Electron spin relaxation in paramagnetic Ga(Mn)As quantum wells

Electron spin relaxation in paramagnetic Ga(Mn)As quantum wells

J. H. Jiang,¹Y. Zhou,¹T. Korn,²C. Schüller,²and M. W. Wu^1,

*

1Hefei National Laboratory for Physical Sciences at Microscale and Department of Physics, University of Science and Technology of China, Hefei, Anhui, 230026, China

2Institut für Experimentelle und Angewandte Physik, Universität Regensburg, D-93040 Regensburg, Germany 共Received 8 January 2009; revised manuscript received 25 February 2009; published 1 April 2009兲 Electron spin relaxation in paramagnetic Ga共Mn兲As quantum wells is studied via the fully microscopic kinetic spin Bloch equation approach where all the scatterings, such as the electron-impurity, electron-phonon, electron-electron Coulomb, electron-hole Coulomb, electron-hole exchange 共the Bir-Aronov-Pikus mecha- nism兲and thes-dexchange scatterings, are explicitly included. The Elliott-Yafet mechanism is also incorpo- rated. From this approach, we study the spin relaxation in bothn-type andp-type Ga共Mn兲As quantum wells.

Forn-type Ga共Mn兲As quantum wells, where most Mn ions take the interstitial positions, we find that the spin relaxation is always dominated by the D’yakonov-Perel’共DP兲mechanism in the metallic region. Interestingly, the Mn concentration dependence of the spin relaxation time is nonmonotonic and exhibits a peak. This is due to the fact that the momentum scattering and the inhomogeneous broadening have different density depen- dences in the nondegenerate and degenerate regimes. Forp-type Ga共Mn兲As quantum wells, we find that the Mn concentration dependence of the spin relaxation time is also nonmonotonic and shows a peak. The cause of this behavior is that thes-dexchange scattering共or the Bir-Aronov-Pikus兲mechanism dominates the spin relaxation in the high Mn concentration regime at low共or high兲 temperature, whereas the DP mechanism determines the spin relaxation in the low Mn concentration regime. The Elliott-Yafet mechanism also contrib- utes to the spin relaxation at intermediate temperatures. The spin relaxation time due to the DP mechanism increases with increasing Mn concentration due to motional narrowing, whereas those due to the spin-flip mechanisms decrease with it, which thus leads to the formation of the peak. The temperature, photoexcitation density, and magnetic field dependences of the spin relaxation time in p-type Ga共Mn兲As quantum wells are investigated systematically with the underlying physics revealed. Our results are consistent with the recent experimental findings.

DOI:10.1103/PhysRevB.79.155201 PACS number共s兲: 72.25.Rb, 75.50.Pp, 71.10.⫺w, 71.70.Ej I. INTRODUCTION

Semiconductors doped with magnetic impurities have in- trigued much interest since the invention of ferromagnetic III-V semiconductors due to the possibility of integrating both the magnetic 共spin兲 and charge degree of freedom on one chip.^1–7 Many new device conceptions and functional- ities based on these materials are proposed, and the material properties together with the underlying physics are exten- sively studied.^4,8–16 Specifically, ferromagnetic Ga_1−xMn_xAs has been used as a highly efficient source to inject spin po- larization into GaAs共Ref. 17兲and magnetic tunneling junc- tions based on ferromagnetic Ga_1−xMnxAs can achieve very high magnetoresistance.¹⁸ Besides, the ability to detect the magnetic moment via Hall measurements^4,19and to control it via gate-voltage²⁰ and laser radiation²¹ opens the way for incorporating optoelectronics with magnetism. Magneto- optical measurements, which could characterize the spin splitting of carriers due to both the external magnetic field and thes-d or p-d exchange field, provide important infor- mation about the microscopic properties of the carriers, such as, thegfactor, thes-dandp-dexchange coupling constants, as well as the electron spin relaxation time共SRT兲. Such mea- surements have recently been performed in dilutely doped paramagnetic Ga共Mn兲As quantum wells.^22–25Although many properties of Ga共Mn兲As have been extensively studied, the electron spin relaxation has not yet been well understood even in dilutely doped paramagnetic phase. This is the aim of

this investigation. We focus on 共001兲 Ga共Mn兲As quantum wells.

Electron spin relaxation in nonmagnetic GaAs has been extensively studied and three main spin relaxation mecha- nisms have been recognized for decades:²⁶ the D’yakonov- Perel’ 共DP兲 mechanism,²⁷ the Bir-Aronov-Pikus 共BAP兲 mechanism²⁸ and the Elliott-Yafet 共EY兲mechanism.²⁹ Usu- ally, the DP mechanism dominates the spin relaxation in n-type quantum wells.^10,30 The BAP mechanism was be- lieved to be most important at low temperature in intrinsic and p-type quantum wells for a long time.^10,26 Recently, Zhou and Wu³¹showed that the BAP mechanism was exag- gerated in the low-temperature regime in previous treatments based on the elastic-scattering approximation, where the Pauli blocking was not considered. It was then found that the BAP mechanism is less efficient than the DP mechanism in intrinsic quantum wells andp-type quantum wells with high photoexcitation density. A similar conclusion was also ob- tained in bulk GaAs very recently.³² Previously, the EY mechanism was believed to dominate the spin relaxation in heavily doped samples at low temperature. However, it was shown to be unimportant in bulk GaAs by our recent investigation.³² Whether this is still true in quantum-well systems remains unchecked. Moreover, in paramagnetic Ga共Mn兲As quantum wells, things are more complicated:共i兲 all the three mechanisms could be important as the material is heavily doped with Mn and the hole density is generally very high.²²共ii兲An additional spin relaxation mechanism due to the exchange coupling of the electrons and the localized

Mn spins共thes-dexchange scattering mechanism兲may also be important.⁸ In this work, we will compare different spin relaxation mechanisms for various Mn concentrations, tem- peratures, photoexcitation densities, and magnetic fields.

