Table Of Content

Theory of Ferromagnetism in Diluted Magnetic Semiconductor Quantum Wells Byounghak Lee1, T. Jungwirth1,2, and A.H. MacDonald1 1Department of Physics, Indiana University, Bloomington, Indiana 47405 2Institute of Physics ASCR, Cukrovarnická 10, 162 00 Praha 6, Czech Republic (February 1, 2008) 0 0 0 We present a mean field theory of ferromagnetism in diluted magnetic semiconductor quantum 2 wells. When subband mixing due to exchange interactions between quantum well free carriers and n magnetic impurities is neglected, analytic result can be obtained for the dependence of the critical a temperature and the spontaneous magnetization on the distribution of magnetic impurities and J the quantum well width. The validity of this approximate theory has been tested by comparing 6 its predictions with those from numerical self-consistent field calculations. Interactions among free 2 carriers,accountedforusingthelocal-spin-densityapproximation,substantiallyenhancethecritical temperature. We demonstrate that an external bias potential can tune the critical temperature ] through a wide range. l l a h - s e m The discovery of carrier-mediated ferromagnetism1–4 the exchange interaction energy J ≈ 0.15 eV nm3. In pd . in diluted magnetic semiconductors (DMS) has opened Eq.( 1) H is the Hamiltonian of the magnetic ions, H t m f a a broadand relativelyunexplored frontier for both basic isthefour-bandLuttingerHamiltonianforfreecarriersin m and applied research. The interplay between collective the valence band, S~ is the magnetic ionspin and~s is the - magnetic properties and semiconductor transport prop- electron-spinoperatorprojectedontothej =3/2valence d erties in these systems presents a rich phenomenology bandmanifoldoftheLuttingerHamiltonian. Weassume n o and offers the prospectof new functionality in electronic here that the 2D free carrier density is sufficiently low c devices. The possibilities for manipulating this interplay that only the lowest energy heavy-hole subband is occu- [ areespeciallyvariedwhenthefreecarriersystemisquasi- pied,andthatthe quantumwellofinterestissufficiently two-dimensional5–7 because ofthe dependence ofsystem narrow that mixing between light-hole and heavy-hole 1 v properties on the subband wavefunction. In this article bands can be neglected. These simplifications lead12 to 1 we address the dependence of the ferromagnetic critical a single parabolic band with the in-plane effective mass 9 temperature Tc of two-dimensional DMS ferromagnets m∗k ≈ 0.11m0 and the out-of-plane mass m∗z ≈ 0.38m0. 3 onthespatialdistributionofmagneticions,thesubband The twospin-statesinthis bandhavedefinite j =±3/2 1 wavefunction of the free carrier system, and on their in- z and definite s = ±1/2; the projection of the transverse 0 z terplay. Our analysis is based on a formulation of the spincomponentsontothisbandiszerosothatthekinetic 0 0 mean-fieldtheoryforcarrier-inducedferromagnetism8 in exchange interaction has an Ising character.13 / a DMS which was developed in earlier work9 and is in- Our mean-field theory9 is derived in a spin-density- at tended to be useful for any spatially inhomogeneoussys- functional theory framework and leads to a set of phys- m tem. We find that Tc is quite sensitive to the relative ically transparent coupled equations. In the absence of - distributions of electrons and magnetic ions and can be external fields, the mean polarization of a magnetic ion d altered in situ by applying a gate voltage. is given by n For definiteness we address the case of a h001i growth o direction 2D hole gas in the Ga1−xMnxAs DMS sys- hmiI =SBS JpdS n↑(R~I)−n↓(R~I) /2kBT ; (2) c tem. Much of our analysis,however,applies equally well (cid:0) (cid:2) (cid:3) (cid:1) : where B (x) is the Brillouin function, v to other interfaces and to other DMS systems with cu- S Xi bic host semiconductors and, with one important caveat 2S+1 2S+1 1 1 B (x)= coth( x)− coth( x) whichwementionbelow,alsotothecaseofa2Delectron S 2S 2S 2S 2S r a gasmediatedferromagnetism. Ourtheoryisbasedonan S+1 (S+1)(2S2+2S+1) ≈ x− x3, x≪1. (3) envelope function description of the valence band elec- 