We study the enhancement of the magnetic transition temperature $T_c$ due to
incident light in ferromagnetic semiconductors such as EuS. The photoexcited
carriers mediate an extra ferromagnetic interaction due to the coupling with
the localized magnetic moments. The Hamiltonian consists of a Heisenberg model
for the localized moments and an interaction term between the photoexcited
carriers and the localized moments. The model predicts a small enhancement of
the transition temperature in semi-quantitative agreement with the experiments.Comment: 5 pages, 5 figure

Afanasev M. M.

Kittel C.

Mishra S.

Oiwa A.

Rudov S. G.

SASHI SATPATHY

SUBODHA MISHRA

English

arXiv

Crossref

PHOTOINDUCED MAGNETISM IN THE FERROMAGNETIC SEMICONDUCTORS

We study the enhancement of the magnetic transition temperature $T_c$ due to incident light in ferromagnetic semiconductors such as EuS. The photoexcited carriers mediate an extra ferromagnetic interaction due to the coupling with the localized magnetic moments. The Hamiltonian consists of a Heisenberg model for the localized moments and an interaction term between the photoexcited carriers and the localized moments. The model predicts a small enhancement of the transition temperature in semi-quantitative agreement with the experiments.This work was supported by the U. S. Department of Energy through Grant No. DE-FG02-00ER45818