In Ga共Mn兲As, the Mn dopants can be either substitutional or interstitial; the substitutional Mn accepts one electron, whereas the interstitial Mn releases two. Direct doping in low-temperature molecular-beam epitaxy growth gives rise to more substitutional Mn ions than interstitial ones, which makes the Ga共Mn兲As ap-type semiconductor.^8,9,22Recently, it was found that in GaAs quantum wells near a Ga共Mn兲As layer, the Mn dopants can diffuse into the GaAs quantum well, where the Mn ions mainly take the interstitial positions, making the quantum well n type.^23,24,33 The experimental results also show interesting features of the SRT.

We apply the fully microscopic kinetic spin Bloch equa- tion 共KSBE兲 approach^34,35 to investigate the spin relaxation in paramagnetic Ga共Mn兲As quantum wells. The KSBE ap- proach explicitly includes all relevant scatterings, such as, the electron-impurity, electron-phonon, electron-electron Coulomb, electron-hole Coulomb, electron-hole exchange 共the BAP mechanism兲, ands-d exchange scatterings. Previ- ously, the KSBE approach has been applied to study the spin dynamics in semiconductor and its nanostructures where good agreement with experiments have been achieved and many predictions have been confirmed by experiments.30–32,34–49In this work, we apply the KSBE ap- proach to bothn- andp-type paramagnetic Ga共Mn兲As quan- tum wells to study the electron spin relaxation. We distin- guish the dominant spin relaxation mechanisms in different regimes and our results are consistent with the recent experi- mental findings.

This paper is organized as follows: in Sec.II, we set up the model and establish the KSBEs. In Sec. III we present our results and discussions. We conclude in Sec.IV.

II. MODEL AND KSBES

We start our investigation from a paramagnetic 关001兴 grown Ga共Mn兲As quantum well of width a in the growth direction共thezaxis兲. A moderate magnetic fieldBis applied along the xaxis共the Voigt configuration兲. It is assumed that the well width is small enough so that only the lowest sub- band of electron and the lowest two subbands of heavy hole are relevant for the electron and hole densities in our inves- tigation. The barrier layer is chosen to be Al_0.4Ga_0.6As where the barrier heights of electron and hole are 328 and 177 meV respectively.⁵⁰ The envelope functions of the relevant sub- bands are calculated via the finite-well-depth model.^30,31

The KSBEs can be constructed via the nonequilibrium Green’s function method⁵¹and read

⳵t␳^ˆk=⳵t␳^ˆk兩coh+⳵t␳^ˆk兩scat, 共1兲

with␳^ˆkrepresenting the single-particle density matrix whose diagonal and off-diagonal elements describe the electron dis- tribution functions and the spin coherence respectively.³⁴The coherent term is given by共ប⬅1 throughout this paper兲

⳵t␳^ˆk兩coh= −i

冋

册

共2兲 in which 关,兴 is the commutator. ge is the electron g factor.

h共k兲represents the spin-orbit coupling共SOC兲, which is com- posed of the Dresselhaus⁵²and Rashba⁵³terms. In symmetric GaAs quantum well with small well width, the Dresselhaus term is dominant⁵⁴ and

h共k兲= 2␥D共k_x共k_y²−具k_z²典兲,k_y共具k_z²典−k_x²兲,0兲. 共3兲 Here 具k_z²典 represents the average of the operator −共⳵/⳵^z兲² over the state of the lowest electron subband and ␥D

= 8.6 eV ų is the Dresselhaus SOC coefficient.³⁰ The mean-field contribution of the s-d exchange interaction is given by

Hˆ

sd

mf= −N_Mn␣具S典␴^ˆ

2, 共4兲

where具S典is the average spin polarization of Mn ions and␣ is the s-d exchange coupling constant. For simplicity, we assume that the Mn ions are uniformly distributed within and around the Ga共Mn兲As quantum well with a bulk densityN_Mn.

⌺ˆ

HF共k兲= −兺q,q_zV_q,q

z兩I共iqz兲兩²␳^ˆk−q is the Coulomb Hartree- Fock共HF兲term, whereI共iqz兲=兰dz兩␰e共z兲兩²e^iqzzis the form fac- tor with␰e共z兲standing for the envelope function of the low- est electron subband.³⁵ V_q,q

z is the screened Coulomb potential. In this work, we take into account the screening from both electrons and holes within the random-phase approximation.³¹

The scattering term ⳵t␳^ˆk兩scat consists of the electron- impurity, electron-electron Coulomb, electron-phonon, electron-hole Coulomb, electron-hole exchange and s-d ex- change scatterings. The expressions of all these terms except the s-d exchange scattering can be found in Ref. 31. How- ever, the expression of the electron-impurity scattering term with the EY mechanism included has not been given in that paper, which we will present later in this paper. The s-d exchange scattering term is given by

⳵t␳^ˆk兩sdscat= −␲^NMn␣^2Is

兺

␩1␩2k⬘

G_Mn共−␩1−␩2兲␦共␧k−␧k⬘兲

⫻关sˆ^␩1␳^ˆk^⬎⬘sˆ^␩2␳^ˆk⬍−sˆ^␩2␳^ˆk^⬍⬘sˆ^␩1␳^ˆk⬎+ H.c.兴. 共5兲

Here ␳^ˆk⬎= 1ˆ−␳^ˆk, ␳^ˆk⬍=␳^ˆk, G_Mn共␩1␩2兲=¹₄Tr共Sˆ^␩1Sˆ^␩2␳^ˆMn兲, and Is=兰dz兩␰e共z兲兩⁴.Sˆ^␩ andsˆ^␩共␩^{= 0 ,}⫾1兲 are the spin ladder op- erators with Sˆ⁰=Sˆz, Sˆ^⫾=Sˆx⫾iSˆy, sˆ⁰= 2sˆz, and sˆ^⫾=sˆx⫾isˆy.

