Geometric Origin of Elliott Spin Decoherence in

PRL 93, 266601 (2004)

PHYSICAL REVIEW LETTERS

week ending 31 DECEMBER 2004

Geometric Origin of Elliott Spin Decoherence in Metals and Semiconductors Yuri A. Serebrennikov* Qubit Technology Center, 2152 Merokee Drive, Merrick, New York 11566, USA (Received 31 August 2004; published 20 December 2004) It has been shown that the Elliott spin-decoherence mechanism in nonmagnetic metals and semiconductors arises from the random acquisition of geometric phases and represents a special case of a more general situation —relaxation of the pseudo-spin-1=2 induced by stochastic gauge fields. The geometric approach gave an opportunity to apply Elliott’s ideas for a wide range of systems and system parameters. DOI: 10.1103/PhysRevLett.93.266601

PACS numbers: 72.25.Rb, 03.65.Vf, 03.67.Lx, 76.60.Es

Recently, an entirely electrical control of an electron spin motion in the absence of external magnetic fields was demonstrated in 2D semiconductor nanostructures [1]. These results open a new pathway to implementation of solid-state quantum computing and spintronics devices, and stimulated theoretical studies of the spin dynamics in a time-varying electric field [2 – 4]. An unavoidable coupling with the environment destroys the entanglement in a spin subsystem that is vital for quantum-logical operations. Hence, successful design and realization of efficient spin-based devices requires a thorough understanding of the nature of the spin relaxation process in zero magnetic fields (ZMFs). Notably, most spin-orbit mechanisms of spin lattice relaxation in semiconductor nanostructures considered so far [5] yield no decay of spin coherence in zero fields. In fact, it is commonly accepted that in the absence of an external magnetic field and hyperfine interaction (i) the time-reversal symmetry of a system prevents direct transitions between Kramers degenerate states even in the presence of spinorbit coupling (SOC), and (ii) adiabatic isolation of the Kramers doublet (KD) will make the spin relaxation ‘‘singularly ineffective.’’ It might therefore seem that in the ZMF the adiabatic motion of the lattice cannot have any effect on the spin relaxation of s-like electrons in the lowest conduction band. This assumption, however, is not correct. Fifty years ago Elliott [6] established that in the absence of external magnetic field the rate of spin relaxation in nonmagnetic semiconductors and metals, 1=TS , is proportional to the rate of the relaxation of the crystal momentum, 1=p : $2 jj1 TS1 ajjg p ;

(1)

$ $ g ge 1^ is the deviation of the ‘‘g tensor’’ where g from the free-electron value and a is a constant in the range 1–10 [7]. The relation (1) was obtained in the tight-binding approximation by introduction the spin-orbit interaction ~ where S~ and L~ denote the electron spin HSO L~ S, and orbital momentum, into the crystal Hamiltonian. The presence of SOC leads to a mixing of spin-up and spin~ in the different ~ : j; down Bloch functions j; ~ ki ~ ki

0031-9007=04=93(26)=266601(4)$22.50

~ where bands with the same crystal momentum k~ : k, ~ ~ is the vector of Pauli matrices and represents the set of parameters necessary to uniquely identify the orientation ~ Because of this mixing, the adiabatic (Ep 1) of k. scattering of a wave vector k~ ! k~0 by fluctuations of the anisotropic part of electron-lattice Coulomb interaction results in a nonzero spin-flip probability P; ~ k~ ! ~ 0 ; k~0 sin2 # ~ ~0 , where # ~ ~0 is the angle between k~ and k~0 . The k;k

k;k

formula (1) is valid to the first order of =E 1, where E is the energy separation from the considered band state ~ and characterize the to the nearest one with the same k, amplitude of the matrix element of HSO between these states. The Elliott relation passed the experimental tests (see, e.g., Ref. [7], and references therein) for many alkali and noble metals, and bulk semiconductors with fairly small interband gap (to satisfy adiabatic conditions E should be larger than 1 p ) and relatively large SOC. Although adiabatic motions do not change the eigenvalues of the Hamiltonian, they affect the eigenvectors, thereby giving rise to phase shifts and to transitions between quantum states that are geometric in nature [8]. It is well recognized that the nontrivial gauge potentials, either Abelian or non-Abelian, which appear in systems that undergo slow loops in the parameter space lead to a geometric phase of a state vector [8]. It has been shown [3,4,9– 12] that due to SOC continuous and coherent adiabatic rotation of an external electric field leads to the difference in the geometric phase acquired by the components of the KD. This phase shift will increase linearly in time, which is equivalent to spin precession or zero-field splitting [3,13]. These transitions between the doubly degenerate Kramers states depend only on the trajectory of the electric field in the angular space. Thereby an intriguing connection between the geometry, gauge fields, and the specific form of spin-electric coupling—spin-rotation interaction —was revealed. Note that a revolving electric field violates the T invariance of the system and, hence, there is no contradiction to Kramers theorem. If adiabatic rotation is incoherent, the resulting transition phase shift will be random and could lead to dephasing, which is equivalent to relaxation [14 –16] of the

