Spin relaxation due to polar optical phonon scattering

Journal of Physics and Chemistry of Solids 73 (2012) 444–447

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Spin relaxation due to polar optical phonon scattering M. Idrish Miah a,b,n, E. MacA. Gray b a b

Department of Physics, University of Chittagong, Chittagong-4331, Bangladesh Queensland Micro- and Nanotechnology Centre, Griffith University, Nathan, Brisbane, QLD 4111, Australia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 21 December 2010 Received in revised form 8 September 2011 Accepted 21 November 2011 Available online 28 November 2011

Spin relaxation due to polar optical phonon scattering in semiconductors was investigated. The relaxation of the electron spin was found to increase with increasing the strength of the electric field. However, a high field completely depolarized the electron spin due to an increase of the spin precession frequency of the hot electrons, suggesting that high field transport conditions might not be desirable for spin-based technology with these semiconductors. It was also found that spin relaxation decreases with increasing moderately n-doping density or decreasing temperature. The results were discussed in comparison with the data available in the literature. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction Because of the increasing interest in spin-sensitive electronic or spintronics, much attention has been observed recently in the transport of spin-polarized carriers in semiconductors [1–3]. Spintronics is a revolutionary new class of electronics based on the spin degree of freedom of the electron in addition to, or in place of, the charge and is a multidisciplinary field involving physics, chemistry, materials science, and engineering [4–6]. However, one of the important requirements necessary in developing spintronic devices is the transportation of the spin-polarized carriers in semiconductors reliably (i.e. without spin relaxation or the loss of spin polarization) over reasonable distances that are comparable to the device dimensions. The determination of spin relaxation, or the spin lifetime, is extremely important, in particular for electronic applications, because if the spins relax too fast, the distance travelled by the spin-polarized current will be too short to serve any practical purpose. Three major spin relaxation mechanisms for semiconductors are discussed in the literature. They are the Elliott–Yafet (EY) [7], Bir– Aronov–Pikus (BAP) [8], and Dyakonov–Perel (DP) [1]. In the EY mechanism, the spin–orbit (SO) interaction leads to a mixing of wave functions of opposite spin, which results in a nonzero electron spin-flip due to impurity and phonon scattering. For a III–V semiconductor (such as GaAs), for example, the EY relaxation is less effective due to the large bandgap (Eg) and low scattering rate. The BAP mechanism, important in either p-doped or insulating semiconductors and at very low injection energies, is due to the electron-hole exchange interaction. For n-doped materials, as holes

n Correspondence address: Department of Physics, University of Chittagong, Chittagong-4331, Bangladesh. E-mail address: m.miah@griffith.edu.au (M.I. Miah).

0022-3697/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2011.11.023

are rapidly recombined with electrons due to the presence of a large number of electrons, spin relaxation due to the BAP mechanism is usually blocked [9]. Finally, the DP mechanism is due to SO coupling in semiconductors lacking inversion symmetry. Sufficiently long spin lifetime and the possibility of manipulate the spin orientation (have control over spin relaxation) are required for the development of practical spintronic devices. Electron spin lifetime ts in semiconductors have extensively been measured by means of the Hanle effect [10], time-resolved photoluminescence [11], and magneto-optical Faraday/Kerr effects [9], which are suitable to measure ts in the absence of holes and are optical methods. Although future spintronics applications, e.g. Datta–Das spin transistor (spin polarization of the carriers is controlled by the electric field applied by the gate voltage, or by the Rashba effect) [12], are likely to depend on drifting transport of spins, very little attention has been paid to the influence of an electric drift field (E) on ts. Based on a Monte Carlo method, the DP electron spin relaxation under drift in GaAs was studied [13]. A very recent Monte Carlo approach was for study the temperature dependences of both ts and the spin polarization length [14]. Scattering from ionized impurities, and optical and acoustical phonons was used to explain the spin relaxation rates in various semiconductors [15]. They studied the doping and temperature dependences of spin relaxation using different scattering mechanisms (BAP, DP, EY). The high-field transport in GaAs was also investigated very recently [16]. Here, an approach, so-called Ehrenreich’s variational approach, has been introduced to investigate the spin relaxation due to longitudinal polar optical phonon (POP) scattering in n-type GaAs, without requirements of the specialized and massive computations. During transport in the electric field, the inclusion of POP in the scattering process is because of increasing the electron temperature via the energy-independent nature of the dominant energy relaxation process, leading to the DP spin relaxation. As,

M.I. Miah, E.M. Gray / Journal of Physics and Chemistry of Solids 73 (2012) 444–447

for the n-doped GaAs, DP is the dominant mechanism, the spin relaxation calculations are performed based on only it via the POP interaction. Other mechanisms are not considered. The calculated results are discussed and are compared with those obtained in theories and experiments.

