MECHANISMS OF CARRIER-INDUCED FERROMAGNETISM IN DILUTED MAGNETIC SEMICONDUCTORS
Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Prospect Na
uki 45, Kiev 252028, Ukraine
arXiv:cond-mat/9908040v1 [cond-mat.stat-mech] 3 Aug 1999
Institute of Physics, National Academy of Sciences of Ukraine, Prospect Nauki 46, Kiev 252028, Ukraine (February 1, 2008) Two di?erent approaches to the problem of carrier-induced ferromagnetism in the system of the disordered magnetic ions, one bases on self-consistent procedure for the exchange mean ?elds, other one bases on the RKKY interaction, used in present literature as the alternative approximations is analyzed. Our calculations in the framework of exactly solvable model show that two di?erent contributions to the magnetic characteristics of the system represent these approaches. One stems from the diagonal part of carrier-ion exchange interaction that corresponds to mean ?eld approximation. Other one stems from the o?-diagonal part that describes the interaction between ion spins via free carriers. These two contributions can be responsible for the di?erent magnetic properties, so aforementioned approaches are complementary, not alternative. A general approach is proposed and compared with di?erent approximations to the problem under consideration.
One of the fruitful ideas of microscopic theory of magnetism is to separate the electrons with non-compensated spins into those localized in the crystalline lattice cites and delocalized (band) ones. The exchange interaction between localized and delocalized electrons is then represented as a spin density of band electrons at the cites occupied by magnetic ions. Vonsovskii realized this idea in1 and calculated band electron energy shift due to s?dexchange interaction. The Hamiltonian of s? d-exchange interaction was further used to study the ferromagnetic metals (2 ,3 ), super?ne interactions in the solid solutions of magnetic metals (4 ), and magnetic impurities in nonmagnetic metals (5 ). The detailed review of this material has been made in6 . Let us note some speci?c features of the carrier-ion exchange interaction. Its diagonal part is represented by the exchange integral J(k, k ′ = k) over the Bloch functions with wave vectors k, k ′ = k and describes so-called magnetizing e?ect (1 ) (or e?ect of redistribution of electronic population (repopulation e?ect),7 ). The latter e?ect is indeed the variation of total spin of the band carriers in the e?ective exchange ?eld GL created by localized spin moments (LSM). The value of this variation is de?ned by Pauli susceptibility with ?eld GL . We emphasize that the e?ect of giant spin splitting (GSS) of band states (8 ) discovered in diluted magnetic semiconductors (DMS) has similar nature. From the standpoint of8 , the repopulation e?ect in metals may be considered as a redistribution the band electrons populations between spin split subbands in the same way as a single Fermi level is established for electrons with opposite spin orientations. The o?-diagonal in k (9 ) part of the carrierion exchange interactions with the J(k, k ′ = k) gives rise to the spin density oscillations (4 ). These oscillations, however, lead to indirect exchange interaction between
the LSMs, known in metal physics as RKKY interaction (10 ,2 ,11 ). E?ective exchange ?eld GL is proportional to LSM magnetization M . Latter, in turn, is de?ned by the sum of external magnetic ?eld B and e?ective exchange ?eld Be , which acts on the LSM by the polarized electron spins. The LSM magnetization M also depends on spinspin interactions between LSM. Below we will present the arguments that both direct spin-spin interaction which is independent on the free carriers (LL-interactions) and indirect one induced by the free carriers (LeL -interactions) contribute to M . In turn, the e?ective ?eld of carriers Be depends on the quantity GL + ge ?B B, determined by electronic spin polarization, where ge is electron g-factor, ?B is Bohr magneton. So, the e?ective exchange ?eld GL is expressed through itself in a self consistent manner. Generally speaking, such self consistency increases the magnetic susceptibility χ = Const(T ? Θ)?1 (12 ,13 ). Thus one can expect a ferromagnetic phase transition if Θ > 0. A possibility of the appearance of dopant-induced ferromagnetism in semiconductors due to exchange interaction of the band and localized electrons was studied for the case of semiconductor with bivalent shallow impurities which reveal both magnetic and electric properties (12 ). Later the expressions for critical temperatures of ferromagnetic transition in the DMS with deep magnetic ion levels were obtained in13 . It should be noted that self consistent procedure was essential for determination of the Be and BL exchange ?elds in both aforementioned works. In recent years, it has been substantial increase of interest to studies of the carriers induced ferromagnetism in the DMS. A number of works (see14 -17 and references therein) were devoted to the proof of the existence of fer1