Mishra, Subodha, 1965-

Satpathy, Sashi Sekhar, 1956-

University of Missouri: MOspace

arXiv:0906.5514v1  [cond-mat.mtrl-sci]  30 Jun 2009Photoinduced magnetism in the ferromagnetic semiconductorsSubodha Mishra and Sashi SatpathyDepartment of Physics & Astronomy, University of Missouri, Columbia, MO 65211, USA(Dated: June 30, 2009)We study the enhancement of the magnetic transition temperature Tc due to incident light in fer-romagnetic semiconductors such as EuS. The photoexcited carriers mediate an extra ferromagneticinteraction due to the coupling with the localized magnetic moments. The Hamiltonian consists ofa Heisenberg model for the localized moments and an interaction term between the photoexcitedcarriers and the localized moments. The model predicts a small enhancement of the transitiontemperature in semi-quantitative agreement with the experiments.PACS numbers: PACS: 75.50.PpThere is a class of materials which are ferromagneticsemiconductors1,2,3,4,5 such as the Europium chalco-genides EuX (X=S, O, Se, Te) and chalcogenide spinelsMCr2Y4 (M=Hg, Cd; Y=S,Se). The ferromagnetismin these materials are generally well described by theHeisenberg model. In this paper, we are concerned withthe effect of incident light on the magnetism, in par-ticular, on the ferromagnetic transition temperature Tc.The basic mechanism of the photoenhanced magnetismis that light excites electron-hole pairs, which mediatean extra magnetic interaction between the local mag-netic moments, on top of any pre-existing magnetic in-teraction, thereby enhancing the Tc. The related phe-nomena, where ferromagnetism occurs only if light ispresent but otherwise the system is paramagnetic, hasbeen studied in connection with the dilute magneticsemiconductors.9,10,11,12,13,14,15,16Here we study the photoinduced magnetism in the fer-romagnetic semiconductors by solving a model Hamil-tonian within a mean-field approach by constructing aBCS-type wave function for the electron-hole pairs,17similar to our analysis developed earlier for the di-lute magnetic semiconductors.14 Our work provides asemi-quantitative explanation for the enhancement of Tc,which has been observed to be as large as 0.16K forEuS6,7,8 under intense laser radiation.The ferromagnetic semiconductor is modeled byisotropic conduction and valence bands interacting withfixed Eu magnetic moments originating from the f elec-trons. The bands are spin-split through an exchangeinteraction with the magnetic ions as indicated in Fig.1. The Hamiltonian consists of a Heisenberg interactionterm between the fixed Eu magnetic moments and a termdescribing the itinerant band electrons and their interac-tion with light. We thus writeH = −∑<ij>JMSi.Sj +HLeh, (1)where the first term is the Heisenberg term and < ij >denotes summation over distinct pairs of nearest neigh-bours. Each Eu atom has twelve nearest neighbor in thefcc lattice of EuS. The second term HLeh describes elec-trons and holes making up the conduction and valencee_h+hω+<S>Euspinse2J<S>h2J<S>JMFIG. 1: A schematic picture of the photoinduced ferromag-netism in EuS. Incident light creates electron-hole pairs thatmediate an extra ferromagnetic exchange interaction betweenthe Eu moments. The net result is an enhancement of thecritical temperature Tc by a fraction of a degree K for typicalparameters.bands and their interaction with lightHLeh = Hkin +Hc +HL(t). (2)The kinetic energy term is given byHkin =∑kσ(Eekσc†kσckσ + Ehkσd†kσdkσ), (3)where c†kσ, ckσ and d†kσ , dkσ are the field operators for theelectrons and holes, respectively, with k being the Blochmomentum and σ, the spin index. Throughout this pa-per, electrons refer to the photoexcited electrons in theconduction bands and holes refer to the lack of an elec-tron in the valence bands. Since the photon has zero spinand negligible momentum, absorption of light involves avertical transition of an electron in the valence bands intothe conduction bands with the spin of the electron con-served thus resulting in the creation of an electron-hole2pair described by c†kσd†−k−σ. Without incident light, wehave the “vacuum state” with no electrons nor holes.The single-particle energies in Eq. 3 include the Zee-man splitting due to the electron interaction with thelocalized Eu spins, so thatEekσ = Eg +~2k22me± Je < S > (4)andEhkσ =~2k22mh± Jh < S >, (5)where ± describes the Zeeman splitting of the up-spinand the down-spin states. Here Je and Jh are the elec-tron and hole exchange interactions with the localized Eumoments, < M >= gµB < S > is the average moment ofa Eu atom, me and mh are the electron and hole masses,respectively, and Eg is the band gap.The Coulomb interaction part consists of three termsHc = Hee +Hhh +Heh, (6)where the electron-electron interaction term isHee =12∑kk′qσσ′Vq c†k+q,σc†k′−q,σ′ck′σ′ckσ , (7)with a similar term for the hole-hole interaction, whilethe electron-hole interaction is given by the expressionHeh = −∑kk′qσσ′Vq c†k+q,σd†k′−q,σ′dk′σ′ckσ. (8)Finally, the last term in Eq. (2) which describesthe coupling to the external electromagnetic field of fre-quency ω, in the “rotating wave approximation” is givenasHL(t) = λ∑kρk[(c†k↑d†−k↓ + c†k↓d†−k↑)e−iωt + h.c.]