3S 90S3 trons and a spin representationof their kinetic-exchange interaction10 with d-electrons11 on the S = 5/2 Mn++ Theelectronspin-densitiesnσ(~r)aredeterminedbysolv- ing the Schrödinger equation for electrons which experi- ions: ence a spin-dependent potential due to kinetic exchange H=H +H +J S~ ·~s δ(~r −R~ ), (1) with the polarized magnetic ions: m f pd I i i I Xi,I 1 ∂2 ∂2 1 ∂2 − + − +v (~r) where i labels a free carrier,I labels a magnetic ion and (cid:20) 2m∗ ∂x2 ∂y2 2m∗∂z2 es k(cid:0) (cid:1) z 1 σ J , the free-carrier in-plane effective mass m∗, the sub- +vxc,σ(~r)− hpd(~r) ψk,σ(~r)=ǫk,σψk,σ(~r); (4) pd k 2 (cid:21) band wavefunction ψ(z), and the magnetic ion distribution function c(z). It is important to realize that the entire systembehaves collectively. It has a single critical n (~r)= f(ǫ )|ψ (~r)|2 . (5) σ k,σ k,σ temperature which depends on the function c(z), rather Xk thanaz-dependentT likethatwhichappearsinanaive c Intheseequationsves(~r)istheelectrostaticpotential,in- adaptationsofthebulk RKKYtheoryforTc toquasi-2D cluding band offset and ionized impurity contributions, systems. v (~r)isanexchange-correlationpotentialonwhichwe An explicit expressionfor the criticaltemperature can xc,σ comment further below, and f(ǫ) is the Fermi distribu- be obtained by linearizing the self-consistent equation tion function. The spin-dependent kinetic-exchange po- (2). ExpandingtheBrillouinfunctiontothefirstorderin tential, itsargumentgivesthehomogeneouslinearintegralequa- tion h (~r)=J δ(~r−R~ )hmi (6) pd pd I I 1 XI hmi(z)= dz′K(z,z′)hmi(z′) (10) kBT Z isnon-zeroonlyintheferromagneticstate. Inthefollow- ing we assume that the magnetic ions are randomly dis- where the kernel tributedanddenseonascalesetbythefreecarrierFermi S(S+1)J2 m∗ wavevector in the xˆ−yˆ plane, and that their density in K(z,z′)= pd k |ψ(z)|2c(z′)|ψ(z′)|2. (11) the growth direction, c(z), can be precisely controlled.7 12 π¯h2 This allowsus to take a continuumlimit where the mag- Since the linearizedequation is satisfied only at the crit- netic ion polarizationand the kinetic-exchangepotential ical temperature, k T is equal to the largest eigenvalue B c depends only on the growth direction coordinate and ofK(z,z′),theeigenvaluecorrespondingtoeigenfunction f(z)∝|ψ(z)|2. It follows that h (z)=J c(z) hmi(z). (7) pd pd In the following we discuss how the ferromagnetic tran- T = S(S+1)Jp2d m∗k dz|ψ(z)|4c(z). (12) sition temperature Tc depends on c(z). c 12 kB π¯h2 Z The T calculationis greatly simplified when the spin- c dependence of the exchange correlation potential is ne- ThecriticaltemperatureisproportionaltoJp2dandm∗k,is glected and the quantum well width w is small enough independent of 2D carrierdensity, and for a system with to make subband mixing due to kinetic exchange inter- uniformly distributed magnetic impurities is inversely actions negligible, i.e., when Jc(z) ≪ ¯h2/m∗w2. We see proportional to the quantum well width, as illustrated z later that the theory which results from these approxi- by the solid line in Fig. 1. The accuracy of Eq. (12) mations is normally quite accurate. With subband mix- has been tested by numerically solving the set of self- ing neglected, the only effect of the mean-field kinetic- consistent Eqs. (2)-(7) together with the Poisson equa- exchange interaction on the 2D carriers is to produce a tion for ves(z) and the local-spin-density-approximation rigid spin-splitting of the 2D bands, given by the first- (LSDA) formula14 for v (z). If exchange and cor- xc,σ order perturbation theory expression: relation effects are neglected (vxc,σ(z) = 0), the self- consistent numerical results represented by circles in ǫ −ǫ =J dz′hmi(z′)c(z′)|ψ(z′)|2 (8) Fig. 1 are identical to those obtained from Eq. (12) for ↓ ↑ pd Z quantumwellwidthsw =10,15,and20nm. Interactions among the 2D carriers substantially enhance the critical where ψ(z) is the growth direction envelope function of temperature and also cause a strong dependence of T the lowestsubband. Forthe caseofa quantumwellwith c on the 2D carrier density N, as illustrated by squares infinite barriers, ψ(z) = 2/wcos(πz/w). The electron (N =1×1011 cm2) and triangles (N = 0.5×1011 cm2) spin-polarizationcanthenpbe obtained by summing over in Fig. 1. Note that unlike the 3D case, where T is occupied free-carrier states. It follows that, as long as c proportional to Fermi wavevector in the non-interacting both spin-↑ and spin-↓ bands are partly occupied, limit8,9 and is an increasing function of density even if m∗ the interactions are included,9 the critical temperature n↑(z)−n↓(z)= 2π¯hk2|ψ(z)|2JpdZ dz′hmi(z′)c(z′)|ψ(z′)|2. fdoernsailtly,thNree=q0u.5an×tu1m011wcemlls2,ofwhFeigre. 