Diffusion equation and spin drag in spin-polarized transport

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1 Downloaded from orbit.dtu.dk on: Apr 7, 29 Diffuion equation and pin drag in pin-polarized tranport Flenberg, Karten; Jenen, Thoma Stibiu; Mortenen, Ager Publihed in: Phyical Review B Condened Matter Link to article, DOI:.3/PhyRevB Publication date: 2 Document Verion Publiher' PDF, alo known a Verion of record Link back to DTU Orbit Citation (APA): Flenberg, K., Jenen, T. S., & Mortenen, A. (2). Diffuion equation and pin drag in pin-polarized tranport. Phyical Review B Condened Matter, 64(24), General right Copyright and moral right for the publication made acceible in the public portal are retained by the author and/or other copyright owner and it i a condition of acceing publication that uer recognie and abide by the legal requirement aociated with thee right. Uer may download and print one copy of any publication from the public portal for the purpoe of private tudy or reearch. You may not further ditribute the material or ue it for any profit-making activity or commercial gain You may freely ditribute the URL identifying the publication in the public portal If you believe that thi document breache copyright pleae contact u providing detail, and we will remove acce to the work immediately and invetigate your claim.

2 PHYSICAL REVIEW B, VOLUME 64, Diffuion equation and pin drag in pin-polarized tranport Karten Flenberg, Thoma Stibiu Jenen, and Niel Ager Mortenen,2 O rted Laboratory, Niel Bohr Intitute fapg, Univeritetparken 5, 2 Copenhagen, Denmark 2 Mikroelektronik Centret, Technical Univerity of Denmark, 28 Lyngby, Denmark Received 6 July 2; publihed 29 November 2 We tudy the role of electron-electron interaction for pin-polarized tranport uing the Boltzmann equation, and derive a et of coupled tranport equation. For pin-polarized tranport the electron-electron interaction are important, becaue they tend to equilibrate the momentum of the two-pin pecie. Thi pin drag effect enhance the reitivity of the ytem. The enhancement i tronger the lower the dimenion i, and hould be meaurable in, for example, a two-dimenional electron ga with ferromagnetic contact. We alo include pin-flip cattering, which ha two effect: it equilibrate the pin denity imbalance and, provided it ha a non--wave component, alo a current imbalance. DOI:.3/PhyRevB PACS number: Ba, Rb, Dc I. INTRODUCTION Recent advance in the fabrication of ferromagneticemiconductor heterotructure and the obervation of pin injection into emiconductor 2 have lead to interet in the tranport propertie of pin-polarized ytem. There ha been coniderable work done in the field of metallic magnetic multilayer, which ha been analyzed in term of tranport equation with pin dependent ditribution function. 3,4 Thee work baed their analyi on diffuion tranport equation. The jutification of uing thee equation wa given by Valet and Fert, 5 who derived a pin diffuion equation from the Boltzmann equation in the limit where the pin cattering length i much longer than the momentum relaxation length. Recently, the tranport equation were utilized to analyze the feaibility of pin injection into emiconductor, with the reult that the crucial parameter i the conductivity mimatch between the emiconductor and the ferromagnet, 6 and to tudy pin-polarized tranport theoretically in inhomogeneou doped emiconductor. 7 None of thee approache took electron-electron (e-e) cattering into account. Clearly e-e interaction play a different role than in uual pin degenerate tranport, where the e-e interaction doe not provide a mechanim for momentum relaxation and hence ha only indirect conequence for tranport coefficient. In pin-polarized tranport the two pin pecie have different drift velocitie, and e-e interaction are intrumental in equilibrating thi difference. Thi lead to a pin drag effect where the pin carrying the larger current will drag along the pin carrying the maller current. Thi drag effect wa recently conidered by D Amico and Vignale 8,9 in three dimenion uing linear repone theory. They found that the pin drag reitivity wa appreciable, and at elevated temperature can be a fraction of the uual reitivity of the metal. In two dimenion the effect of e-e interaction on pin diffuion wa conidered theoretically by Takahahi et al., Uing a quantum kinetic equation approach, previouly utilized in 3 He 4 He olution, 2 they tudied the pin diffuion coefficient in two-dimenional electron gae. In order to tudy thi pin diffuion they ued variational function, but did not include pin-relaxation