␳^ˆMnis the Mn spin density matrix.␧k=k²/2m^ⴱis the electron kinetic energy with m^ⴱ denoting the effective mass. The equation of motion for Mn spin density matrix consists of three parts ⳵t␳^ˆMn=⳵t␳^ˆMn兩coh+⳵t␳^ˆMn兩scat+⳵t␳^ˆMn兩rel. The first part describes the coherent precession around the external magnetic field and the s-d exchange mean field, ⳵t␳^ˆMn兩coh=

−i关gMn␮BB·Sˆ−␣兺kTr共^␴₂^ˆ␳^ˆk兲·Sˆ,␳^ˆMn兴. The second part repre- sents the s-d exchange scattering with electrons ⳵t␳^ˆMn兩scat=

−^␲␣₄²兺_␩1␩2k␦共␧k−␧k⬘兲Tr共sˆ^−␩2␳^ˆk^⬍⬘sˆ^−␩1␳^ˆk⬎兲关共Sˆ^␩1Sˆ^␩2␳^ˆMn

−Sˆ^␩1␳^ˆMnSˆ^␩2兲+ H.c.兴. The third term characterizes the Mn spin relaxation due to other mechanisms, such as, p-d ex- change interaction with holes or Mn spin-lattice interactions, with a relaxation time approximation, ⳵t␳^ˆMn兩rel= −共␳^ˆMn

−␳^ˆMn

0 兲/␶Mn. Here ␳^ˆMn

0 represents the equilibrium Mn spin density matrix.␶Mnis the Mn spin relaxation time, which is typically 0.1–10 ns.⁵⁵ In our calculation we take␶Mn= 1 ns.

Att= 0, the Mn spin density matrix is chosen to be the equi- librium one␳^ˆMn

0 . The Mn spins can be dynamically polarized via thes-dexchange interaction, and feedback to the electron spin dynamics. However, we find that this process affects the electron spin dynamics little. Hence, the choice of␶Mndoes not affect our discussions on electron spin dynamics.

The s-d exchange scattering ␶sd can be obtained analyti- cally. In the absence of an external magnetic field, the spin polarization of the electron system is always along the z direction. As thes-dexchange interaction conserves the spin polarization of the total system, the spin polarization of Mn ions共which is assumed to be zero initially兲can only be along the z direction. By keeping only the diagonal term of␳^ˆMn, from the Fermi Golden rule, the spin relaxation time due to the s-d exchange scattering can be obtained directly,

␶sd=

再

冎

共6兲 whereS= 5/2 is the spin of the Mn ion. It is evident that␶sd

is independent of temperature and electron density, but is inverse proportional to Mn concentration N_Mn, the square of thes-dexchange coupling␣^{2 andI}s=兰dz兩␰e共z兲兩⁴.Isis deter- mined by the confinement of the quantum well. For an infi- nite depth well Is=_2a³, thus ␶sd is proportional to the well widtha.

After incorporating the EY mechanism, besides the ordi- nary spin-conserving term, there are spin-flip terms. For electron-impurity scattering these additional terms are

⳵t␳^ˆk兩eiEY= −␲ⁿi

兺

k⬘␦共␧k−␧k⬘兲关U_k−k^共1兲 _⬘共⌳ˆ

k,k⬘

共1兲 ␳^ˆk^⬎⬘⌳ˆ

k⬘^,k

共1兲

⫻␳^ˆk⬍−⌳ˆ

k,k⬘

共1兲 ␳^ˆk^⬍⬘⌳ˆ

k⬘^,k

共1兲 ␳^ˆk⬎兲+U_k−k^共2兲 _⬘共⌳ˆ

k,k⬘

共2兲 ␳^ˆk^⬎⬘

⫻⌳ˆ

k⬘^,k

共2兲 ␳^ˆk⬍−⌳ˆ

k,k⬘

共2兲 ␳^ˆk^⬍⬘⌳ˆ

k⬘^,k

共2兲 ␳^ˆk⬎兲+ H.c.兴, 共7兲

where ni=N_Mn^S + 4N_Mn^I +n_i0 with N_Mn^S , N_Mn^I , and n_i0 being the densities of substitutional Mn, interstitial Mn and nonmagnetic impurities, respectively, due to different charges. U_k−k^共1兲 _⬘=^␭c

2

4兺q_zV_k−k⬘,q

z

2 兩I共iqz兲兩²q_z² and U_k−k^共2兲 _⬘=

−␭c

2兺q_zV_k−k⬘,q

z

2 兩I共iq_z兲兩². Here ␭c=_3m^{␩共1−␩/2兲}

cE_g共1−␩/3兲 with ␩⁼_⌬SO^⌬SO+E_g. Eg and⌬SO are the band gap and the spin-orbit splitting of the valence band, respectively.²⁶ The spin-flip matrices are given by ⌳ˆ

k⬘^,k

共1兲 =关共k+k

⬘

k,k⬘

共2兲 =关共k, 0兲

⫻共k

⬘

k⬘^,k

共1兲 and⌳ˆ

k⬘^,k

共2兲 contribute to the out-of-plane and in-plane spin relaxations, respectively. They are generally different and therefore the spin relaxation due to the EY mechanism in quantum wells is anisotropic. The EY mechanism can be incorporated into other scatterings

similarly.³²However, we find that the EY mechanism can be important only when the impurity density is high, where the electron-impurity scattering is most important. Therefore, for simplicity, we include only the EY spin-flip processes asso- ciated with the electron-impurity scattering.