266601-1

 2004 The American Physical Society

PRL 93, 266601 (2004)


pseudo-spin-1=2 that represents the KD in ZMF (spin is not a good quantum number in the presence of SOC). Because of enormous generality and nonmodel character of the geometric results, many old problems appear in a new light. Here, we will show that the Elliott’s relation, Eq. (1), arises from the random acquisition of geometric phases and represents a special case of a more general situation —relaxation of the pseudo-spin-1=2 induced by stochastic gauge fields. In the Elliott mechanism of spin relaxation, the loss of coherence occurs only in the short time intervals during collisions. To describe the evolution of the KD throughout the particular scattering event, it is convenient to transform the basis into the moving (M) frame of reference that ~ M t Rt L t, follows the adiabatic rotation of k, where is the instantaneous eigenvector of the total, nontruncated Hamiltonian of the crystal, H. The rotation ~ maps the space-fixed laboraoperator R expin^~ J tory (L) or reference frame into the actual orientation of the M frame at time t. Recall that in the presence of SOC the Bloch functions are not factorizable into the orbital and spin parts; hence, the total electron angular momentum, ~ is included into the transformation Rt. The J~ L~ S, quantization axis of the system is chosen along the direction of k~ and corresponds to the Z axis of the M frame. Then, the unit vector n^~ denotes the instantaneous axis of the k~ ! k~0 rotation, and the angle of this rotation is denoted as . Note that in the M frame the total Hamiltonian of the _ problem must be replaced with H M R1 HR iR1 R. The first term is static, while the second one is the additional gauge potential that appears in the M frame. In crystals with inversion symmetry in ZMF, the s-like electron Bloch function is doubly degenerate at any instanta~ neous orientation of kt. Thus, for the Bloch KD adiabatically isolated from the rest of the band structure, transformation into the rotating M basis yields the following Schro¨dinger-type equation (for details, see Refs. [3,12]): M M i _ M KD t Heff t KD t;

(2)

M t : 1=2!t ~ $M ~ M iAM Heff WZ t:

(3)

M t : To simplify notations, we introduce KD M ~ j~ ; kt; ci, where c is a conduction-band index, and h has been set equal to unity. The AM WZ is the Wilczek-Zee non-Abelian gauge potential [8], !t ~ is an instantaneous angular velocity of the M frame relative to the L frame at time t. The ‘‘tensor’’ $M is defined by the expression [12] M $M M : 1 ~L 1=2 ~ PM KD R tJ RtPKD ;

(4)

where PM KD is the projector onto the complex 2D Hilbert space spanned by the KD. The Schro¨dinger-type Eq. (2)


and the expression (3) depend on a choice of gauge that specifies the reference orientation, i.e., the orientation in which the M frame coincides with some space-fixed frame. At the moment t 0, this orientation may always be chosen such that $M is diagonal and that the main axis Z of this tensor represents the quantization axis of the : ~ M =2. Clearly, transformapseudospin operator S~M eff tion into the M frame is the gauge transformation and is responsible for the appearance of the gauge potential M AWZ t in Eq. (3). Suppose that during a short time of a collision, "tc , the plain of the k~ ! k~0 rotation remains constant. In this situM ation, the gauge potential AWZ t lost its non-Abelian M character (Heff commutes with itself at different moments t) and Eq. (2) is readily integrable [3]: ~ X ~ M u~ ZM t Trf~ M Z expi!t? =2 Z g cos!t? : M Here the axis of rotation n^~ is assigned to X, ? : $ XX $M YY , ! is assumed to be constant, and we introduce the ~ M , where polarization vector, u~ M t : Tr&M KD t M M M &KD t : j KD tih KD tj is the corresponding density operator. Now, to describe the evolution of the KD during a collision in the local reference frame, we have to perform a reverse rotation of the basis compensating for the rotation of the M frame, thereby closing the path by the geodesic which yields the following result: u~ L ~ L ~ X ~ L Z t Trf Z expi!t"? =2 Z g cos!t"? ;