cattering. In thi paper, we ue the Boltzmann equation to tudy pin-dependent tranport and pin diffuion. We retrict ourelve to the tudy of collinear magnetization, and our goal i to derive a et of tranport equation in the emiclaical limit. For thi purpoe the Boltzmann equation i adequate. For the noncolinear cae, where phenomena uch a damped tranvere pin mode can occur, one mut go beyond the preent approach; ee, e.g., Ref. and and reference therein. We include impurity cattering, both pin independent and pin flip cattering, a well a e-e cattering. We how that a hifted Fermi-Dirac SFD ditribution, i a valid olution at low temperature TT F, and without pin-flip cattering. Thi i alo the cae for weak e-e cattering, where the problem in abence of pin-flip reduce to the ordinary Coulomb drag ituation. 3,4 We then go on to dicu the general cae at higher temperature, general interaction trength and finite intrinic pin-flip cattering. Uing a SFD anatz, for an iotropic ytem we find the following macrocopic tranport equation: J e n f, a e e J D e f, J J. b Here J i the current carried by electron with pin, i the local pin-dependent electrochemical potential, i the conductivity of the pin electron ga, f i a pin lifetime due to intrinic pin-flip procee, f, i a pin current converion conductivity ariing from the angle dependence of the pin-flip cattering, and n /n i the relative pin denity, ee Eq. 35 and 39 for definition. Finally D i the pin drag conductivity, given by D 2 de 2 n n dq dq 2 eq 2 2 d Im q,im q,, 2 k B T inh 2 /2k B T /2/6424/245387/$ The American Phyical Society

3 FLENSBERG, JENSEN, AND MORTENSEN PHYSICAL REVIEW B where Im i the polarization function: Im q, dk 2 d f kq f k kq k. Formula 2 i well known from Coulomb drag. 3 At low temperature it i ( D ) T 2 in two and three dimenion, while in one dimenion it i proportional to T; ee, e.g., Ref. 5. The firt tranport equation Eq. a i the continuity equation, which expree the conervation of current in the preence of pin-flip procee. The econd equation Eq. b i a generalized Ohm law. The firt term on the righthand ide i Ohm law, while the econd term how that a momentum imbalance between the two pin direction give rie to an additional reitance if there i a mechanim for converion of the pin current. There are two uch procee poible. Thi firt one i the pin drag effect mentioned above, where e-e cattering make a tranfer of momentum poible. The econd one i due to the elatic pin-flip cattering on, for example, magnetic impuritie, which can convert a current with one pin polarization to a current of the oppoite polarization, if the pin-flip matrix element ha an angular dependence. For example if the pin-flip predominant catter forward, thi mean that pin-flip cattering i accompanied by a tranfer of momentum. In contrat if the pin-flip cattering, i purely -wave cattering the momentum tranfer between the pin channel i on average equal to zero. Thi can be een mathematically from the expreion for f in Eq. 35c. The derivation of thee two term i the main reult of the preent paper. Two conequence of the pin current relaxation term can immediately be read off. Firt, they give rie to an increaed reitivity in the cae where the current i pin polarized. For example, taking J, the effective reitivity for electron with pin become (/ / D / f, ), and hence i an enhanced reitivity. Second, from Eq. we obtain a diffuion equation for the electrochemical potential difference where 2 e2 l f f 3 2, 4 2 l f n f, D. Thi how that the intrinic pin relaxation length i decreaed by the pin-drag- and angle-dependent pin-flip effect. Similarly, we obtain that the following weighted um of electro chemical potential mut vanih, where 5 2 c c, 6 c n f, D n n. Below, we derive Eq. a and b and etimate the pin drag contribution. For the two-dimenional cae, we alo perform the integration of Eq. 2 numerically. II. BOLTZMANN EQUATION FOR COLLINEAR SPIN TRANSPORT We bae our analyi on the Boltzmann equation for tranport through a ytem with lifted pin degeneracy. We take the current to run in the x direction, and denote the nonequilibrium ditribution function by f (k) and the equilibrium Fermi-Dirac ditribution function by f, f k e ( k ), 8 where i the chemical potential and, a uual, the invere temperature. The eigenenergie are denoted k, where i the pin quantum number and k the quantum number labeling the relevant tate croing the Fermi level. For implicity, we aume a parabolic diperion and write k 2 k 2 2m, where i the band offet which can be pin dependent if the material i ferromagnetic. The linearized Boltzmann equation then read v x (k) f k,x ee x x f k k x f k,x t 7 9. coll. We take the colliion integral to include elatic cattering and e-e cattering, f k t H f kh f f, f kh e-e f, f k coll. H e-e f, f k, where H i the cattering from impuritie or quaielatic phonon cattering, giving rie to a momentum relaxation H f k dk 2 d W k,k f k f k k k, 2 and where H f decribe elatic cattering procee that flip the pin: H f f, f k dk 2 d W fk,k f k f k k k. 3 Finally, the e-e cattering i after the linearization given by