III. RESULTS AND DISCUSSIONS

By solving the KSBEs numerically, we obtain the tempo- ral evolution of the single-particle density matrix␳^ˆkand then the spin polarization along thezaxis, i.e.,sz. From the decay of s_z, the SRT is extracted. The initial spin polarization is chosen to beP= 4%. The well widtha= 10 nm. The external magnetic fieldBis zero unless otherwise specified. We usex to denote the Mn density, where N_Mn=xN₀ with N₀=⍀⁻¹

= 2.22⫻10²² cm⁻³ 共⍀ is the volume of the unit cell in GaAs兲. The other material parameters used are listed in Table I.^56–58

The value of the s-d exchange coupling in III共Mn兲V ma- terials is still in dispute. In bulk Ga共Mn兲As, first-principles calculation gives the value N₀␣⬇0.25 eV.⁵⁹ However, the experimental measurements in Ref. 22 show thatN₀␣ ^{is in} the range of 关−0.21, −0.07兴 eV varying with quantum well width. In this paper, we choose N₀␣= −0.15 eV unless oth- erwise specified.

A. Spin relaxation inn-type Ga(Mn)As quantum wells In this subsection, we study the electron spin relaxation in n-type Ga共Mn兲As quantum wells, where the Mn dopants mainly take interstitial positions. For simplicity, we neglect the substitutional Mn’s and assume that electrons from Mn donors are all free electrons. We will discuss the situations that the quantum wells are either undoped orn-doped before Mn-doping.

For quantum wells which are undoped before Mn-doping, Ne=N_e^Mn+N_exwhereN_e^Mnis the density of electrons from Mn donors and N_ex is the photoexcitation density. We choose N_ex= 10¹⁰ cm⁻²which is usually smaller thanN_e^Mn. The SRTs due to various mechanisms are plotted as function of x in Fig. 1共a兲. N₀␣ is chosen to be −0.25 eV, which is smaller 共i.e., thes-dexchange interaction is stronger兲than the value measured by experiments.²²However, even for such a strong exchange coupling, the spin relaxation due to the s-d ex- change scattering mechanism isstill much weaker than that due to the DP mechanism. It is further seen from Fig. 1共a兲

TABLE I. Material parameters used in the calculation.

␬0 12.9 ␬⬁ 10.8

D 5.31⫻10³ kg/m³ e₁₄ 1.41⫻10⁹ V/m v_st 2.48⫻10³ m/s v_sl 5.29⫻10³ m/s

⌶ 8.5 eV ␻LO 35.4 meV

⌬SO 0.341 eV E_g 1.55 eV

⌬E_LT 0.08 meV a₀ 146.1 Å

g_e −0.44 m^ⴱ 0.067m₀

g_Mn 2 S 5/2

that the BAP and EY mechanisms are also unimportant.

Therefore, the SRT is determined by the DP mechanism. In- terestingly, the SRT due to the DP mechanism ␶DP first in- creases then decreases with increasingx. The ␶^{-xcurve thus} has a peak. The underlying physics is that the SRT has dif- ferent x共density兲 dependence in the nondegenerate and de- generate regimes. Similar behavior has been found in bulk nonmagnetic III-V semiconductors in Ref.32 very recently.

Let us first recall the widely used expression, ␶DP

= 1/关具兩h共k兲兩²典␶p兴 共具¯典denotes the ensemble average兲, which is derived within the elastic-scattering approximation and is only correct qualitatively.³²The expression contains two key factors of the DP spin relaxation: 共i兲 the inhomogeneous

broadening from thek-dependent spin-orbit field⬃具兩h共k兲兩²典; 共ii兲the momentum scattering time␶p. The SRT due to the DP mechanism increases with increasing momentum scattering but decreases with increasing inhomogeneous broadening. It should be mentioned that for this system, N_e⬇n_i/2 = 2xN₀ 共Note that the charge numberZof the Mn ion is included in n_i as Z². For interstitial Mn, which acts as a double donor, Z= 2.兲In the smallx共low density兲regime, the electron sys- tem is in the nondegenerate regime, and the distribution is close to the Boltzmann distribution. Therefore, the inhomo- geneous broadening of the k dependent spin-orbit field

⬃具兩h共k兲兩²典changes little with electron densityNe共hencex兲.

On the other hand, in the nondegenerate regime the electron- electron scattering increases with increasing electron density^60,61共thusx兲. Moreover, the electron-impurity scatter- ing also increases with increasing x as the impurity density increases. Therefore, ␶DP increases with increasing x 共mo- tional narrowing兲. In largex共high density兲regime, the elec- tron system is in the degenerate regime, the inhomogeneous broadening changes as k_F²⬀Ne⬀x共k_F⁶⬀N_e³⬀x³兲 if the linear 共cubic兲 term dominates the SOC. On the other hand, the electron-electron scattering decreases with increasing elec- tron density 共thus x兲 in the degenerate regime.^60,61 Besides, the electron-impurity scattering increases slower than Ni⬀x because the scattering cross section decreases as the electron 共Fermi兲energy increases. Thus for both the linear- and cubic- term dominant cases,␶DP decreases with increasingx in the large x regime. Consequently, ␶DP first increases then de- creases with increasing xand a peak is formed in the cross- over regime whereT⬃T_F^e共T_F^e is the electron Fermi tempera- ture兲. It is seen from Fig.1共a兲that for bothT= 30 and 200 K cases, the peaks indeed appear atT⬃T_F^e. It should be pointed out that the situation here is different from that in Ref. 42, where the density dependence of the SRT also has a peak in intrinsic quantum wells at room temperature. In that case, the impurity density is rather low and the relevant momentum scatterings are the carrier-carrier Coulomb and electron- phonon scatterings. In the situation here, the impurity density is extremely high共ni= 2Ne兲and the relevant momentum scat- tering is the electron-impurity scattering.