romagnetic transition in the DMS P b1?x?y Sny M nx T e, induced by strong exchange interaction of the Mn ions with the band holes with wide range (up to 2 · 1021 cm?3 ) of concentrations. Ferromagnetism of the Mn ions was found to be due to their interaction with band holes con?ned in two-dimensional quantum wells on the base of DMS Cd1?x M nx T e (18 ). The carrier-induced ferromagnetism was also observed in the structures A3 M nx B 5 1?x with x about few percents. The holes in these structures are also associated with the Mn ions (19 ). However, the approach used in the aforementioned works (17 ,20 ,21 ) was di?erent from that of12 and13 . Namely, the role of the band carriers were reduced to induction of the RKKY interaction only. So, self-consistent contribution of diagonal part of carrier-ion exchange interaction was not taken into account. Note, that authors of Ref. [D2] carried out a special analysis of magnetic susceptibility peculiarities in the assumption that role of carrier-ion interaction is reduced to the repopulation e?ect only. In the spirit of the works12 and13 , they used the self consistent procedure for the exchange ?elds GL and Be . The ferromagnetic phase transition temperature ΘMF , obtained by this procedure was shown to coincide with transition temperature ΘRKKY , calculated with the help of RKKY interaction, considered as the sole reason for carrier-induced ferromagnetism. Such coincidence can be obtained only under following additional assumptions: (i) RKKY interaction can be represented in terms of Curie-Weiss ?eld, (ii) the magnetic ions spatial distribution corresponds to the ideal gas of particles. The obtained coincidence ΘRKKY = ΘMF may cause an illusion of equivalence and interchangeability of these two approaches. It already follows from aforementioned discussion that the equality ΘRKKY = ΘMF is not strictly asserted for real systems because at least the spatial distribution of magnetic ions corresponds to the lattice gas rather then to ideal gas. Moreover, it is possible to imagine a situation when the non random spatial distribution results in antiferromagnetic transition due to oscillating nature of RKKY interaction with ΘRKKY < 0 while inequality ΘMF > 0 is true for any spatial distribution of the magnetic ions. Formally speaking, the equality ΘRKKY = ΘMF by itself cannot be used as an evidence of the statement that either self-consistent exchange ?elds consideration or carriers-induced LeL interaction are the equivalent descriptions of the same interactions. This stems from the aforementioned fact that they are described by di?erent parts of Hamiltonian. Thus, both self-consistent mean exchange ?elds, leading to the repopulation e?ect, and Curie-Weiss ?eld stemming from the carriers-induced spin-spin interaction, have to be taken into account simultaneously (22 ). They are complementary to each other in the analysis of both carrierinduced ferromagnetism in DMS and dopant-induced ferromagnetism in metals (see23 and a remark to this point in13 ). In spite of the fact, that latter conclusion follows 2
both from the analysis of classical works1 ,10 and from detailed discussion of the di?erences between manifestations of diagonal and o?-diagonal parts of interaction in4 , it is not completely clear if it is applicable to the case of carrier-induced ferromagnetism in DMS. In our view this question can be clari?ed by the calculation of phase transition temperature Tc from the ?rst principles by means of exactly solvable model. In this case, we avoid the auxiliary models for self-consistent ?elds GL , Be or Curie-Weiss ?eld of LeL-interacting LSM. We also avoid the discussion of the equivalence or complimentarity of the approaches under consideration. It is impossible to calculate exact value of Tc in real experimental cases14 ,18 ,19 . However, this is not necessary since our aim is to clarify the relative role of aforementioned mechanisms of magnetic phase transitions. Below we give an illustrative example of exact solution for the magnetic phase transition temperature Tc . This example allows also to separate contributions from self-consistent ?elds GL and Be and carriers-induced LeL spin-spin interaction. The Hamiltonian of our model is similar to that applied in aforementioned works and comprises a sum of LSM , electrons and their interaction Hamiltonians: H = Hm + He + Hem , where Hm = gm ?B B He =
j j SZ ≡ ω m M L ,
(εb + ωe σ) a ?b,k,σ ab,k,σ , Ab,b′ (σM ) a ?b,k,σ ab′ ,k,σ .