. (9)As mentioned earlier, the light induces a vertical tran-sition with no net spin change and λρk is the cou-pling strength for the creation of such an electron-holepair. The coupling strength is given as λρk = eE <ψvk|eˆ · ~p|ψck > /mω where ψck, ψvk are conduction andthe valence state wave functions, E is the strength of theelectric field and eˆ is the electric polarization vector and ~pis the momentum operator. The matrix element may becomputed once the wave functions are known, but here,for simplicity, we neglect the momentum dependence ofthe matrix elements, so that ρk = 1 and the parameter λindicates the strength of the interaction. For strong laserradiation available in the laboratory, a broadly estimatedvalue used by earlier authors16,18 is λ ≈ 0.1 eV , whichwe use here in our numerical calculations.The time-dependence of Eq. (9) can be eliminated14,18by an appropriate unitary transformation e−iSHeiSwhere S = −ωt2∑kσ(c†kσckσ + d†kσdkσ) and the Hamil-tonian becomesH˜ = −∑<i,j>JMSi.Sj+∑kσ(Eekσ − ~ω/2)c†kσckσ + (Ehkσ − ~ω/2)d†kσdkσ+Hc + λ∑k{ρk(c†k↑d†−k↓ + c†k↓d†−k↑) + h.c.}. (10)Thus the time has been eliminated and the transformedHamiltonian may be interpreted as a “quasi free en-ergy” with the chemical potential for the creation of anelectron-hole pair given by µ = ~ω. One can thus writethe transformed Hamiltonian in the form H˜ = HQE−µN,where N =∑kσ c†kσckσ is the number of electron-holepairs.The Coulomb interaction term appearing in the Hamil-tonian (7) and (8) is given by Vq = 4πe2/(ǫq2), but in ourdiscussions below, we replace it, for the sake of simplic-ity, by a contact interaction term V (~r) = V0δ(~r) withoutchanging the qualitative physics. Its typical magnitudein semiconductors is V0 ≈ 20 eV.A˚3.14 Typical valuefor the exchange parameter is: JM = 2.19 × 10−5 eV ,which is calculated using the expression for the criti-cal temperature for the Heisenberg Hamiltonian kBTc =(ν/3)JMS(S+1). For EuS, Tc = 16K, S = 7/2, and thenumber of nearest neighbours ν = 12. The magnitudeof J = Je − Jh = −0.036 eV as obtained from estimat-ing Je and Jh from the splitting of the spin-up and thespin-down bands in the band structure of EuS.1We compute the free energy of the system by using avariational wave function similar to the Bardeen-Cooper-Schrieffer (BCS) wave function with zero net momentumand spin. The variational wave function consists of asuperposition of the electron-hole pairs19,20 generated bythe coupling to light|Ψ >=∏kσ(ukσ + vkσc†kσd†−k−σ)|0 >, (11)where |0 > is the intrinsic vacuum state (filled valenceand empty conduction bands), vkσ is the probability am-plitude for the creation of an electron-hole pair with theelectron momentum and spin of kσ and a hole with theopposite momentum and spin, while ukσ is the proba-bility amplitude for the absence of such pair excitation.The normalization condition is, as usual, u2kσ + v2kσ = 1,and the coefficients ukσ and vkσ may be chosen as real.In the present case the localized moments break the sym-metry between the spin up and down electrons leadingto a net spin polarization of the electron and holes. Thisin turn affects the net magnetic moment, which then af-fects the energies of the particles through the Zeemanterm, so that all quantities have to be determined in aself-consistent manner.We now calculate the expectation value of the “freeenergy” from Eq. (10), which may be rewritten in terms3of the gap parameter to yield14F0 =∑kσ[(ξkσ +Ωσ/2)v2kσ − (∆σukσvkσ/2 + λukσvkσ)]+ν2JM < S >2 ×N , (12)where∆σ = 2V0∑k′uk′σvk′σ + 2λ (13)andΩσ = −2V0∑k′v2k′σ. (14)Here, ν is the number of nearest neighbour (12 for the fcclattice) and the factor of 1/2 in the last term of Eq. 12appears to avoid double counting. N is the total numberof sites in the crystal and ξkσ is the energy of the electron-hole pair excitation with a spin-up electron and a spin-down hole measured with respect to the photon energyξkσ = Eek,σ + Ehk,−σ − ~ω= Eg +~2k22m∗± J < S > −~ω, (15)where J = Je−Jh and the effective mass of the electron-hole pairm∗−1 = m−1e +m−1h . Minimizing the free energywith respect to vkσ , ∂F0/∂vkσ = 0, and using the normal-ization conditions u2kσ + v2kσ = 1, we get the occupationamplitudes ukσ and vkσ, which readu2kσ =12[1 +ξkσ +Ωσ√∆2σ + (ξkσ +Ωσ)2] (16)andv2kσ =12[1−ξkσ +Ωσ√∆2σ + (ξkσ +Ωσ)2]. (17)We solve the equations (16) and (17) numerically whichyield the results for ukσ and vkσ , from which the freeenergy F0 is computed using Eq. (12). The typical pair-occupation amplitudes uk and vk have been shown in Fig.