4 DIFFUSION EQUATION AND SPIN DRAG IN SPIN-... PHYSICAL REVIEW B H e-e f, f 7(k) kt 2 dk dq 2 d 2 d U q, k kq 2 k k kq k qf k f k f kq f k q k k kq kq, 4 where the deviation from equilibrium i expreed in the function through f (r,k) f (k) f k k r,k. 5 The interaction U i the Coulomb interaction between two electron with pin and. It can in principle depend on the relative direction of the pin if exchange i included. Thi et of integral equation cannot be olved in general, and one mut either olve them numerically for example a in Ref. 7, or proceed with approximate method. However, one implification i poible from ymmetry. Becaue of the cylindrical ymmetry the function (k only depend on the angle between k and the direction of the current, which we here chooe to be in the x direction. Denoting thi angle by, we have co k"xˆ/k, and we can write r,k x,k,. 6 It i convenient to expand the ditribution function in harmonic of the angle a x,k, g (n) x,kco n, n which we utilize in Sec. III. III. SPIN DRAG WITHOUT SPIN-FLIP PROCESSES FOR T T F 7 In thi ection we tudy the Boltzmann equation in the preence of e-e interaction, but in the abence of pin-flip procee, i.e., H f. Furthermore, becaue a lowtemperature expanion allow for a olution of the Boltzmann equation, we tart by examining thi limit, and later we dicu the validity of thi olution even at elevated temperature. It turn out that the olution in the low-temperature regime correpond to a SFD ditribution. In the low-temperature limit, we ee from Eq. that the econd term on the left-hand ide the driving term retrict k to lie cloe to the Fermi level, uch that the deviation (k) need only to be evaluated at k F. Thi i therefore alo true for the ditribution function in the elatic colliion term, H. Due to the Pauli principle, thi will alo be the cae for the inthee-e colliion integral, which i een a follow. Uing tandard trick ee, e.g., Ref. 3, we rewrite the e-e colliion term a H ee f, f (k) 2 where 2 dk 2 d dq 2 d d U q, k kq 2 kt inh 2 /k B T Im k,q; Im k,q; k k kq kq, Im k,q; f kq f k kq k. 8 9 Now, at low temperature the factor /inh 2 retrict the integral to mall of order kt, and hence kq in Eq. 9 deviate from k by an amount of order kt, and we expand Im a Im k,q; f k k kq k. 2 From thi we conclude that both k and k and hence alo kq and k q ) are within a hell of order k BT from the Fermi level. To leading order in kt/ F, we can therefore neglect the dependence on k and keep only the angular dependence of. Therefore, in the following we replace k k F,, 2 where k F i the Fermi wave vector for the pin direction. Now we expand the function in harmonic of the angle a in Eq. 7. Inerting Eq. 5 and 7 into the Boltzmann equation give, for the left-hand ide, k x m f k k x x ee n k co m f k k co n g (n) x ee, 22 and for the right-hand ide we have two term. The firt one i the pin conerving impurity cattering term, which become n tr, H f n (n) cong n tr f k k, 23 where we defined tranport time of order n, dk 2 W d k,k co n k,k k k,

5 FLENSBERG, JENSEN, AND MORTENSEN PHYSICAL REVIEW B and where k,k i the angle between k and k. The econd term i the one with the e-e cattering. When expanion 7 i inerted into the e-e interaction term, different n do not couple; ee, for example, the derivation in Ref. 7. The trick i to write, for example, the angle of kq a co n k q,x co n ( k q, k k,x ) co n k q,k co n k,x in n k q,k in n k,x, and note that the in term vanih due to ymmetry. Therefore we can expre the e-e colliion term a H e-e f, f (k, Ä n cong (n) J (n) g (n) I (n), 25 where J (n) correpond to the firt and third term in Eq. 8, while I (n) correpond to the econd and fourth term. (n) Now a et of equation for the coefficient g can be extracted by multiplying the Boltzmann equation by co n and integrating over, while uing that d co n co n nn. The left-hand ide of Eq. 22 i expanded in harmonic, uing that co co n 2co (n) co (n). We find the following et of equation: (2) k g () m x g where g () 2 e k 2 m x g (n) g (n), 26a x () g g () J () I () tr () g () I, 26b (n) (n) g n g (n) J (n) I (n) g (n) I, tr