Ultrafast magnetization dynamics in diluted magnetic

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Home Search Collections Journals About Contact us My IOPscience. Ultrafast magnetization dynamics in diluted magnetic semiconductors. This article has been downloaded from IOPscience Please scroll down to see the full text article. 2009 New J Phys 11 073010,http iopscience iop org 1367 2630 11 7 073010. View the table of contents for this issue or go to the journal homepage for more. Download details,IP Address 130 79 210 55,The article was downloaded on 29 05 2011 at 13 24. Please note that terms and conditions apply, New Journal of Physics The open access journal for physics. Ultrafast magnetization dynamics in diluted magnetic. semiconductors,O Morandi1 3 P A Hervieux2 and G Manfredi2. INRIA Nancy Grand Est and Institut de Recherche en Math matiques. Avanc es 7 rue Ren Descartes F 67084 Strasbourg France. Institut de Physique et Chimie des Mat riaux de Strasbourg 23 rue du Loess. F 67037 Strasbourg France,E mail morandi dipmat univpm it.
New Journal of Physics 11 2009 073010 12pp,Received 13 March 2009. Published 3 July 2009,Online at http www njp org,doi 10 1088 1367 2630 11 7 073010. Abstract We present a dynamical model that successfully explains the. observed time evolution of the magnetization in diluted magnetic semiconductor. quantum wells after weak laser excitation Based on the pseudo fermion. formalism and a second order many particle expansion of the exact p d. exchange interaction our approach goes beyond the usual mean field. approximation It includes both the sub picosecond demagnetization dynamics. and the slower relaxation processes that restore the initial ferromagnetic order in. a nanosecond timescale In agreement with experimental results our numerical. simulations show that depending on the value of the initial lattice temperature a. subsequent enhancement of the total magnetization may be observed within the. timescale of a few hundred picoseconds,1 Introduction 2. 2 Pseudo fermion formalism 3,3 Time evolution model 4. 4 Spin evolution in DMS 6,5 Conclusion 10,Acknowledgments 11.
References 11, Author to whom any correspondence should be addressed. New Journal of Physics 11 2009 073010, 1367 2630 09 073010 12 30 00 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. 1 Introduction, Ultrafast light induced magnetization dynamics in ferromagnetic films and in diluted magnetic. semiconductors DMS is today a very active area of research Subsequent to the observation of. the ultrafast dynamics of the spin magnetization in nickel films 1 and the analogous processes. in ferromagnetic semiconductors 2 special interest has been devoted to the development of. dynamical models able to mimic the time evolution of the magnetization on both short and. long timescales In III V ferromagnetic semiconductors such as GaMnAs and InMnAs a small. concentration of Mn ions is randomly substituted by cation sites so that the Mn Mn spin. coupling is mediated by the hole ion p d exchange interaction allowing the generation of. a ferromagnetic state with a Curie temperature of the order of 50 K 3 The magnetism can. therefore be efficiently modified by controlling the hole density through doping or by excitation. of electron hole pairs with a laser pulse In particular unlike metals in a regime of strong laser. excitation total demagnetization can be achieved 4. In the Zener model 5 which was originally developed to describe the magnetism of. transition metals the d shells of the Mn ions are treated as an ensemble of randomly distributed. impurities with spin 5 2 surrounded by a hole gas or an electron gas Unlike ferromagnetic. metals III Mn V ferromagnetic semiconductors offer the advantage that they provide a clear. distinction between localized Mn impurities and itinerant valence band hole spins thus allowing. the basic assumptions of the Zener theory to be satisfied Based on this hypothesis a few. mean field models have been successfully applied for modelling the ground state properties. of DMS nanostructures In particular within the framework of the spin density functional. theory at finite temperature relevant predictions of the Curie temperature have been obtained. 6 7 Ultrafast demagnetization in DMS is a phenomenon where the p d exchange interaction. causes a flow of spin polarization and energy from the Mn impurities to the holes which is. subsequently converted to orbital momentum and thermalized through spin orbit and hole hole. interactions 8 Since energy and spin polarization transfer is a many particle effect the mean. field Zener approach cannot provide a satisfactory explanation for the ultrafast demagnetization. regime that has been observed in DMS 4 9, A phenomenological approach able to take into account this energy flux is given in 1 10. where a model based on three temperatures is derived More recently a study of the coupling. of the electromagnetic laser field with the hole gas revealed the possibility of an ultrafast. demagnetization during the femtosecond optical excitation due to light hole entanglement 11. A model capable of describing the dynamics of carrier ion spin interactions is provided in. 12 13 This model generalizes the stationary theory of 10 and takes into account the. picosecond demagnetization evolution that occurs in a strong excitation regime but neglects. the slow in time evolution of the spin dynamics The mean p d interaction is averaged out over. the randomly distributed positions of the Mn ions, In this work we derive a dynamical model based on a many particle expansion of the p d.