(2). We note that in the limit of small interaction (V = 0and λ = 0), the free energy (Eq. (12)) boils down to theenergy of the noninteracting system of electron-hole pairstates occupied up to the chemical potential (i.e., v2k↑ andv¯k↓2 equal to one below ~ω and zero above).Once the free energy is computed, the magnetizationof the Eu moments is calculated within the mean-fieldapproximation. Each Eu moment experiences an effectivemagnetic field Heff given byHeff = −∂F0(M)/N∂M, (18)so that the magnetization is given by the expressionM(T ) = gµBSBJ (βgµBSHeff (M)), (19)100.330occupationh2k2/2m* (eV)uk2vk2FkFIG. 2: Electron-hole pair amplitudes uk↑ and vk↑ as a func-tion of momentum k. Pairs of spin up electrons and spindown holes are filled up to the Fermi momentum kF↑ with-out any interactions. The step function is in the limit of zerointeractions (V0 = 0, λ → 0), while both the Coulomb inter-action and the coupling to light smear out the Fermi surface.Here parameters are: λ = 0.1 eV , m∗ = 1.5me, V0 = 20eV.A˚3 and ~ω − Eg = 0.33 eV, JM = 2.19 × 10−5 eV andJ = −0.036 eV . The pair amplitudes for the other spin, uk↓and vk↓, (not shown in the figure) are slightly shifted to theleft, since kF↓ is slightly lower than kF↑.where the BJ is the Brillouin function and β = (kβT )−1and M(S) = gµBS. The above two equations are solvedself-consistently for a given temperature using the freeenergy expression Eq. (12).Note that in our formulation, we have neglected thethermal excitation of the electrons from the valence to theconduction bands, because the number of such carriersis much smaller than the number of the photogeneratedcarriers. The latter number, obtained by integrating v2kσover all momentum, is typically ∼ 10−4/ A˚2, while thenumber of intrinsic thermal carriers is many orders ofmagnitude smaller,23 when the semiconductor bandgapis as large as ∼ 1eV , as is the case for EuS.76543210 0  4  8  12  16  20M(T) (µΒ)T(K)FIG. 3: Dependence of the magnetization M as a functionof temperature T with (dashed line) and without (solid line)the incident light. Parameters are: λ = 0.1eV , m∗ = 1.5,V0 = 20 eV.A˚3, ~ω −Eg = 1.5eV , JM = 2.19× 10−5 eV , andJ = −0.036 eV .The calculated magnetization and the number of pho-4Tcnn4.52 0  5  10  15  20number of e-h pairs (10-4 /A3 )T(K)FIG. 4: Pair occupancies as a function of temperature. n↑denotes the number of electron-hole pairs with spin up elec-trons and spin down holes, while n↓ denotes the number ofpairs with opposite spins. Below the critical temperature, n↑is larger than n↓ because of the lower Zeeman energy of theformer pairs. Above Tc, there is no net megnetization, so thatthe energy of the electron is independent of the spin (zero Zee-man splitting), so that the pair occupancies are independentof the spin of the electron or hole. Parameters are the sameas for Fig. 3. 0 0.6 1.2-0.5  0  0.5  1  1.5  2  2.5∆Tc (K)hω-EgFIG. 5: Photoinduced increase of the magnetic transitiontemerature as a function of the photon energy. Parametersare the same as in Fig. 2. A larger photon energy leads to alarger number of carriers which add to the ferromagnettoexcited electron-hole pair as a function of the temper-ature are shown in the Figs. 3 and 4 respectively. Theseare the two main results of this paper. As seen fromthe figures, there is a transition from the paramagneticto the ferromagnetic state as temperature is decreased.Fig. 3 shows that with the incident light and for typ-ical parameters, the photoinduced increase in the tran-sition temperature Tc is of the order of a fraction of adegree, ∆Tc ∼ 0.2K, in line with the experimental value8∆Tc ∼ 0.16K for the light frequency of ~ω−Eg = 0.33eV .Fig. 4 shows the number of pairs of spin-up electronand spin-down hole and pairs of spin-down electron andspin-up holes as function of temperature. While the num-ber of the electron-hole pairs is controlled by the chemicalpotential (µ = ~ω), the distribution of the pairs betweenthe two different spin types (spin-up electron and spin-down hole or vice versa) is determined by the magnitudeof 〈S〉, which affects the pair energy as indicated fromEqs. (15) and (19). The effective magnetic field expe-rienced by the Eu spins is roughly proportional to thedifference of the number of the two spin types, as may beseen by keeping the first line in the free energy Eq. (12)and using Eqs. (15) and (18), so that Heff ∝ (n↑ − n↓).This in turn implies that this number difference and themagnetization M follow one another, both ultimately go-ing to zero at the transition temperature Tc as seen fromFigs. 