2. Calculations, results and discussion The origin of the field-dependent efficient electron spin relaxation in n-doped GaAs is discussed based on the DP spin relaxation mechanism [1]. DP mechanism is due to SO coupling in semiconductors lacking inversion symmetry. In III–V semicon! ductors, the degeneracy in the conduction band is lifted for k a 0 due to the absence of inversion symmetry. Without inversion symmetry the momentum states of spin-up (m) and spin-down (k) electrons are not degenerate, i.e. E! a E! , where km kk E! ðE! Þ is the momentum-dependent electron energy with km kk spin m(k). The resulting energy difference, for electrons with the ! same k but different spin states, plays the role of an intrinsic magnetic field, known as effective magnetic field [9], i !! a_2 h 2 2 h ð k Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kx ðky kz Þx^ þ CP , 2em*Eg

ð1Þ

where CP stands for cyclic permutations, _ ¼ h=2p is the reduced Planck constant and a is a dimensionless, material-specific parameter which gives the magnitude of the SO splitting and is approximately given by 4Z mn a C pffiffiffiffiffiffiffiffiffiffi , 3Z mcv

ð2Þ

where Z ¼ D=ðEg þ DÞ, D is the SO splitting of the valence band, mn is the electron’s effective mass and mcv is a constant close in magnitude to free electron mass m0, induced by the presence of the Dresselhaus (due to the bulk inversion asymmetry [17]) SO interaction in a zinc-blende structure, acting on the spin with its ! magnitude and orientation depending on k . This effective magnetic field results in spin precession (spin relaxation) with ! intrinsic Larmor frequency os ð k Þ during the time between ! ! ! collisions, according to the relation d P =dt ¼ os ð k Þ  P , where ! !! ! os ð k Þ ¼ ðe=mn Þ h ð k Þ and P is the electron spin polarization vector. The corresponding Hamiltonian term due to spin-orbital splitting of the conduction band describing the precession of electrons in the conduction band is ! ! _! HSO ð k Þ ¼ s dos ð k Þ, ð3Þ 2 ! where s is the vector of Pauli spin matrices. In a quantum well (QW), for example, DP Hamiltonian is composed of the Dresselhaus [17] and Rashba terms [18]. The Rashba term appears if the self-consistent potential within a QW is asymmetric along the growth direction and is therefore referred to as a structural inversion asymmetry contribution. The increased electron momentum at higher electric fields !! brings about a stronger h ð k Þ and, consequently, the electron ! precession frequency os ð k Þ becomes higher. The effective magnetic field depends on the underlying material, on the geometry ! of the device, and on k . Momentum-dependent spin procession described by the DP Hamiltonian of Eq. (3), together with momentum scattering characterized by momentum relaxation time leads to DP spin relaxation. Since the magnitude and the ! direction of k changes in an uncontrolled way due to electron scattering with the environment, this process contributes to spin

445

relaxation, given by [9] 1

ts

¼g

a2 _

3 E! tp k , Eg

ð4Þ

where tp ðE!Þ is the momentum relaxation time, E! ¼ kB T e is the k k thermal energy, Te is the electron temperature, kB is the Boltzman constant and g is a dimensionless factor that ranges from 0.8 to 2.7 depending on the dominant momentum relaxation process. For example, for scattering by POP or piezoelectric phonons g E0.8, while scattering by ionized impurities gives g E1.5, and scattering by acoustic phonons g E2.7 [9]. DP spin relaxation in a bulk zinc-blende structure occurs due !! to the spin precession about h ð k Þ induced by the presence of the Dresselhaus SO interaction. During transport in the electric field, electrons are accelerated to higher velocities at higher fields, where the Te increases sharply due to the energy-independent nature of the dominant energy relaxation process via the longitudinal POP scattering (g E0.8) [19]. The resulting high Te leads 3 to enhanced DP spin relaxation (1=ts  E! tp ) because they have k large kinetic energy between successive collisions. We want to evaluate ts using the calculated value of the drift velocity (vd) from the momentum-balance equation incorporating the hot phonon effect [20,21]. We obtain the solution of the momentum-balance equation as pffiffiffiffi 3 peto E vd ¼ ½fð1 þf ÞeG þ f gB1 eG=2 þ ffð1 þ f ÞeG f gB0 eG=2 g1 : m*G3=2 ð5Þ