Hem = ?
j Here SZ is Z-component of j-th LSM spin while ML = j j SZ , j = 1...Nm , Nm is the number of LSM in the system, gm is the LSM g-factor, ωm = gm ?B B is LSM Zeeman splitting in the ?eld B and ωe is delocalized electrons Zeeman splitting in the ?eld B. Three quantum numbers can be attributed to electrons: band number b, intraband quantum number k and projection of spin σ =±1/2; a?b,k,σ and ab,k,σ are the creation and annihilation operators; J is a constant of carrier-ion exchange interaction; normalization factor N0 corresponds to one half of the number of electronic states in each of bands b; Ab,b′ (σM ) is an interband transition matrix element. The structure of Hamiltonian (1) is similar to that in12 21 . The di?erence is both in the dispersions of band carriers, εb,k = εb , that formally corresponds to ?at bands, and in the lack of intraband exchange scattering. The exchange scattering between bands b and b′ with electron spin ?ip is taken into account by the matrix element Ab,b′ . If we restrict ourselves to only two electronic bands b = 1, 2, the diagonalization of the Hamiltonian becomes trivial. Eigenenergy E is de?ned by the redistribution of electrons (with spins projections σ) within bands b = 1
and 2 as well as by the magnetic ions (with spin projecj tions SZ ) distribution or, more precisely, by normalized amount of magnetic ions ? = ML /Nm . For simplicity we assume Ab,b′ = 1. Thus, the energy of unit volume reads: Eb = 1 nb ?E + (GL + ωe ) (nb+ + nb? ) ± 2 1 nb ?E 2 1+ GL ?E
obey a Boltzmann statistics. Such approach is evidently realized in the limit Ne ? N0 . Thus, the partition function Z (and hence the magnetic susceptibility ) is calculated by straightforward integration in Eq. (3) with aforementioned assumptions. After some algebra we arrive at following ?nal result: χ0 = 1?
2 3 S(S
+ ωm nm ?.
χ0,L J 2 x2 Ne 1) 4T ?E Nm ? 2 S(S 3
+ 1) J x2 8T
The signs ”-” or ”+” before the square root sign in the equation (2a) correspond to b = 1 or b = 2, ?E is an energy interval between these bands, the nb+ and nb? corresponds to concentration of electrons with the spin projection σ = +1/2 and ?1/2 in the band b, the total concentration is nb = nb+ + nb? , GL = Jx?, x = Nm /N0 being a fraction of magnetic cations in the crystal, Nm is their concentration. Since value of GL is in?nitesimal at T > Tc , the square root in expression (2a) can be expanded over the small parameter (GL /?E)2 up to ?rst nonvanishing term. j j′ This term is proportional to ?2 = (Nm )?2 j,j ′ SZ SZ . Therefore, it can be considered as a contribution to the energy from LeL spin-spin interaction induced by the band electrons. We will also assume, that the only lowest energy band b = 1 is ?lled , i.e. E ? kT . Then, the energy spectrum assumes following form: E = ne Jx?σe ? ne GL 2?E
where χ0,L = 2 S(S + 1) nm is a paramagnetic suscepti3 T bility of noninteracting LSSM with concentration nm Expression (5) allows an easy interpretation. In the case of band electrons absence, Ne = 0, the χ0 describes an ideal paramagnet. If the electrons are present, the interband exchange scattering in?uence can be excluded from consideration as ?E → ∞ . Clearly, this corresponds to taking into account the self-consistent exchange ?elds GL and Be only. General Eq. (5) gives for this case Tc = ΘMF = 1 S(S + 1)J 2 ?2 nm ne , 0 12 (6)
+ ωm nm ?.