3 and 4. In the paramagnetic phase, the photoex-cited electron-hole pairs are present, but the two spintypes are the same in number.In Fig. 5 we plot the increase in Tc vs. the inci-dent light frequency, which as discussed earlier effectivelyserves as the chemical potential. We see that as the fre-quency increases, more and more photocarriers are cre-ated and hence ∆Tc increases. The figure shows thateven when ~ω − Eg < 0, there is an increase in Tc. Thismeans that we have photoinduced magnetization even forthe sub-bandgap light frequency, an issue that has beenstudied in detail by previous authors.16,24 Note that wehave neglected the relaxation of the excited carriers, sothat in our case carriers are filled up to the chemical po-tential µ = ~ω at equilibrium. This relaxation effect willdiminish the total number of photoexcited carriers, re-sulting in a somewhat lower value of ∆Tc than predictedfrom our analysis. Such relaxation effects must be takeninto account if a more quantitative description is desired.In summary, we studied the photoinduced enhance-ment of magnetization in the ferromagnetic semiconduc-tors by solving a model Hamiltonian which takes intoaccount the Coulomb interaction, light-matter interac-tion, the exchange interaction between the carriers andthe fixed magnetic moments, as well as the direct ex-change interaction among the localized moments. Fromthe calculated free energy, we computed the magnetiza-tion of the system as a function of temperature. There isan enhancement of Tc by a fraction of a degree K due tocarriers created by light, in semi-quantitative agreementwith the observed experimental results for EuS.This work was supported by the U. S. Department ofEnergy through Grant No. DE-FG02-00ER45818.1 W. Muller and W. Nolting, Phys. Rev. B 66 085205 (2002).2 P. Larson and W. L. Lambrecht, J. Phys.:Condensed Mat-ter 18 11333 (2006)3 W. Lems, P. J. Rijnierse, P. F. Bongers and U. Enz, Phys.Rev. Lett. bf 21 1643 (1968)4 N. Sanford, R. W. Davies, A. Lempicki, W. J. Miniscalcoand S. J. Nettel, Phys. Rev. Lett. 50 1803 (1983)5 S. G. Rudov and V. G. Veselago, Sov.Phys.Solid State 2151875 (1979)6 E. L. Nagaev, Phys. Status Solidi B 145, 11 (1988)7 M. M. Afanasev, M. E. Kompan and I. A. Merkulov, Zh.eksper. teor. Fiz. 71, 2068 (1976)8 M. M. Afanasev, M. E. Kompan and I. A. Merkulov, Zh.eksper. teor. Fiz.,Pisma 23, 621 (1976)9 H. Krenn, W. Zawadzki and G. Bauer, Phys. Rev. Lett.55, 1510 (1985)10 S. Koshihara, A. Oiwa, M. Hirasawa, S. Katsumoto, Y.Iye, C. Urano, H. Takagi and H. Munekate, Phys. Rev.Lett. 78, 4617 (1997)11 A. Oiwa, Y. Mitsumori, R. Moriya, T. Slupinski and H.Munekata, Phys. Rev. Lett. 88, 137202-1 (2002)12 Y. Mitsumori, A. Oiwa, T. Slupinski, H. Maruki, Y.Kashimura, F. Minami and H. Munekata, Phys. Rev. B.69, 033203 (2004)13 J. Wang, I. Cotoros, K. M. Dani, X. Liu, J. K. Furdyna,and D. S. Chemla, Phys. Rev. Lett. 98, 217401 (2007)14 S. Mishra, G. S. Tripathi, and S. Satpathy, Phys. Rev. B77, 125216 (2008)15 S. Mishra, Photoinduced Ferromagnetism (VDM VerlagDr. Mueller e. K., Saarbru¨cken, 2008)16 J. Fernandez-Rossier, C. Piermarocchi, P. Chen, A. H.MacDonald and L. J. Sham, Phys. Rev. Lett. 93, 127201(2004)17 J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev.B. 108,1175 (1957)18 C. Comte and G. Mahler, Phys. Rev. B 34, 7164 (1986);C. Comte and G. Mahler, Phys. Rev. B 38, 10517 (1988)19 Th. Harbich and G. Mahler, Phys. Status Solidi B 104,185 (1981)20 Th. Harbich and G. Mahler, Phys. Status Solidi B 117,635 (1983)21 F. Bassani and G. Pastori Parravicini, Electronic Statesand Optical Transitions in Solids (Pergamon, New York,1975)22 V. I. Litvinov, K. A. Valuev, and K. D. Tovstyuk, Sov.Phys. JETP 55, 478 (1982)23 C. Kittel, Introduction to Solid State Physics, Seventh Ed.(Wiley, 1995)24 D. Fro¨hlich, A. No¨the, and K. Reimann, Phys. Rev. Lett.55, 1335 (1985)

Photoinduced magnetism in the ferromagnetic semiconductors

Photoinduced magnetism in the ferromagnetic semiconductors

Abstract

Similar works

Full text

Available Versions

Crossref

University of Missouri: MOspace