Now replacing the phonon occupancy f ðoÞ by its thermodynamic equilibrium value f 0 ðoÞ, obtained from the steady-state solution of the equation describing the rate of change of f ðoÞ in the relaxation time approximation: f¼

f 0 þðtp =t0 Þðn=Ds GÞeG , 1 þðtp =t0 Þðn=Ds GÞð1eG Þ

we obtain pffiffiffiffi      3 petef f E tp n vd ¼ 1 þf 0 þ eG þ f 0 B1 eG=2 3=2 t0 Ds G m*G i1 G G=2 , þ ffð1þ f 0 Þe f 0 gB0 e g

ð6Þ

ð7Þ

where Ds(Te) is the effective density of states, to is the characteristic time for POP interaction, G ¼ _o=kB T e and B0 ðG=2Þ and B1 ðG=2Þ are the zero- and first-order Bessel functions. The constant to is the characteristic time for the polar interaction and is given by rffiffiffiffiffiffiffiffiffiffi  1 e2 o mn e0 ep ¼ , ð8Þ t0 h 2_o e0 ep where e0 (e1 ) is the static (high-frequency) permittivity. Using parameter values for GaAs [19], we obtain t0 E0.1 ps. The POP effect manifests itself in the effective time constant teff given by

tef f ¼ t0 þ tp ðn=Ds GÞð1eG Þ:

ð9Þ

The power dissipated per electron in a drifted Maxwellian distribution [21], after incorporating hot phonons by eliminating f ðoÞ, can be obtained as pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2=p_o

eEvd ¼ ð1 þ f 0 ÞeG f 0 G=2B0 eG=2 : ð10Þ

to

Eqs. (7) and (10) establish a relation between electric field and electron temperature, or a relation between electric field and drift velocity.

M.I. Miah, E.M. Gray / Journal of Physics and Chemistry of Solids 73 (2012) 444–447

where tE! ðE!Þ is the energy relaxation rate and the angular k k brackets denote the averaging over the thermal distribution. In the calculation of ts, we first evaluate tp including the contribution from the longitudinal POP scattering. According to the Ehrenreich’s variational approach [23], tp is obtained as rffiffiffiffiffiffiffi  4_ m0 e0 e1 eYPOP 1 tp ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G ex ð12Þ n YPOP POP 3 pR0 kB T e m e0 e1 where YPOP ¼ yPOP =T e , yPOP is the longitudinal POP frequency (in units of temperature), R0 is the Rydberg constant and GPOP ðe, yÞ in general is a factor depending on the scattering mechanisms, the average inertial masses associated with the mobility and the sign of the charge carrier. Here, e ¼ x=kB T e , x is the Fermi energy and y ¼ _yPOP =kB T e ¼ _YPOP =kB . The function GPOP exp ðxÞ is calculated, as in Ref. [24], as a function of the carrier temperature and density. Values of the material parameters used in the calculation are taken from Ref. [19] for GaAs. Figs. 1–3 show the results. As can be seen from Fig. 1, the spin lifetime decreases with E, while spin relaxation frequency increases. It is also seen that the spin relaxation frequency is rapid at higher fields and is almost infinite for field higher than 300 mV mm  1, which might be as a result of increase of the electron temperature at higher fields and, consequently, the higher DP spin relaxation frequency. For electric fields higher than 300 mV mm  1, it is relevant to note that in GaAs a significant reduction of the spin lifetime is caused by the spin-orbit coupling in L-valleys stronger with respect to that in G-valley [25]. This field-dependence agrees well with those of the transport experiment reported in Refs. [26,27], where authors showed that the photo-generated spins could travel without losing their initial spin orientation as long as E was below 100 mV mm  1 and the spin relaxation rate increased rapidly with E and the polarization disappeared at  350 mV mm  1. The results are also consistent with the results of the simulation performed by others [13]. They showed that for relatively low fields up to 100 mV mm  1, a substantial amount of spin polarization is preserved for several

1/ττs (THz)

0.6

0.4

0.2

0.0

100

200

300

E (mV

400

500

μm-1)

Fig. 1. Spin relaxation frequency vs. drift field for n¼1.5  1016 cm  3 at TL ¼ 300 K.