For brevity, we introduce a total concentration of electrons ne = nb=1 and an average projection of electronic spins σe = (n+ ? n? )/(n+ + n? ). The ?rst term in Eq.(2b) corresponds to diagonal part of interaction (1) while the second one represents the contribution from o?-diagonal (with respect to quantum number b) part; the electronic g- factor is assumed to be equal to zero. For magnetic susceptibility calculations we need the partition function. This function has the form Z= UNm (ML ) UNe (MB ) exp ? E kT dML dMB (3) and can be immediately calculated with the help of Eq. (2b). The full projections (ne yasno) of LSM ML = Nm ? and the band electrons MB = Ne σe are introduced in Eq. (3). Beyond magnetic saturation, the statistical weights UN (M ) are de?ned by Gaussian distribution in thermodynamic limit (Nm → ∞, Ne → ∞ ) (24 ): UN (M ) = M2 (2S + 1)N √ exp ? ?S π?S , (4)
where ?0 is a crystal unit cell volume. It is interesting to note that in spite of the extreme simplicity of the model under consideration, Eq. (6) reproduces a result of13 obtained in terms of self-consistent exchange ?elds for more realistic situation. It allows to consider the third term in the denominator of Eq. (5) as a contribution from selfconsistent exchange ?elds Ge and GL . As it was mentioned above, the second term is due to o?-diagonal part of carrier-ion exchange Hamiltonian. With respect to this part only, we have to omit the last term in denominators of Eq.(5). In this case we obtain Tc = Θind ≡ 1 J 2 ?2 n m n e 0 S(S + 1) . 6 ?E (7)
where ?S =(2/3)S(S+1)N, S is the LSM value. Eq.(4) is also applicable for band electrons with S=1/2 if electrons 3
It is seen that Eq. (7) is not similar to Eq. (6), neither quantitatively nor qualitatively. It gives possibility to make a following general statement. For the problems of magnetic phase transitions, as well as for the calculations of magnetic susceptibility, magnetization, etc. it is important to simultaneously take into account both parts (diagonal and o?-diagonal) of carrier-ion exchange interaction. Therefore, the omission of any one of the aforementioned terms in the Hamiltonian leads, generally speaking, to signi?cant inaccuracy or even to qualitative changes. We shall present now the way of consideration of the problem of spontaneous magnetic transitions induced by band carriers in DMS, which we consider to be correct. We choose the Hamiltonian in the form (1), but will incorporate there the LL spin-spin interaction between LSM HLL (13 ) to the Hm and the intraband exchange scattering between Bloch electron states (25 ).