exchange interaction in the pseudo fermion framework This formalism originally developed. by Abrikosov 14 to deal with the Kondo problem introduces unphysical states in the Hilbert. space for which impurity sites are allowed to be multiply occupied Following the work of. Coleman 15 a suitable limit procedure is applied to our dynamical model in order to recover. the correct physical description of the magnetic impurities. Our approach extends the Zener model beyond the usual mean field approximation It. includes both the sub picosecond demagnetization dynamics and the slower cooling processes. New Journal of Physics 11 2009 073010 http www njp org. that restore the initial ferromagnetic order which is achieved in a ns timescale Moreover in. agreement with recent experimental results 9 our simulations show that depending on the. initial lattice temperature a subsequent enhancement of the total magnetization is observed. within the timescale of 100 ps,2 Pseudo fermion formalism. We consider a volume V containing N h V holes with spin S h 1 2 strongly coupled by. spin spin interaction with N M V randomly distributed Mn impurities with spin S M 5 2 We. assume that the exchange interaction between localized ions and heavy holes dominates both the. short range antiferromagnetic d d exchange interaction between the ions and the s d exchange. interaction between electrons in the conduction band and Mn ions typical values for the s d and. the p d interactions in a GaAs are 0 1 and 1 eV respectively 16 Furthermore electron hole. radiative recombination carrier phonon interactions and interactions leading to the hole spin. relaxation in the hole gas are included phenomenologically The time evolution of the system is. governed by the Hamiltonian,H k s ak s, where ak s ak s is the creation annihilation operator of a hole with spin projection s and. quasi momentum k In the parabolic band approximation the kinetic energy of the holes reads. k s E h h 2 mk where E h is the valence band edge The Kondo like exchange interaction Hpd. is given by,Hpd Jm 0 m s 0 s b m,0 b m ak 0 s 0 ak s ei k k R. where the sum is extended over all indices is the p d coupling constant and and J are. the spin matrices related to S h and S M respectively The ion spin operator is represented in. the pseudo fermion formalism 14 15 in which b m b m denotes the creation annihilation. operator of a pseudo fermion with spin projection m and spatial position R. The Hpd Hamiltonian reproduces the correct ion hole exchange interaction provided that. the ion sites are singly occupied i e n m S M b m b m 1 Following 14 15 this. constraint may be taken into account by adding a fictitious ionic chemical potential. to the original Hamiltonian and letting go to infinity at the end of the calculation. The grand canonical expectation value of a pseudo fermion operator A related to the total. Hamiltonian H H reads,Tr H e n A,n H e n A n m, New Journal of Physics 11 2009 073010 http www njp org. where Z Tr H e n H e H and 1 kB T h with kB being the Boltzmann. constant and T h the hole temperature n m r n 11 n 2S. n r 2S 1 denotes all, possible occupation numbers n k 0 or 1 for r ion sites Since each site has 2S M 1 available.
pseudo fermion states the system will contain at most 2S M 1 r pseudo particles The correct. expectation value of the operator A is obtained using the limit 15. Z z 0 z 1 zr,where Z lim z 0 z 1 z Z r, In the next section we will show that the time evolution of the spin of the ion hole. system may be expressed in terms of the expectation value of the pseudo fermion operator. b m b m 1 b m, 0 b m 0 with m 6 m and evaluated in the mean magnetic field S generated by. the holes We have the general relationship which also applies when the system is driven far. from equilibrium,lim b m b m 1 b m 0 b m 0 lim b m. When the system approaches thermal equilibrium the quantity hb m b m i becomes the usual. spin thermal distribution Using equation 1 we obtain. e Q nm0 0 1 P n m0 1 with Q m 0 6 e m S n 0,where Z Q e. In order to derive equation 3 we have used,Tr H e 0 0 n 0 n m Q.
n m e S mn Q,z 0 z 1 zr,P m n 0 1 m n 1,with H e S. m and n m b m b m,3 Time evolution model, The Heisenberg equations P of motion lead to a hierarchy. P of time evolution equations for the,mean densities n hs N1h k hak s ak s i and n M. NM hb m b m i,d k hak s ak s i X,NhNM Ws s m 1 m 1 4. d hb m b m i,NMNh Ws1 s1 m m 5, Jm 01 m s10 s Cem 01 m s10 s Jm m 01 s s10 Cem m 01 s s10.
