Eur. Phys. J. B 49, 403–406 (2006)
DOI: 10.1140/epjb/e2006-00088-x
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Softening of spin-wave stiffness near the ferromagnetic phase
transition in diluted magnetic semiconductors
Shih-Jye Sun1,a and Hsiu-Hau Lin2
1 Department of Applied Physics, National University of Kaohsiung, Kaohsiung 811, Taiwan, ROC 2 Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan
Physics Division, National Center for Theoretical Science, Hsinchu 300, Taiwan Received 1st November 2005
Published online 31 March 2006 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2006 Abstract. Employing the self-consistent Green’s function approach, we studied the temperature depen-dence of the spin-wave stiffness in diluted magnetic semiconductors. Note that the Green’s function ap-proach includes the spatial and temperature fluctuations simultaneously which was not possible within conventional Weiss mean-field theory. It is rather interesting that we found the stiffness becomes dramati-cally softened as the critical temperature is approached, which seems to explain the mysterious sharp drop of magnetization curves in samples within diffusive regime.
PACS. 75.30.Ds Spin waves – 75.40.Gb Dynamic properties (dynamic susceptibility, spin waves, spin diffusion, dynamic scaling, etc.) – 75.50.Dd Nonmetallic ferromagnetic materials
1 introduction
Spintronics [1] brings out an industrial renaissance in the past decades for its powerful usage of the extra spin degrees of freedom in many electronic applications. For instances, giant magneto-resistance (GMR) had been ap-plied to read-write head of computer hard disk and tunnel-ing magneto-resistance (TMR) to the newest nonvolatile memory for MRAM. The huge success kicked off the in-tense investigations on possible realizations of similar de-vices in semiconducting materials, which can be directly integrated with the existing industrial techniques. The idea has enjoyed its primary success in the so-called di-luted magnetic semiconductors (DMS), where the mag-netic ions are doped into the host semiconductors and lead to a ferromagnetic phase.
The key issue at current stage is how to raise the criti-cal temperature so that the ferromagnetic order is robust even at room temperature. Taking the well-known ma-terial (Ga, Mn)As as example [2–4], it was demonstrated that the critical temperature can be raised significantly by thermal annealing. However, the highest critical temper-ature at the time of writing is around 160 K [5], which is still far from the goal for room-temperature DMS. Based on Zener model, Dietl et al. proposed to look for room-temperature DMS in wide bandgap semiconductors and oxides such as GaN, ZnO, TiO2 [6–8]. While the critical temperatures in these materials are typically higher com-pared with (Ga, Mn)As, clustering seems to be a serious
a e-mail: sjs@nuk.edu.tw
problem which prevents their potential usage in realistic devices.
While the issue of raising up the critical temperature seems to lie in experimentalists’ hands, we believe a better understanding of the ferromagnetic phase would also help. After intense theoretical investigations, the origin of ferro-magnetism in DMS is believed to be carrier mediated [9]. Typically, the doped transition metal ions provide the lo-calized impurity spins with small direct exchange (often antiferromagnetic) among themselves. Furthermore, the experimental results show that only 10 ∼ 30% of the doped magnetic ions contribute itinerant carriers [10] into the host semiconducting bands. Through double-exchange mechanism, the itinerant carriers mediated the indirect exchange interactions between the localized impurity spins and the ferromagnetic order sets in when the system is cooled below the critical temperature.
While the origin of the ferromagnetic phase is more or less clear, some of its physical properties remain puzzling. To achieve a full understanding, it is crucial to include the exchange coupling between itinerant and localized spin densities, thermal fluctuations, the random locations of the doped ions, the realistic band structure and the re-pulsive interaction between itinerant carriers. Since it is almost impossible to incorporate all effects in single for-malism, one needs to glue up piecewise information from different approaches.
In this paper we employ the self-consistent Green’s function method [11], which includes the spatial and ther-mal fluctuations simultaneously, to study the temperature
404 The European Physical Journal B dependence of spin-wave stiffness in DMS. Since our goal
is to demonstrate the peculiar temperature dependence, we simplify the realistic band structure by the single-band approximation, at price of sacrificing the quantitative de-scription for realistic materials. However, within single-band approximation, we were able to establish the close tie between the softening of stiffness and the sudden drop in magnetization curves [2]. Moreover, our self-consistent Green’s function approach also shows the appearance of concave magnetization curve [12] in the regime where itin-erant carriers are dilute. This implies a smooth crossover from the diffusive regime to the localized regime with strong disorder. Since previous studies including the full six bands only renormalize various physical parameters and twist the phase boundaries [13], we expect our results to be qualitatively robust.