We now turn to the situation that the quantum wells aren doped before Mn-doping. In this case, Ne=N_eⁱ+N_e^Mn+N_ex, where N_eⁱ denotes the density of the electrons from other dopants which is chosen to be 10¹¹ cm⁻². We assume that the other dopants are far away from the quantum wells, so that they contribute little to the electron-impurity scattering, corresponding to the genuine case of modulation doping.

However, the Mn ions are doped in the quantum wells.^23,24 The photoexcitation density is N_ex= 10¹⁰ cm⁻². The results are plotted in Fig. 1共b兲. As the density of the photoexcited holes is much smaller than the electron density, the BAP mechanism is obviously negligible and thus not plotted in the figure. From the figure, it is noted that the EY and s-d ex- change scattering mechanisms are also insignificant. Conse- quently, the spin relaxation is still dominated by the DP mechanism. Similar to that in Fig. 1共a兲, the SRT due to the DP mechanism ␶DP first increases then decreases with in- creasingx. For the case ofT= 200 K, the peak of the SRT is still around T=T_F^e. However, for the case of T= 30 K, the peak moves to a largerxvalue compared to that in Fig.1共a兲.

10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷

10^-7 10^-6 10^-5 10^-4 10¹ 10² 10³ 40 4

0.4 0.04

x

N_e^Mn ( 10¹¹cm^-2)

τ(ps)

(a)

TFe (K)

10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷

10^-7 10^-6 10^-5 10^-4 10¹ 10² 10³ 40 4

0.4 0.04

x

N_e^Mn ( 10¹¹cm^-2)

τ(ps)

(b)

TFe (K)

(b) (a)

FIG. 1. 共Color online兲 SRT ␶ due to various mechanisms in n-type Ga共Mn兲As quantum wells which are 共a兲 undoped or 共b兲 n-doped before Mn-doping as function of Mn concentrationxat 30 K共쎲兲 and 200 K共䊐兲. Red solid curves: the SRT due to the DP mechanism␶DP; green dotted curves: the SRT due to the EY mecha- nism ␶EY; brown dashed curves: the SRT due to the BAP mecha- nism ␶BAP; blue chain curve: the SRT due to the s-d exchange scattering mechanism␶sd. The Fermi temperature of electronsT_F^e is plotted as black curve with䉭共the scale ofT_F^e is on the right-hand side of the frame兲and T_F^e=T for bothT= 30 and 200 K cases are plotted as black dashed curves. We also plot the scale of the elec- tron density from Mn donorsN_e^Mnon the top of the frame.

This can be understood by noting that the electrons have two sources: the Mn donors and other dopants. For the case of T= 200 K, the crossover of the nondegenerate and degener- ate regimes takes place aroundT⬃T_F^e, where the correspond- ing x is 10⁻⁵. At suchx, electrons are mainly from the Mn ions rather than from other dopants. Thus the situation is the same as that in Fig. 1共a兲 and the peak appears at T⬃T_F^e. However, in the case ofT= 30 K, for allxin the figure,T_F^eis larger than T and the situation is hence different. The ␶-x behavior in this case can be understood as follows: for x

⬍10⁻⁶, electrons are mainly from the other dopants andNe

changes slowly with x, thus the inhomogeneous broadening varies slowly with x. On the other hand, the electron- impurity scattering increases as the impurity density in- creasesni⬇4xN₀. At low temperature共T⬍T_F^e兲, the electron- impurity scattering can be important even when ni⬍Ne.^30,32 Therefore, the momentum scattering increases withxsignifi- cantly. Consequently,␶DP increases with increasingx. Forx

⬎10⁻⁵, electrons mainly come from the Mn donors. The sce- nario becomes the same as that in the case of Fig.1共a兲and the SRT decreases with increasing x as T_F^e is much larger than T. Consequently, the peak is formed in the range 10⁻⁶

⬍x⬍10⁻⁵atx= 3⫻10⁻⁶, which is larger than that in the case of T= 30 K in Fig.1共a兲.

It should be mentioned that our results are consistent with the latest experimental finding that in the low Mn concentra- tion regime the SRT increases with increasing x.^23,24

B. Electron spin relaxation inp-type Ga(Mn)As quantum wells In this section, we discuss the electron spin relaxation in p-type Ga共Mn兲As quantum wells. Both substitutional and in- terstitial Mn ions exist in the system. Each substitutional Mn donates one hole, whereas each interstitial Mn compensates two holes. For simplicity, we assume that all the holes are free. The ratio of the hole densityN_hto the Mn densityN_Mn is obtained by fitting the experimental data in Ref. 22, as shown in Fig. 2. From these densities, according to charge

neutrality, the densities of substitutional MnN_Mn^S and inter- stitial MnN_Mn^I are determined. The photoexcitation density is chosen to be N_ex= 5⫻10¹⁰ cm⁻² unless otherwise specified.