Ws s m m 6, New Journal of Physics 11 2009 073010 http www njp org. In the last equation the mean correlation function reads. i X k 0 k1,Cm 0 m 11 s 0 s1 ei k1 k1 R,Cem 0 m 1 s10 s1 h M. where Cm 0 m s 0 s hb 0 m 0 b m ak 0 s 0 ak s i The time evolution equation of this quantity is given by. dCm0 m s0 s,1 10 Jm 01 m 1 s10 s1 BA At Bt ei k1 k1 R 1. ih 1E MF Cm0 m s0 s 8,m1 m1 s1 s10,where the compact notations m m s k s B bm. 0 bm1 bm0 bm B bm0 bm bm0 bm1, A as 0 as1 as 0 as and At as 0 as as 0 as1 have been employed.
The mean field contribution to the total energy is given by 1E MF s 0 s M. m 0 m S where M N M m S M,M m n m and S N,s S h s n s are the mean magnetic field. generated by the ions and by the holes respectively. The use of equation 8 combined with equations 4 and 5 leads to a non Markovian time. evolution of the macroscopic dynamical variables such as the density and the magnetization. By assuming an instantaneous spin spin interaction the Markov approximation can be easily. recovered For further details of the justification of the Markovian approximation in a DMS. excited by a laser pulse we refer the reader R t to 12. By using the Dirac identity 17 e i t t h dt 0 h ih P 1 where P denotes. the principal value the integration of equation 8 with respect to the time leads to. i k10 k1 R 1 k 0 k R 0,Cm 0 m s 0 s i 1EMF Jm 01 m 1 s10 s1 BA B A e. m1 m1 s1 s1 k k,where 1EMF k 0 k 1E MF, Since the matrix operators J are real it is clear from equation 6 that the imaginary part. gives no contribution to the equation of motion, The many particle expansion of the correlation function Ceallows us to express equation 9. in terms of the single particle density matrix elements n hs and n M m By using the commutation. rules of the creation and annihilation operators we obtain. m1 m m m1 s1 s s1 s,0 bm0 as as as 0 as0, Furthermore as a closure hypothesis we have assumed that the non diagonal matrix elements.
of the density like operators as 0 as and bm 0 bm with respect to the indexes and k vanish From. the above approximations and using the definition 4 and equation 9 we get. Ws s m 1 m 1 Jm 01 m 1 s s1 Jm 1 m 01 s1 s,h N S N M V. m1 s1 m 1 m 1,1EMF 5m 1 m 01 s s1 5m 01 m 1 s1 s 10. New Journal of Physics 11 2009 073010 http www njp org. 5 m 1 m 01 s s1,1 ak 1 s1 ak1 s1,b m 0 b m 0,ak s ak s 11. A similar expression can be found for s1 Ws1 s1 m m in equation 5 By using equation 2 we. recover the fermionic limit of 5 namely,5 m 1 m 1 s s1. k s k s 1 a,k1 s1 k1 s1 12, In order to evaluate the time derivative of n hs equation 12 can be solved numerically In the.
following paragraphs we show that equation 12 may actually be further simplified According. to the Zener model the ground state of the system can be estimated by taking into account only. the mean field interaction between the holes and the magnetic ions The hole gas experiences. a mean magnetic field equal to M and in turn generates a mean field acting on the ions system. equal to S By converting the sum over k and k1 in equation 12 into the corresponding integral. with respect to the energy variable E k we obtain,M M kB T h. N n m 01 e f a h E E 1E MF dE 13,s M kB T h,where f a has as i 1 has1. as1 i h 1 e s11M kk kB T h and denotes the hole density of states. Ultrafast magnetization dynamics in diluted magnetic semiconductors O Morandi1 3 P A Hervieux2 and G Manfredi2 1 INRIA Nancy Grand Est and Institut de Recherche en Math matiques Avanc es 7 rue Ren Descartes F 67084 Strasbourg France 2 Institut de Physique et Chimie des Mat riaux de Strasbourg 23 rue du Loess F 67037 Strasbourg France

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