n2 26c f k k. 27 The olution of thee equation i g (n) for n2. Thi i due to the fact that x g (), which decouple the equation for n2 from the firt two equation. Equation 26a expree current conervation within each pin pecie. If we include pin-flip cattering in the equation, then the equation couple becaue x g (). Now we note that etting g (n) for n2 correpond preciely to a linearized hifted Fermi-Dirac ditribution f SFD k f kk, from which we read off for k in the x direction 28 g (), g () co v x k g () 2 k F m k. 29 From Eq. 26, one can now determine g and g. They correpond to the change of the local charge denitie and to the local current, repectively. We will ee thi in Sec. IV, where we ue the SFD anatz to tudy the general cae. IV. MACROSCOPIC TRANSPORT EQUATIONS Above we aw that at low temperature the exact olution of the Boltzmann equation in the abence of pin-flip wa a hifted Fermi-Dirac function. The ame concluion applie to the ituation of arbitrary temperature but weak e-e cattering, becaue thi limit correpond to the uual Coulomb drag regime. However, thi i no longer necearily true at arbitrary e-e cattering, when the temperature increae, or when pin-flip procee are included. Neverthele, we hall aume in the following that the SFD ditribution i a good approximation for the exact ditribution function. The argument for doing thi i a follow: the e-e interaction will drag the ditribution function toward hifted Fermi-Dirac ditribution, becaue the interpin channel e-e colliion term vanih for f f SFD, i.e., H e-e f SFD, f SFD. Since the e-e cattering rate increae a e-e ( F /)(kt/ F ) 2, increaing the temperature actually help. Furthermore, ince the energy dependence of the elatic cattering i important in determining the actual hape of the ditribution function, and becaue we do not go into detail of thi ort, we view the SFD ditribution function a reaonable parametrization of the true ditribution function. Our tarting point i thu an anatz ditribution function given by f k f k f k k x f k k v x k x. 3 Here correpond to a change of the local chemical potential, and hence alo to the local denity, while k decribe a hift of the ditribution function in k pace and thu give a finite drift velocity. Inerting thi into the Boltzmann equation give, for the left-hand ide, Lv x f k k x v x k ee x, 3 and for the right-hand ide we have three term. The pinconerving colliion term become H f v x k tr, f k k, 32 where the uual tranport time i

6 DIFFUSION EQUATION AND SPIN DRAG IN SPIN-... PHYSICAL REVIEW B tr, dk 2 W d k,k co k,k k k. 33 The econd term on the right-hand ide i the pin-flip cattering term, which become H f f, f v x k f,tr k k f f k k v x f, 34 where the three different pin-flip cattering time are given by dk 2 W fk,k d k k, f 35a f,tr dk 2 W fk,k co k,k d k k, f dk 2 W fk,kco k,k d k k. 35b 35c Finally, the e-e cattering i given by H e-e f, f. But, in accordance with detailed balance, the e-e cattering between two identical Fermi-Dirac ditribution i zero, H e-e f, f, and we are left with H e-e f, f 2 m 2 dk 2 d dq 2 d d U, q, k kq 2 kt inh 2 /k B T Im k,q; Im k,q;q x k k. 36 The final form of the Boltzmann equation i thu Next we find the current and the denity. They are given by J e dk 2 d v xf (k e dk 2 d v x 2 f k k k e m n k, 38a e dk e dk 2 d 2 df k f k f k k e n. 38b We find two tranport equation for the current and charge denity or chemical potential by integrating Eq. 37 and alo Eq. 37 multiplied by v x with repect to k, and we arrive at Eq., where n n, 39a n e 2 m e tr, f,tr, 39b f, n e 2 f m. 39c In Eq. a we introduced the drag conductivity defined in Eq. 2. In deriving the drag term, we have made ue of the reult obtained for Coulomb drag in, e.g., Ref. 3. Furthermore, the local electrochemical potential ha been defined a e, where i the electrical potential. V. EVALUATION OF THE SPIN DRAG RESISTIVITY 4 A. One dimenion The polarization function i in one dimenion at mall temperature, where we can perform an expanion, given by Im q, m2 f q/2, 4 3 q 2 q/2,. 4 Inerting thi into the formula for D, performing the integration for the cae of a nonmagnetic conductor k k, and uing that f () 2 (6kT) ( F ), we find D kt k F U2k F. 42 F 64 e 2 2 F The pin drag reitance i thu proportional to temperature and dependent on the Coulomb backcattering matrix element. Clearly, thi contribution can be very large at finite temperature. However, in trictly one dimenion, where Fermi-liquid theory i not expected to apply, the Boltzmann equation i not a correct tarting point, and one hould be omewhat careful about drawing firm concluion from thi. Neverthele, thi Fermi golden rule reult i indicative of e-e interaction being very important for pin tranport in one dimenion