1. Mn concentration dependence of the SRT

We first study the Mn concentration dependence of the SRT. In Fig.3, we plot the SRTs due to various mechanisms and the total SRT as function of the Mn concentration x at T= 5, 50, and 200 K. It is noted that the total SRT first in- creases and then decreases with increasing xand there is a peak atx⬃3⫻10⁻⁵. Remarkably, the spin relaxation at large x is not dominated by the DP mechanism, but by the s-d exchange scattering 共or the BAP兲 mechanism at low 共or high兲 temperature. At medium temperature, the EY mecha- nism also contributes for large x. The SRT due to the DP mechanism increases with increasingx, whereas those due to the s-d exchange, EY, and BAP mechanisms decrease with increasing x. Consequently a peak is formed. It should be pointed out that the underlying physics here is different from that in the case ofn-type Ga共Mn兲As quantum well where the peak is solely due to the electron density dependence of the DP spin relaxation. It should be mentioned that the peak position isx⬃10⁻⁴, which is consistent with that observed in Ref. 22.

Let us now turn to the x dependence of the SRT due to various mechanisms. The increase in␶DPwith increasingxis due to the increase in the electron-impurity and electron-hole scatterings共motional narrowing兲. For the EY mechanism, the SRT decreases as the spin-flip scattering increases with in- creasing impurity density 关see Eq. 共7兲兴. The s-d exchange scattering increases with increasing x as the Mn density in- creases 关see Eq. 共5兲兴. The x dependence of the BAP spin relaxation is more complicated. To facilitate the understand- ing, we plot␶BAPfrom the full calculation and that from the calculations without the Pauli blocking of electrons 共holes兲 in Fig.4. It is seen that forx⬍10⁻⁶,␶BAPchanges little with x. This is due to the fact that the holes from Mn dopants are much fewer than those from photoexcitation, and hence N_h changes little with x. So does ␶BAP. For larger x,␶BAP first decreases then increases a little and finally saturates with increasing x at T= 5 K. It is noted that without the Pauli blocking of holes, ␶BAPdecreases with increasing xrapidly, which indicates that the slowdown of the decrease and the saturation of␶BAPare due to the Pauli blocking of holes. It is further shown that the Pauli blocking of electrons is also important asT_F^e= 20 K is larger thanT= 5 K. For the case of T= 200 K, the effect of the Pauli blocking of electrons is negligible as TⰇT_F^e. The Pauli blocking of holes becomes visible only for x⬎10⁻⁴, where the hole Fermi temperature becomes larger thanT= 200 K.

From Eq.共5兲, one can see that the spin relaxation due to the s-d exchange scattering is independent of temperature.

However, the spin relaxation due to the BAP mechanism increases with increasing temperature because the Pauli blocking of electrons and holes decreases with increasing temperature. Moreover, the matrix element of the BAP mechanism increases with the center-of-mass momentum of the interacting electron-hole pair, of which the ensemble av- erage hence increases with increasing temperature.³¹ The 0.15

0.2 0.25 0.3 0.35

0 0.02 0.04 0.06 0.08 0.1 0.12 0 1 2 3 4 5

x ( 10^-2)

Nh/NMn Nh(1012 cm2 )

FIG. 2. 共Color online兲Ratio of the hole density to the Mn den- sityN_h/N_Mnvs the Mn concentrationxinp-type Ga共Mn兲As quan- tum wells. The black dots represent the experimental data. The red solid curve is the fitted one. The hole densityN_his also plotted共the blue dashed curve兲. Note that the scale ofN_his on the right-hand side of the frame.

spin relaxation due to the EY mechanism increases with in- creasing temperature too, as the spin-flip matrices关⌳ˆ

k⬘^,k

共1兲 and

⌳ˆ

k⬘^,k

共2兲 兴 in Eq. 共7兲 increase with increasing k. Consequently, the BAP and EY mechanisms eventually become more effi- cient than the s-d exchange scattering mechanism at high temperature.

The appearance of the peak in the ␶^-x curve has been observed in a recent experiment at 5 K.²²However, the SRTs we obtain are much larger than the experimental value under the same conditions. The deviation may come from preter- mission of localized holes. At such low temperature 共5 K兲, the localization of holes is not negligible.^26,62 The localized holes act as exchange interaction centers located randomly in the sample, which thus lead to spin relaxation similar to the s-d exchange scattering mechanism. As there is no Pauli blocking of the localized holes, the spin relaxation can be very efficient.²⁶It should be mentioned that, however, recent studies have also shown that there is some compensation of the s-d exchange interaction and the electron-hole exchange interaction as holes are always localized on the Mn acceptors.^63,64 This leads to a longer spin relaxation time⁶³ and smaller measured 共by magneto-optical techniques兲 s-d exchange coupling constant.⁶⁴ However, for the high- temperature case, the localization is marginal and our con- sideration is close to the genuine case. The predicted ␶^-x dependence should be tested experimentally.

2. Temperature dependence of the SRT

We now discuss the temperature dependence of the SRT.

In Fig.5, we plot the SRT as function of Mn concentrationx for different temperatures. It is seen that for each case the␶^-x curve shows a peak. It is further noted that the temperature 10²

10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹

10^-7 10^-6 10^-5 10^-4 10^-3 200 20

2 0.2 0.02

x

N_Mn ( 10¹¹cm^-2)

T = 5 K

τ(ps)

(a)

τ_tot τ_DP τ_EY τ_BAP τ_sd

10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸

10^-7 10^-6 10^-5 10^-4 10^-3 200 20

2 0.2 0.02

x

N_Mn ( 10¹¹cm^-2)

T = 50 K

τ(ps)

(b)

τ_tot τ_DP τ_EY τ_BAP τ_sd

10² 10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸

10^-7 10^-6 10^-5 10^-4 10^-3 200 20

2 0.2 0.02

x

N_Mn ( 10¹¹cm^-2)

T = 200 K

τ(ps)

(c)

τ_tot τ_DP τ_EY τ_BAP τ_sd (b)

(a)

(c)

FIG. 3.共Color online兲SRT␶due to various mechanisms and the total SRT inp-type Ga共Mn兲As against the Mn concentrationxat共a兲 T= 5,共b兲50, and共c兲200 K. We also plot the scale ofN_Mnon the top of the frame.