(5)

where "? : ? 1. We would like to emphasize that Eq. (5) is applicable only during the collision (t "tc , !t '), in the local reference frame that reflects the geometry of the particular scattering event. This simple form can be easily rationalized. The effecM tive spin-Hamiltonian Heff , Eq. (3), can be viewed as a generic Zeeman Hamiltonian of a spin-1=2 particle in an ‘‘effective’’ time-dependent magnetic field !t ~ $M that appears during the collision in the frame that follows the ~ In the local reference frame, the adiabatic rotation of k. L differential action of Heff is proportional to the angle of $M ^ rotation, j!t ~ 1jdt, i.e., to the distance in the angular space, which reveals the geometric character of the phenomenon. Accordingly, as long as the reorientation of a crystal momentum represents an adiabatic perturbation to the system, the evolution of the spinor L KD depends ~ only on the path traveled by k in the angular space and is independent of the way the system moves along that path. In other words, the geometric Berry-phase shift acquired by the spin-up and spin-down components of the KD during a collision depends only on the angular distance,

266601-2

PRL 93, 266601 (2004)


#k;~ k~0 !"tc . The polarization vector follows the reorientation of the lattice momentum, but is generally falling somewhat behind ("? #k;~ k~0 ). We are ready to address the effect of random collisions that result in a stochastic acquisition of Berry’s phase. First, we consider 2D crystals, where the axis X of the ^~ is constant and can be chosen to local reference frame (n) be at right angle to the lateral plane. Suppose that an average angle of the in-plain k~ ! k~0 rotation is small (#k;~ k~0 1). Then the stochastic scattering process of a wave vector can be safely modeled by the one-dimensional diffusion in the angular space. In this case, the gauge potential in Eq. (3) is ‘‘Abelianized’’ and Eq. (5) can be easily averaged, h i, over the stochastic ensemble with the probability density function P#k;~ k~0 ; t 4'Dt1=2 exp#k;2~ k~0 =4D1 t. Integration over all possible angles 2 #k;~ k~0 gives hu~ L Z ti exp"? D1 t, i.e., an exponential decay of spin coherence with the rate

1=TS 2D "2? D1 ;

(6)

proportional to the one-dimensional diffusion coefficient D1 . The 3D case is generally more complex since n^~ can change its direction in time, so the elementary rotations in the local and the mesoscopic reference frames may not commute (non-Abelian case). Note that all relevant information about the crystal Hamiltonian, which comprises the actual physical problem, is now represented by the ‘‘ tensor.’’ The original, nontruncated H serves only to determine the gauge group and the principal values of $M . The explicit form of PM KD and, thus, $M depends on the problem at hand. Examples $ of ‘‘ tensor’’ calculations can be found in Ref. [12]. It has been shown [12] that, generally in the case of weak SOC to the first order in =E, $ M

$M 1^ g :

In this situation, it is natural to assume that j!t ~ $M ^ 1j"t ~ k~0 1. From the geometric point c 1, even for #k; of view, this means that on the average the state of a ~ quantum system, represented by huti, is independent of the particular position in the angular space. Equivalently, the rate of transitions between m 1=2 levels of the Bloch KD is much slower than the inverse correlation time of the fluctuating effective Hamiltonian, Eq. (3). Thus, the 3D problem reduces to the traditional calculations of the relaxation operator in the ‘‘fast motional’’ limit that allows us to carve the following result (see [17] for details): 1=TS 3D 4=3"2? h!2 i! 4=3g2? 1 p ;

(7)