The organization of the paper is the following: in the Section 2 of this paper, we derive the formalism for the self-consistent Green’s function approach. In Section 3, we show our numerical results and discuss connections to other approaches in the literature.
2 Theory
The sd model is proper to describe the DMS systems in-cluding a strong exchange interaction between local spins, which come from electrons in d orbits of transition atom, and itinerant spins around whole system, which come from impurity doping donation. The Hamiltonian of the sd model can be expressed by
H = H0+ J
d3rS(r) · σ(r), (1)
the first term H0 is the kinetic energy of the itinerant carriers and the second term is the exchange interaction between itinerant carrier spins and the localized spin mo-ments, where the spin density of the localized moments is
S(r) =Iδ3(r − RI)SI and the itinerant spin density is
σ(r) = ψ†(r)(τ /2)ψ(r). In momentum representation the Hamiltonian is expressed as H = Hk+ HJ = k,σ εkc†k,σck,σ− cJ2 k Sk+σ−k −cJ 2 k Sk−σ+k − cJ k Skzσz−k, (2)
where the constant c is the density of the magnetic ions in DMS and J is the magnetic coupling integral in the unit of eV nm3. Since the RKKY interaction dominates the magnetism in DMS, to investigate the magnon dis-persion is necessary for studying in microscopic. In order to obtain the magnon dispersion of local spin in DMS we have to calculate the retarded local spin Green’s function defined by
Gi,j(t) = Si+(t); Sj−(0)y
=−iθ(t)[Si+(t), Sj−(0)], (3)
where θ(t) is a step function of time t, · · · represent-ing the expectation value and [· · · ] is the commutator. It is more convenient to derive this Green’s function in momentum space. Through the Fourier transformation to obtain the spin Green’s function in momentum represen-tation, we get
G(q, t) = Sq+(t); S−(0, 0). (4)
Employing the equation of motion to equation (4) makes the spin Green’s function calculation reduced to
id
dtG(q, t) = ϕ + [S
+
q (t), H]; S−(0, 0), (5)
where function ϕ comes from the derivative of time for step function. Following commutation rules in RPA (ran-dom phase approximation) are we used in this paper, there are [Sq+, Sk−] = 2 N ei(q−k)·RSz = 2Szδq,k, (6) [Sq+, Skz] =− 1 NS + q+k, (7) [σk+, σk−] = p (c†p+k,↑cp−k,↑− c† p+k+k,↓cp,↓) (8) and [σ+k, σzk] =−1 Nσ + k+k, (9)
where Sz is the expectation value of magnetization for local spin. Employing these commutation rules to equa-tion (5) results to the equaequa-tion
ωG(q, ω) = ϕ − cJSzσq+; S−(0, 0)
+cJσzS+q ; S−(0, 0), (10)
where σz is the expectation value of the magnetization for itinerant carriers. We found the derivation process for local spin Green’s function resulting to another new Green’s function ζ(k, ω) = σ+k; S−(0, 0) in the mean-while. It is clearly the new resulted Green’s function re-veals the physics that the exchange interaction between local spins needs the itinerant carriers’ mediation. There-fore we need to calculate the new Green’s function by the same way,
id dtσ
+
q ; S−(0, 0) = [σ+q, H]; S−(0, 0). (11)
The commutator in equation (11) results to [σq+, Hk] = p (εp− εp+q)c†p+q,↑cp,↓ (12) [σq+, HJ] =−cJ 2 k [σq+, σk−]Sk+− cJ k [σ+q, σzk]Skz, (13)
Shih-Jye Sun and Hsiu-Hau Lin: Softening of spin-wave stiffness near the ferromagnetic phase transition in diluted... 405 where εkis the electrons kinetic energy with momentum k.