10² 10³ 10⁴ 10⁵

10^-7 10^-6 10^-5 10^-4 10^-3 200 20

2 0.2 0.02

x

N_Mn ( 10¹¹cm^-2)

τBAP(ps)

FIG. 4. 共Color online兲SRT due to the BAP mechanism␶BAPas function of the Mn concentration atT= 5共쎲兲and 200 K共䊐兲. Red solid curves: ␶BAP from the full calculation; green dotted curves:

␶BAPfrom the calculation without the Pauli blocking of electrons;

blue dashed curves: ␶BAP from the calculation without the Pauli blocking of holes.

dependences of the SRT are different for small 共e.g., x

= 10⁻⁷兲 and large x 共e.g., x= 10⁻³兲. To make it more pro- nounced, we further plot the temperature dependence of the SRT for x= 10⁻⁷, 3⫻10⁻⁵, and 10⁻³ in Fig. 6. For x= 10⁻⁷, the SRT first increases then decreases with increasing tem- perature and there is a peak around 20 K. It is understood that for such a small x, the electrons and holes are mainly from the photoexcitation. For such system, the electron- electron and electron-hole Coulomb scatterings are most im- portant. It is shown in Ref. 30 that the nonmonotonic tem- perature dependence of the electron-electron Coulomb scattering leads to a peak in the ␶-T curve. In the situation here, the electron-hole Coulomb scattering also contributes to the formation of the peak. For the case ofx= 3⫻10⁻⁵, all spin relaxation mechanisms are relevant and the most impor- tant momentum scattering is the electron-impurity scattering.

In this case the SRT due to the DP mechanism decreases with increasing temperature monotonically as the increase in the inhomogeneous broadening dominates.³⁰ Moreover, the

SRTs due to the EY and BAP mechanisms also decrease with increasing temperature. Consequently, the total SRT de- creases with increasing temperature monotonically. For the case of large x共x= 10⁻³兲, the spin relaxation is dominated by the s-d exchange scattering共or the BAP兲mechanism at low 共or high兲temperature. As thes-dexchange scattering mecha- nism is independent of the temperature, the temperature de- pendence is rather weak in the low-temperature regime. As the temperature increases, the EY and BAP mechanisms be- come more and more important, which leads to a fast de- crease in the SRT with temperature.

3. Photoexcitation density dependence of the SRT We now study the photoexcitation densityN_exdependence of the spin relaxation. In Fig.7the SRT is plotted against the 10²

10³ 10⁴

10^-7 10^-6 10^-5 10^-4 10^-3 200 20

2 0.2 0.02

x

N_Mn ( 10¹¹cm^-2)

τ(ps)

50 K5 K 100 K 200 K

FIG. 5. 共Color online兲SRT␶as function of Mn concentrationx at different temperatures.

10² 10³ 10⁴

10 100

T ( K )

τ (p s)

x = 10^-7 3×10^-5 10^-3

FIG. 6. 共Color online兲SRT␶as function of temperatureT for different Mn concentrations.

10² 10³ 10⁴ 10⁵

10^-7 10^-6 10^-5 10^-4 10^-3 200 20

2 0.2 0.02

x

N_Mn ( 10¹¹cm^-2)

T = 5 K

τ(ps)

(a) ^0.1×10

11cm^-2 0.5×10¹¹cm^-2 1×10¹¹cm^-2

10² 10³

10^-7 10^-6 10^-5 10^-4 10^-3 200 20

2 0.2 0.02

x

N_Mn ( 10¹¹cm^-2)

T = 200 K

τ(ps)

(b)

0.1×10¹¹cm^-2 0.5×10¹¹cm^-2 1×10¹¹cm^-2

(b) (a)

FIG. 7. 共Color online兲SRT␶as function of the Mn concentra- tion for different photoexcitation densities at 共a兲T= 5 and 共b兲200 K. Red solid curve with 쎲: N_ex= 0.1⫻10¹¹ cm⁻²; green dotted curve with 䊐: N_ex= 0.5⫻10¹¹ cm⁻²; blue dashed curve with 䉭: N_ex= 1⫻10¹¹ cm⁻².

Mn concentrationxfor three photoexcitation densities at low 共5 K兲and high共200 K兲temperatures. It is noted that the SRT exhibits very different photoexcitation density dependences at low and high temperatures. Moreover, the photoexcitation density dependence varies withx. Let us divide the variation of x into three regimes: the small x共x⬍3⫻10⁻⁶兲 regime where the DP mechanism is dominant; the medium x共3

⫻10⁻⁶⬍x⬍10⁻⁴兲regime where the DP mechanism is com- parable with the other mechanisms; the largex共x⬎10⁻⁴兲re- gime where the DP mechanism is irrelevant.

In smallxregime, the DP mechanism dominates the spin relaxation. The photoexcitation dependence of the DP spin relaxation is different in the degenerate and nondegenerate regimes. Similar to the case of n-type Ga共Mn兲As quantum wells 共see Sec. III A兲, in degenerate 共low temperature兲 re- gime, the density dependence of the SRT is dominated by the increase in the inhomogeneous broadening with increasing density, and hence the SRT decreases with increasing den- sity. In nondegenerate共high temperature兲regime, the density dependence of the SRT is dominated by the increase of the electron-electron and electron-hole Coulomb scatterings with density, and hence the SRT increases.