To calculate the magnetic susceptibility with the help of modi?ed Hamiltonian (1), we shall carry out the approximate diagonalization of Eq. (1) by elimination of its o?-diagonal (in k and k ′ ) components by canonical transformation method (26 ) in second order of perturbation theory. As a result, the operator of e?ective LeL spin-spin interaction assumes the form: HLeL = Jef f Rj,j ′ S j S j ,
where Rj,j ′ is a radius-vector joining the pairs of magnetic ions in the crystal lattice sites j and j ′ . The structure of HLeL is similar to the HLL in magnetic Hamiltonian Hm and can be added to it. The speci?c form of Jef f Rj,j ′ is de?ned by electronic gas degeneration (27 ), in?uence of magnetic ?eld (28 ), e?ect to casual anisotropy (Ref. [IK]), structure of energy band of semiconductor (17 ) and dimensionality of the system (20 ,30 ). The diagonal part of the operator Hem can be written down in the form of LSM’s Zeeman energy in the e?ective ?eld Be = J?0 ne σe /gm ?B . This part can be added to the Zeeman term of magnetic Hamiltonian Hm . One more standard step is the transformation of spin-spin interactions in Hm and HLeL (8) to Curie-Weiss ?elds. Such approach, as it is well known, reduces the thermodynamical treatment of interacting spin to the consideration of isolated spins with the e?ective temperature Tef f = T ?Θ. The parameter Θ = ΘLL +ΘLeL is de?ned by both LL interaction and LeL interaction (Eq.(8)). As a result the free energy can be presented in terms of electronic and ionic parts only (31 ): F = Fe (σe ) + Fm (B + Be , T ? Θ) , (9)
where Fm (B + Be , T ? Θ) is a contribution of noninteracting (isolated) spins subjected to the uniform magnetic ?eld B + Be at the temperature T ? Θ. Note that the Eq. (9) takes into account both diagonal part of carrierion exchange interaction (term Be ) and its o?-diagonal part (term Θind ). The electronic polarization σe is calculated by minimization of functional (9). Further substitution of σe obtained in such manner to Eq. (9) de?nes completely thermodynamic characteristics of the system: magnetization Mα = ??F/?Bα , susceptibilityχαβ = ?? 2 F/?Bα ?Bβ , α, β = x, y, z and the temperature of magnetic phase transition. Speci?c form of free energy functional (9) depends on aforementioned and many other peculiarities of our system. As an illustration, we consider now the most popular case of degenerated electronic gas in a simple isotropic band of semiconductor. We consider the magnetic transition temperature Tc based on the preceding results (12 ,13 ,14 ,17 ,20 ,23 ,31 ). The Eq. (9) permits to obtain following equation for the critical temperature point: (Tef f )c ? ΘMF = 0. (10) 4
Here, ΘMF is de?ned by corresponding formulas of the13 ,20 , where only the diagonal part of the interac′ tion operator Hem is taken into account and (Tef f )c = Tc ? ΘLL ? ΘRKKY . Parameter ΘRKKY coincides with the ΘMF only under the conditions, mentioned in the introduction to this paper(20 ). Parameter ΘLL should be taken from the experiment, ΘLL = ?T0 , where T0 > 0 corresponds to antiferromagnetic LL exchange interaction realized in majority of experimental situations for DMS (Ref. [15,19,20]). So, for Tc we can obtain Tc = 2ΘMF ? T0 . If one takes into account only self-consistent exchange mean ?elds or RKKY interactions, the value of Tc is determined by other expression: Tc = ΘMF ? T0 . This di?erence can be important to the prediction of conditions for carriers-induced ferromagnetism realization in di?erent experimental situations. We had shown that neglecting of any part of interactions Hem leads to essential reduction of predicted Tc value even in the simplest models. Moreover, considered example shows that such neglecting leads to the qualitatively di?erent results. Present work shows that both diagonal and o?-diagonal in k parts of carrier-ion exchange interaction are important. Their simultaneous consideration is shown to change the part of earlier results quantitatively or even qualitatively. Nevertheless, main conclusion of preceding works remains valid: carrier induced ferromagnetic transition in DMS is possible under high enough carriers concentrations and reduction of the system dimensionality enhances this e?ect. Let us ?nally note that we have considered the necessity of simultaneous consideration of both contributions of carrier-ion exchange interaction. One can see that effect is essential. It is clear, that similar approach have to be applied to many other cases. For instance, it would be useful to take into account the LeL exchange interaction of the LSM via conduction electrons in the problem of free or bound magnetic polaron.
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