 71. [1] T. Hassenkam, S. Pedersen, K. Baklanov, A. Kristensen, C. B. Sorensen, P. E. Lindersen, P. E. Lindelof, F. G. Pikus, and G. E. Pikus, Phys. Rev. B 55 (1997) 998.