In the largexregime, the spin relaxation is mainly due to the EY, BAP, and s-d exchange scattering mechanisms. At low temperature, the s-d exchange scattering mechanism is dominant. As␶sd is independent of the electron density, the photoexcitation density dependence of the SRT 共which mainly comes from the EY mechanism兲 is weak. At high temperature, the BAP mechanism dominates. As holes are mainly from the Mn dopants and the electron system is non- degenerate, the SRT also changes little with photoexcitation density.

In the mediumxregime, all the four mechanisms contrib- ute to the spin relaxation. As ␶sdis independent of the elec- tron density, the density dependence comes from the other three mechanisms. At low temperature, besides the DP mechanism, the BAP mechanism is also important. However, the BAP spin relaxation changes slowly with electron共hole兲 density as the Pauli blocking is important at low temperature 共see Fig.4兲. The spin relaxation due to the EY mechanism also increases with increasing N_ex as the spin-flip matrices 关⌳ˆ

k⬘^,k

共1兲 and⌳ˆ

k⬘^,k

共2兲 兴in Eq.共7兲increase with increasingk. How- ever, the EY mechanism is usually less efficient than the DP mechanism forx⬍10⁻⁴. As the DP spin relaxation increases with increasing photoexcitation density while other relevant mechanisms change slowly or less important than it, the peak moves to largerxwith increasing photoexcitation density as indicated in Fig. 7共a兲. Moreover, the total SRT decreases with increasing photoexcitation density. At high temperature, as the electron system is nondegenerate, the inhomogeneous broadening changes slowly withN_ex. However, the screening increases with increasingN_exas the carrier density increases.

Hence the momentum scattering 共mainly from the electron- impurity scattering兲 decreases with increasing N_ex. There- fore, the SRT due to the DP mechanism decreases with in- creasing N_ex. The EY mechanism is less important than the DP mechanism in this regime 关see Fig. 3共c兲兴. Moreover, as the hole system is nondegenerate, the spin relaxation due to the BAP mechanism increases with hole density. Therefore,

the SRT also decreases with increasing photoexcitation den- sity.

4. Effect of magnetic field on the SRT

We now study the effect of magnetic field on the SRT.

The magnetic field is applied parallel to the quantum-well plane, which is perpendicular to the initial electron spin po- larization 共the Voigt configuration兲. In traditional nonmag- netic n-type quantum wells, where the electron spin relax- ation is dominated by the DP mechanism, the magnetic field in the Voigt configuration has dual effects on spin relaxation:

共i兲 elongating the spin lifetime by a factor of 关1 +共␻L␶p兲²兴 共␻Lis the Larmor frequency,␶pis the momentum scattering time兲;²⁶ 共ii兲 mixing the in-plane and out-of-plane spin relaxations,^35,65e.g.,¹_␶=¹₂共_␶¹_z+_␶¹_储兲when␻L⬎₂¹共_␶¹_z−_␶¹_储兲.⁶⁵Usu- ally, effect共i兲is weak, but effect共ii兲is more important. Dif- fering from the case of nonmagneticn-type quantum wells, there are several new scenarios in the p-type Ga共Mn兲As quantum wells: 共i兲 the magnetic field can polarize the Mn spins, which alters the spin relaxation due to the s-d ex- change scattering mechanism. 共ii兲 The nonequilibrium Mn spin polarization can be induced during the evolution through the s共p兲-d exchange interaction with both electrons and holes. It precesses around the magnetic field and pro- duces the Mn beats. This has been studied both experimen- tally and theoretically in II-VI magnetically doped quantum wells.^66,67However, we find that the induced nonequilibrium Mn spin polarization is rather small共much smaller than the electron spin polarization兲 and affects the spin dynamics marginally. This is consistent with the fact that the Mn beats are not observed in p-type Ga共Mn兲As quantum wells.²²共iii兲 The spin relaxation due to the EY mechanism is also aniso- tropic because the out-of-plane spin relaxation comes from

⌳ˆ

k⬘^,k

共1兲 while the in-plane relaxation from ⌳ˆ

k⬘^,k

共2兲 关see Eq.共7兲兴. At large xwhen the EY mechanism is important, this aniso- tropy may show up.

In Fig. 8, we plot the SRT as function of the magnetic field with different Mn concentration atT= 5 and 200 K. For the case of smallx共x= 10⁻⁶兲, it is seen that the SRT increases abruptly when the magnetic field varies from 0 to 0.2 T and is almost a constant forB= 0.2 to 6 T. This abrupt increase in the SRT originates from the mixing of the out-of-plane and in-plane electron spin relaxations in the presence of magnetic field. For smallx, the spin relaxation is dominated by the DP mechanism. For the DP spin relaxation, the in-plane spin relaxation is slower than the out-of-plane one, as only part of the inhomogeneous spin-orbit field h共k兲 contributes to the in-plane spin relaxation. After the magnetic field is applied, the spin relaxation rate becomes ¹_␶=¹₂共_␶¹_z+_␶¹

储兲. The condition for this relation is␻L⬎¹₂共_␶¹_z−_␶¹

储兲. In the situation considered here, it is Bⲏ0.005 T共0.02 T兲 for low 共high兲 temperature case. Therefore, the variation in the SRT with the magnetic field seems abruptly.

For the case of largex共x= 10⁻³兲, the relevant spin relax- ation mechanisms at high temperature are the BAP, EY, and s-d exchange mechanisms. The BAP and s-d exchange mechanisms are isotropic. However, the EY mechanism is anisotropic. Our calculation indicates that the ⌳ˆ

k⬘^,k

共1兲 term is