7 FLENSBERG, JENSEN, AND MORTENSEN PHYSICAL REVIEW B B. Two dimenion For the two-dimenional cae we tart by deriving a lowtemperature reult, and go on to compare it with a full numerical integration of D. At mall and q the imaginary part of the polarization function i given by m 2 Im q,, 2 3 q 3 k F and the creened tatic Coulomb interaction i 43 Uq e2 2 r qq TF. In thi approximation, the q integral become dqq 3 qq TF, which i clearly not convergent, and therefore we et the upper limit to be 2k F, becaue Im i zero for a momentum exchange larger than 2k F. With thee input, we arrive at the approximate expreion D kt ln, 46 F 2 2 e where 2k F /q TF and q TF me 2 /2 r 2 i the twodimenional invere Thoma-Fermi creening length. Typically i of order. Thi mean that the pin drag reitivity can be equal to a fraction of the quantum reitance, and hould therefore indeed be meaurable for tandard highmobility quantum well. We have alo integrated the pin drag formula numerically; ee Ref. 4 for detail. The reult i hown in Fig. for realitic number for a two-dimenional GaA electron ga. The integration i done uing the full dynamically creened interaction for a quantum well with finite thickne. The approximate formula Eq. 46 i een to overetimate the pin drag effect lightly. VI. CONCLUSIONS We have derived a et of tranport equation for pinpolarized drag which incorporate e-e cattering. Thi ha been done within the framework of the Boltzmann equation. Firt we howed that in the abence of pin-flip cattering and at low temperature the exact olution of the Boltzmann FIG.. The pin drag reitivity in the two-dimenional cae a a function of temperature. The thick line i the numerical integration of Eq. 2 for a two-dimenional quantum well of thickne nm and electron denity 2 5 m 2. We have ued typical parameter for GaA-baed heterotructure. The rightmot thin line i the approximate expreion in Eq. 46, while the left thin line i the reult of integrating Eq. 2, but uing the T expreion for (q,). equation correpond to two hifted Fermi-Dirac ditribution function. Furthermore, if the interaction i weak, one can ue perturbation theory and arrive at the ame concluion following the line of argument from Coulomb drag. Having oberved that the hifted Fermi-Dirac ditribution i correct at low temperature or weak e-e cattering, we go on to the general cae, which i olved approximately by uing the SFD a an anatz, which allow for a olution of the coupled Boltzmann equation. The main concluion from thi i that e-e interaction introduce a pin drag term, which tend to drag the pin current to be equal. There are two uch mechanim, namely, e-e interaction, which i temperature dependent, and angular-dependent elatic pin-flip cattering, which i temperature independent. Therefore, if a pin-polarized current i driven through the ytem, the pin drag will give rie to an additional reitivity. Thi reitivity increae with temperature. We have olved for the pin drag reitivity numerically in two dimenion, which how that it can become coniderable and even exceed the ordinary impuritycattering-induced reitivity. The pin drag hould thu be meaurable in, for example, a tructure combining a twodimenional electron ga with ferromagnetic material or for one-dimenional ytem, e.g., fabricated by nanotechnology in emiconductor or by contacting nanotube to ferromagnetic contact. P. M. Levy, Solid State Phy. 47, See, e.g., in Proceeding of Firt International Conference of the Phyic and Application of Spin-related Phenomena in Semiconductor, edited by Katayama Phyica E 2. 3 M. Johnon and R. H. Silbee, Phy. Rev. B 35, P. C. van Son, H. van Kempen, and P. Wyder, Phy. Rev. Lett. 58, T. Valet and A. Fert, Phy. Rev. B 48, G. Schmidt, D. Ferrard, L. W. Molenkamp, A. T. filip, and B. J. van Wee, Phy. Rev. B 62, R I. Zutik, J. Fabian, and S. Da Sarma, cond-mat/ I. D Amico and G. Vignale, Phy. Rev. B 62, I. D Amico and G. Vignale, Europhy. Lett. 55, Y. Takahahi, K. Shizume, and N. Mauhara, Phy. Rev. B 6,

8 DIFFUSION EQUATION AND SPIN DRAG IN SPIN-... PHYSICAL REVIEW B Y. Takahahi, K. Shizume, and N. Mauhara, Phyica E, W. J. Mullin and J. W. Leon, J. Low Temp. Phy. 88, A.-P. Jauho and H. Smith, Phy. Rev. B 47, K. Flenberg and B. Y.-K. Hu, Phy. Rev. B 52, B. Y.-K. Hu and K. Flenberg, in Hot Carrier in Semiconductor, edited by K. He Plenum Pre, New York, 996, p H. Smith and H. H. Jenen, Tranport Phenomena Clarendon Pre, Oxford, B. Y.-K. Hu and K. Flenberg, Phy. Rev. B 53,