80

τs (ps)

In order to evaluate the DP spin relaxation quantitatively, we carry out a calculation of ts as a function of E. We first calculate the drift-field and lattice temperature (TL) dependences of Te using the relation [20,22]

E! 2evd E k * +, ð11Þ Te ¼ TL þ 3kB E!=tE! k k

75

70

0

100

200

300

TL (K) Fig. 2. Spin relaxation time as a function of the lattice temperature for n ¼ 1.5  1016 cm3 at E ¼100 mV mm  1.

80

τs (ps)

446

70

60

0

10x1015 n (cm-3)

20x1015

Fig. 3. Spin relaxation time as a function of the doping at E¼ 100 mV mm  1 and TL ¼300 K.

microns at TL ¼300 K and the DP spin relaxation frequency increased rapidly for fields higher than 150 V mm  1. Spin relaxation time as a function of the lattice temperature is shown in Fig. 2. As can be seen, ts increases with decreasing TL, in consistence with the experimental [28,29] as well as the theoretical [13,14] findings reported earlier. The spin relaxation time was also found to increase the moderately n-type doping (Fig. 3). The introduction of n-type dopants in semiconductors increases ts, because the electronic spin polarization in these systems survives for longer times. Studies of spin precession in GaAs reveal that moderately n-type doping yields significantly extended tS and show that tS increases with decreasing TL [5]. A similar trend was observed in another experiment [30], which studied the bias dependence of the spin-polarization of electrons and showed that several percentage of polarization was increased by increasing n-type doping.

3. Conclusions Spin relaxations via the polar optical phonon scattering in dependences of moderately doping density and temperature were investigated in n-type GaAs. The spin lifetime of electrons was

M.I. Miah, E.M. Gray / Journal of Physics and Chemistry of Solids 73 (2012) 444–447

found to decrease with increasing the strength of the electric field, while a high field completely destroyed the electron spin polarization due to an increase of the spin precession frequency of the hot electrons. It was also found that the spin lifetime increases with increasing moderately doping density or decreasing temperature. The results were discussed and were compared with those obtained in theoretical investigations and photospintronics experiments. References [1] M.I. Dyakonov, V.I. Perel, Zh. Eksp. Teor. Fiz. 60 (1971) 1954; M.I. Dyakonov, V.I. Perel, Sov. Phys. JETP 33 (1971) 1053. [2] M.I. Dyakonov, A.V. Khaetskii, Spin Hall Effect, in: M.I. Dyakonov (Ed.), Spin Physics in Semiconductors, Springer-Verlag, Berlin, 2008. [3] M.W. Wu, J.H. Jiang, M.Q. Weng, Phys. Rep. 493 (2010) 61. [4] S.D. Sarma, Am. Sci. 89 (2001) 516. [5] D.D. Awschalom, D. Loss, N. Samarth (Eds.), Semiconductor Spintronics and Quantum Computation, Springer, Berlin, 2002. [6] M. Idrish Miah, J. Optoelectron. Adv. Mater. 10 (2008) 2487–2493. [7] R.J. Elliott, Phys. Rev. 96 (1954) 266. [8] G.L. Bir, A.G. Aronov, G.E. Pikus, Sov. Phys. JETP 42 (1976) 705. [9] G.E. Pikus, A.N. Titkov, in: F. Meier, B.P. Zakharchenya (Eds.), Optical Orientation, Modern Problems in Condensed Matter Science, vol. 8, NorthHolland, Amsterdam, 1984. [10] R.R. Parsons, Phys. Rev. Lett. 23 (1969) 1152. [11] A.P. Heberle, W.W. Ruhle, K. Ploog, Phys. Rev. Lett. 72 (1994) 3887.

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