2 where we introduce 1 p h! i! , and ! is the correlation time of !. ~


The way we derived Eq. (7) reveals the geometric origin of Elliott spin-decoherence mechanism, Eq. (1). It is fundamentally connected to the admixture of ‘‘fast’’ orbital (spatial) degrees of freedom to electron spin wave functions. The states that are coupled by SOC to form the Bloch KD have energy separations of electron interband excitation. Therefore, high frequencies will characterize the time-dependent response of the system to motionally induced perturbation acting towards the change in the mixing coefficients of the zero order wave functions. As a result, during the ‘‘slow’’ scattering event (E 1 ! ) electron pseudospin will adiabatically follow the rotation of the crystal momentum. Fundamentally, this effect can be described as a manifestation of a relevant gauge potential and can be represented in purely geometric terms as a consequence of the corresponding connection [8]. Our results, Eqs. (6) and (7), have a nonmodel geometric origin and, hence, do not depend on the system being electronic or nuclear. The geometric dephasing in the context of zero-field nuclear quadrupole resonance experiments was considered by Jones and Pines [15], who derived a decoherence rate of 131 Xe nuclei induced by thermal collisions with the walls of a toroidal container. They start from the dynamic evolution of the nuclear KD adiabatically isolated from the rest of a spin multiplet (nuclear spin I > 1=2), which can be described by Eqs. (2) and (3) with [3,8,12] jj 1, ? I 1=2. It is easy to see that for 131 Xe (I 3=2) nuclear pseudospin (jmj 1=2) just follows the rotation of the Z axis of the L M-frame Heff t ! ~ X t~ L ~ Y t~ L X ! Y =2, which in this case coincides with the main axes of the tensor of quadrupolar interaction. The one-dimensional diffusion model utilized by Jones and Pines then yields 1= TI3=2;jmj1=2 2D D1 , which represents a special case of the generalized Elliott’s relation, Eq. (6), that is valid beyond the constraint of original theory ( =E 1). To summarize, it has been shown that the Elliott spindecoherence has the geometric origin and, thus, is model independent. This result shows that there is the fundamental upper bound on spin coherence time of s-like electrons in the lowest conduction band. The geometric approach gives an opportunity to apply Elliott’s ideas for a wide range of systems and system parameters in solid-state or atomic scenarios. I wish to thank Professor Ulrich Steiner for numerous stimulating discussions that inspired the study presented here.

*Electronic address: [email protected] [1] Y. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Nature (London) 427, 50 (2004); Y. Kato, R. C. Myers, D. C. Driscoll, A. C. Gossard, J. Levy, and D. D. Awschalom, Science 299, 1201 (2003); J. B. Miller, D. M. Zumbuhl, C. M. Marcus, Y. B. Lyanda-Geller,

266601-3

PRL 93, 266601 (2004)

[2] [3] [4] [5]

[6] [7] [8]

[9]


D. Goldhaber-Gordon, K. Campman, and A. C. Gossard, Phys. Rev. Lett. 90, 076807 (2003). E. I. Rashba and Al. L. Efros, Phys. Rev. Lett. 91, 126405 (2003). Yu. A. Serebrennikov, Phys. Rev. B 70, 064422 (2004). B. A. Bernevig and S.-C. Zhang, quant-ph/0402165. V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev. Lett. 93, 016601 (2004); Y. G. Semenov and K. W. Kim, Phys. Rev. Lett. 92, 026601 (2004); A. V. Khaetskii and Yu. V. Nazarov, Phys. Rev. B 64, 125316 (2001); L. M. Woods, T. L. Reinecke, and Y. B. Lyanda-Geller, Phys. Rev. B 66, 161318 (2002); B. A. Glavin and K. W. Kim, Phys. Rev. B 68, 045308 (2003). R. J. Elliott, Phys. Rev. 96, 266 (1954). I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). M. V. Berry, Proc. R. Soc. London A 392, 45 (1984); F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984); A. Zee, Phys. Rev. A 38, 1 (1988). C. A. Mead, Phys. Rev. Lett. 59, 161 (1987); J. Segert, J. Math. Phys. (N.Y.) 28, 2102 (1987).


[10] A. V. Balatsky and B. L. Altshuler, Phys. Rev. Lett. 70, 1678 (1993). [11] A. G. Aronov and Y. B. Lyanda-Geller, Phys. Rev. Lett. 70, 343 (1993). [12] U. E. Steiner and Yu. A. Serebrennikov, J. Chem. Phys. 100, 7503 (1994); U. E. Steiner and D. Bursner, Z. Phys. Chem. Neue Folge 169, 159 (1990); Yu. A. Serebrennikov and U. E. Steiner, Mol. Phys. 84, 627 (1995). [13] R. Tycko, Phys. Rev. Lett. 58, 2281 (1987); J. W. Zwanziger, M. Koenig, and A. Pines, Phys. Rev. A 42, 3107 (1990); S. Appelt, G. Wackerle, and M. Mehring, Phys. Rev. Lett. 72, 3921 (1994). [14] Yu. A. Serebrennikov and U. E. Steiner, J. Chem. Phys. 100, 7508 (1994); Chem. Phys. Lett. 222, 309 (1994). [15] J. A. Jones and A. Pines, J. Chem. Phys. 106, 3007 (1997). [16] F. Gaitan, Phys. Rev. A 58, 1665 (1998). [17] J. McConnell, The Theory of Nuclear Mafnetic Resonance in Liquids (Cambridge University Press, Cambridge, 1987).

266601-4