1thiesilnatregrearctaiotnlsowareer (9) stronger.15 Eq. (9) can be used to eliminate the free-carrier spin- polarization from Eq. (2) and to obtain a self-consistent equation for the function hmi(z). The equation depends only on the kinetic-exchange coupling constant 2 ForexampleweshowinFig.2aplotofthedependenceof 125 thecriticaltemperatureonthefractiondofthequantum well occupied by magnetic ions. When the integrated d 2D density of magnetic ions (c d) is fixed, the critical 100 temperature increases as magnetic impurities are trans- w ferred toward the center of the quantum well where the 75 free-carrierwavefunctionhasits maximum. Forfixed3D K) analytic ( Hartree density c the number of magnetic impurities decreases T c 50 LLSSDDAA,, NN==10.×51×01101 1c1m cm−2−2 with decreasing d. This effect is stronger than the increase of the overlap between the carrier wavefunction and magnetic ion distribution resulting in the decrease 25 of T , as shown in the inset of Fig. 2. In both cases the c analytic results of Eq. (12) are in qualitative agreement 0 with the full numerical LSDA calculations. 10.0 12.5 15.0 17.5 20.0 w(nm) 35 FIG. 1. Ferromagnetic critical temperature of a quasi-2D 15 DMSasafunctionofquantumwellwidthwforfixed3Dmag- 30 neticion densityc=1nm−3;inset: schematic diagram of the K) 10 quantumwellwiththeshadedportionofwidthddopedwith 25 T (c 5 magnetic ions. In this figure d = w. The non-interacting 0 free carrier results were calculated both by solving the full ) 20 0 10 20 K self-consistent equations (circles) and by using the approxi- ( d (nm) mateTc expressionderivedinthetext(solidline),confirming T c15 the accuracy of the latter approach. Squares and triangles d 10 represent the full numerical LSDA calculations for 2D carrier densities N = 1×1011 cm2 and N = 0.5×1011 cm2, respectively. 5 w analytic LSDA, N=1011 cm−2 0 Before discussing the implications of the expression 0 5 10 15 20 (12) for T we comment on its relation to the RKKY c d (nm) theory8 of carrier induced ferromagnetismin DMS’s. As FIG. 2. Dependence of Tc on the central portion d of applied to bulk 3D electron systems the two approaches the quantum well (w = 20 nm) occupied by magnetic are fundamentally equivalent, although there are differ- ions. The main graph is for fixed 2D magnetic ion density, ences in detail which can be important. The principle c d = 20 nm−2, while the inset is for the case of fixed 3D differenceisthattheRKKYtheorytreatsthefree-carrier magnetic ion density,c=1 nm−3. magnetic-ioninteractionperturbatively. Asaresultboth approaches give the same value for the critical temper- One remarkable feature of quasi-2D ferromagnetic ature where the magnetization vanishes; differences ap- DMS’sisthepossibilityoftuningT throughawiderange c pear at lower temperatures where the electron system is insitu, by the applicationofa gate voltage. InFig. 3 we stronglyor completely spin-polarizedanda perturbative illustrate this for the case of a w =10 nm quantum well treatmentofitsinteractionswiththemagneticionsfails. withmagneticionscoveringonlyad=3nmportionnear Both approaches fail to account for the retarded charac- oneedge. Evenforsuchanarrowquantumwell,thecriti- terofthefree-carriermediatedinteractionbetweenmag- caltemperaturecanbevariedoveranorderofmagnitude neticionsintheselowdensitysystems,andforcorrelated byapplyingabiasvoltagewhichdrawselectronsintothe quantum fluctuations in magnetic ion and free carrier magnetic ion region. subsystems which are important for some properties.16 InclosingweremarkthatforseveralreasonsDMSfer- Both approaches are able to account