Apply the Fourier transformation and RPA for carriers density to the new Green’s function ζ resulting to
ωζ(q, ω) = p (εp− εp+q)c†p+q,↑cp,↓; S−(0, 0) −cJ 2 p (c†p+q,↑cp+q,↑ − c†p,↓cp,↓) ×G(q, ω) + cJSzζ(q, ω). (14)
Abstracting momentum p from ζ(q, ω) in equation (14) results to a relation c†p+q,↑cp,↓; S−(0, 0) = cJ 2 c† p,↓cp,↓ − c†p+q,↑cp+q,↑ ω − εp+ εp+q− cJSz ×G(q, ω). (15)
Combining equations (10) and (14) results to a closed form of Green’s function equation G(q, ω),
(ω − cJσz + cJ 2 2 S z p c†p,↓cp,↓ − c†p+q,↑cp+q,↑ ω − εp+ εp+q− cJSz ) ×G(q, ω) = ϕ, (16)
where the c†p,σcp,σ = fp,σ = (βεp,σ+ 1)−1 is the
car-rier density for spin σ and β = 1/KBT . The poles in
equation (16) represent magnon excitations. In the diluted limitation, the minority is the itinerant carriers which could be seen as a free carrier gas with effective mass
m∗= 0.5me[9], where meis the free electron mass, and in
the ferromagnetic state the majority of local magnetic ions produce an effective magnetic field causing both spins to split with Zeeman energy cJSz. The kinetic energies of different spins are εk,σ=2k2/2m∗∓ σ1/2cJSz
respec-tively. Therefore the magnon excitation energy for each momentum q is ωq =Jσz −cJ 2 8π2S z × ∞ 0 km∗fk↑ 2q × ln ωq− cJSz −2m2q∗2 − 2kq m∗ ωq− cJSz −2m2q∗2 + 2kq m∗ dk + ∞ 0 km∗fk↓ 2q ln ωq− cJSz + 2q2 2m∗ − 2kq m∗ ωq− cJSz +2m2q∗2 + 2kq m∗ × dk . (17)
Finally, we could utilize the Callen’s arbitrary spin for-mula,
Sz =
[S − Φ(S)][1 + Φ(S)]2S+1+ [S + 1 + Φ(S)][Φ(S)]2S+1 [1 + Φ(S)]2S+1− [Φ(S)]2S+1
(18) to obtain the magnetization values, where Φ(S) =
1
N
q(exp(βωq)− 1)−1 is the magnon number.
Fig. 1. We take the exchange coupling and the effective mass are fixed at typical values J = 0.15 eV nm3, m∗ = 0.5me
and the ratio of itinerant and localized spin densities fixed at
c∗/c = 0.1 to calculation resulting to T
C= 45 K. In the insert
of the figure shows a sharp drop of magnetization in the vicinity ofTC. In the sharp drop region the magnon shows a softening effect.
3 Results and discussions
The anomalous temperature dependence of magnetization of sharp drop in the vicinity of TC represents in our
the-oretical result as shown in the insert of Figure 1. is con-sistent with experimental result implying an important interaction existing in the magnetization collapse region. According to the RKKY mechanism the magnetism is es-tablished by itinerant carriers mediation. The sharp drop of magnetization reveals a possibility of carrier-magnon decoupling in the vicinity of TC. This decoupling effect
reduces the effective magnetic interaction and the magne-tization disappears in decoupling completely.
The dispersion of spin wave derived from the conven-tional spin wave theories [14] is temperature independent, which is an intrinsic characteristic for many kinds of mag-nets. The Figure 1 exhibits the dispersion of magnon from our theoretical calculation showing a temperature inde-pendent dispersion at far from TCs, which reveals a result
hat the temperature independence of normal spin wave existing at robust magnetism region, meanwhile it reveals an obvious magnon softening effect in the vicinity of TC
leading to the magnetization falling down sharply. Inter-estingly, this softening effect starts from small magnon momentum qs then extending to whole dispersion region eventually. From the conventional spin wave theory as the
q , the magnon dispersion relation has ω(q) = Dq2, where the stiffness constant D ∝ J and J is the mag-netic coupling integral between two separated spins. From the linear response theory we have derived [15] before, the coupling J is∝J2, where the J is the coupling between itinerant spins and local spins. Therefore this softening ef-fect results to D decreasing, which reveals a fact that the effective coupling J reduces and gives an implication with carrier-magnon decouple in system.
406 The European Physical Journal B In the conclusion, by our theoretical study the
anoma-lous magnetization sharp drop in the vicinity of TCcomes
from the magnon softening effect, and this softening effect possibly comes from the carrier-magnon decoupling. We thank the support of National Center of Theoretical Sci-ence of Taiwan.
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