for interaction ef- romagnetism mediated by 2D electron systems is likely fects in the free carrier system, which increase the ten- to occur only at inaccessibly low temperatures. One im- dencytowardferromagnetismasillustratedinFig.1,and portantfactor is the smaller value of the exchange inter- in the density-functional approach appear in the spin- action parameter in the conduction band case. For ex- dependence of the exchange correlation potential. It is ample for the typical n-type II-VI DMS quantum well3, in inhomogeneous cases, such as the reduced dimension (Zn1−xCdxSe)m−f(MnSe)f,7 the s−d exchange Jsd ∼ situationconsideredhere,thatthedensityfunctionalap- 4×10−3 eV nm3. Our theorythen givesT ∼10 mK for c proach has an advantage. Our approach provides a sim- a 10 nm wide quantum well. Equally important, how- ple description of the collective behavior of the quasi- ever, is the weak magnetic anisotropy expected in the 2D ferromagnetand is able to handle geometric features conduction band case, which will lead to soft spin-wave which can be engineered to produce desired properties. 3 collective modes17 and substantial Tc suppression. ductors, edited by Michel Averous and Minko Balkanski This work was supported by the National Science (Plenum, New York, 1991); Tomasz Dietl in Handbook on Foundation under grants DMR-9714055 and DGE- Semiconductors, Chap. 17 (Elsevier, Amsterdam, 1994). 9902579,andbytheGrantAgencyoftheCzechRepublic 11Weassumefreecarrierscreeningmakesthecentralcellcor- under grant 202/98/0085. rectionsrequired for isolated Mn acceptors inGaAs unim- portant. See A.K. Bhattacharjee and C. Benoit á la Guil- laume,SolidStateCommun.113,17(2000)andworkcited therein. 60 Fg 12See for example Weng W. Chow, Stephan W. Koch, and Murray Sargent III, Semiconductor Laser Physics, 50 (Springer-Verlag, Berlin, 1994). 13More generally there is a delicate interplay between size- 40 quantization controlled heavy-light subband mixing and ) kinetic-exchange interaction anisotropy which can be ac- K T(c30 Hartree, N=1011 cm−2 craotuenlytekdnfoowrnif.thegeometry of a particular sample is accu- 20 LSDA, N=1011 cm−2 14S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58, 1200 (1980). 10 15The LSDA form for the exchange-correlation potential would diverge in the limit of infinitely narrow 2D sys- 0 tem. For the 10 nm wide quantum we expect the LSDA 0 5 10 15 to overestimate the effect of interactions among carriers F (meV/nm) and, hence, to overestimate the critical temperature. Ac- g curate theories of valence-band exchange and correlation FIG. 3. Dependence of Tc on bias voltage applied across effects are an important challenge for theory. the w = 10 nm wide quantum well partially occupied by 16J. Koenig, Hsiu-hau Lin and A.H. MacDonald, cond- magnetic ions over a distance of 3 nm near one edge of the mat/0001320, submitted to Phys. Rev.Lett. (2000). quantum well. Circles (squares) represent results of the full 17T. Jungwirth, J. König, and A.H. MacDonald, unpub- numerical self-consistent calculations without (with) the ex- lished. change-correlation potential. 1T.Story,R.R.Galazka,R.B.Frankel,andP.A.Wolff,Phys. Rev.Lett. 56, 777 (1986). 2H.Ohno,H.Munekata,T.Penney,S.vonMolnarandL.L. Chang, Phys. Rev.Lett. 68, 2664 (1992). 3H. Ohno, A. Shen, F. Matsukura, A. Oiwa, A. Endo, S. Katsumoto and Y. Iye,Appl.Phys. Lett. 69, 363 (1996). 4For a thorough recent review see Hideo Ohno, J. Magn. Magn. Mater. 200, 110 (1999). 5P.Lazarczyk,T.Story,M.Arciszewska,andR.R.Galazka, J. Magn. Magn. Mater. 169, 151 (1997) and references therein. 6S.A. Crooker, D.A. Tulchinsky, J. Levy, D.D. Awschalom, R. Garcia, and Samarth, Phys. Rev. Lett. 75, 505 (1995); S.A. Crooker, D.D. Awschalom, J.J. Baumberg, F. Flack, and N. Samarth, Phys.Rev.B 56, 7574 (1997). 7I.P. Smorchkova, N. Samarth, J.M. Kikkawa, and D.D. Awschalom, Phys. Rev.Lett. 78, 3571 (1997). 8T.Dietl,A.Haury,andY.Merled’Aubigne,Phys.Rev.B 55, R3345 (1997). 9T. Jungwirth, W.A. Atkinson, B.H. Lee, and A.H. Mac- Donald, Phys. Rev.B 59, 9818 (1999). 10J.K. Furdyna, J. Appl. Phys. 64, R29 (1988); Semi- magnetic Semiconductors and Diluted Magnetic Semicon- 4