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Phase Transition of Diluted Magnetic Semiconductor
M. K. Li, S. J. Lee, S. U. Yuldashev, G. Ihm, and T. W. Kang
Citation: AIP Conference Proceedings 1399, 717 (2011);
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Published by the American Institute of Physics
Phase Transition of Diluted Magnetic Semiconductor
M. K. Li1, S. J. Lee*1, S. U. Yuldashev1, G. Ihm2, T. W. Kang1
1. Quantum-functional Semiconductor Research Center, Dongguk Univ-Seoul, Seoul 100-715, South Korea
2. Dept. of Physics, Chungnam National Univ., Daejeon 305-764, South Korea
Abstract. Three types of phase transitions in diluted magnetic semiconductor, first-order, second-order and mixedorder, are found in theory. Especially the mixed type transition shows two steps transition and novel specific heat
property. Specific heat properties disclose a possible meta ferromagnetic phase confirmed by the experimental
qualitative result.
Keywords: Diluted Magnetic Semiconductor, Specific Heat, GaMnAs, Phase Transition
PACS: 75.40.Mg
is the ions concentration. The free energy of the hole
The discovery of carrier mediated ferromagnetism
in diluted magnetic semiconductor (DMS) opens a
new way to develop a kind of new devices combining
information processing and storage functionalities
simultaneously.[1-4] There have been several
theoretical approaches to the study of the
ferromagnetic mechanism, Tc, and magnetic
moment.[4,5] However, few works have been
investigated on the study of specific heat (CV) of the
DMS. [6,7] Multi-step magnetic phase transition has
been observed in CeRh3Si2.[9] This kind of phase
transition also exists in DMS.
3 e2
(3π 2 )1/3 n 4/3 [(1 + ξ ) 4/3 + (1 − ξ ) 4/3 ]
8π ε0
After putting all parameters for GaMnAs, which
are m=0.5m0, J=0.15 eV nm3, S=5/2, 0=12.2, the
dependent of Tc with the Mn and holes concentration
is shown in Fig. 1. The low Mn (holes) concentrations
lead to low Tc due to only partial holes (Mn)
participating in the interaction. Tc is independent on
the phase transition type, especially for the first (Green
region in Fig. 1) and second (White region in Fig. 1)
order transition. The mixed-order phase transition is
shown in up panel of Fig. 2 (dotted line). At this case,
ξ jumps from a high spin polarization state to a low
spin polarization state at the temperature slightly lower
∂2 F
∂T 2
sinh[h( S + 1 / 2) / k B T ]
Here Fion = − k B Tc log
sinh[h / 2k B T ]
where h = Jnξ / 2 , the J is the hole-ion exchange
ξ = (n − n ) / n
where m is the
where 0 is the dielectric constant of DMS. For any
certain temperature, the free energy is the function of
ξ. The ξ is determined by minimizing the free energy
with respect to ξ. And, the free energy is obtained.
specific heat can be derived from CV = −T
ε F↑↓ = (6π 2 n↑↓ ) 2/3 = 2 / 2m ,
Fxc = −
In this paper, we investigate the phase transition
and the specific heat of DMS without external
magnetic field. To obtain the specific heat of DMS, we
write the free energy F=Fion +Fhole +Fxc . Then, the
hole effective mass. We apply the Hartree-Fock
approximation [9,10]:
coupling constant, n
3 ↑↓
5π 2 k B T 2 π 4 k B T 4
= ∑ nε F [1 −
( ) + ( ↑↓ ) ]
12 ε F↑↓
48 ε F
↑↓ 5
is the holes concentration,
is the spin polarization of holes, c
S. J. Lee, e-mail:, Tel: +82-2-2260-3953, Fax: +82-2-2260-3945
Physics of Semiconductors
AIP Conf. Proc. 1399, 717-718 (2011); doi: 10.1063/1.3666578
© 2011 American Institute of Physics 978-0-7354-1002-2/$30.00
when the ξ changes from a high polarized state to a
partially polarized state, the CV jumps to a high level.
The second step transition occurs at Tc, where CV falls
discontinuously to the specific heat of hole gas. We
compare a GaMnAs sample, shown in the inset of Fig.
2, grown by MBE on GaAs substrate with 1.6% Mn
concentration and 2.7×1019cm-3 of holes concentration
measured at room temperature. CV of this sample
shows the similar behavior of mixed type transition.
By fitting the curve, c=0.195nm-3 and kF=0.59nm-1 are
obtained. It means that only 55% Mn participates into
the ferromagnetic interaction. Also small holes
concentration from fitting is expected due to low
temperature compensation. The experiment result
qualitatively confirms the two-steps phase transition.
FIGURE 1. The Curie temperature (K) (broken line) and
three types of phase transition, first (Green), second (White)
and mixed (Yellow) order transitions are mapped with the
Mn (c) and holes (kF) concentration. The three typical
parameters for these transitions to exhibit phase transition
properties are marked as A (c=0.15 nm-3, kF=0.4 nm-1), B
(c=0.15 nm-3, kF=0.6 nm-1), and C (c=0.3 nm-3, kF=0.6 nm-1).
GaMnAs shows three types of phase transition.
The mixed-order transition shows two steps transition.
The first step transition shows a novel spike-like peak
of specific heat, qualitatively confirmed by the
experiment result.
than Tc and then continuously changes to 0 at Tc.[11]
There are two phase transition points. The c and kF
dependences of the mixed-order transition are shown
in Fig. 1 (Yellow region). The region of mixed-order
transition adjoins the region of the first-order transition
and they are not overlapped. It implicates that the
mixed-order transition cannot be concluded into the
first-order transition. It should be a meta ferromagnetic
phase.[8] With increasing Mn or holes concentration,
the abrupt change of ξ decreases gradually to 0 at the
boundary of white and yellow regions.
The CV of the first and second order transition,
shown in the bottom panel of Fig. 2, are similar
although CV for A is only one tenth of CV for C due to
small Mn-hole interaction of low Mn and hole
This work was supported by the Korea Science and
Engineering Foundation (KOSEF) through the
Quantum Functional Semiconductor Research Center
(QSRC), Dongguk University and by the research
funds of Dongguk University.
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FIGURE 2. The spin polarization (ξ) of the hole gas and
specific heat (CV) depend on the reduced temperature (TTc)/Tc. Inset shows CV of a MBE grown GaMnAs
sample of a mixed-order transition (Black square) and
theory fitting (Red line).
concentration of A. However, CV of the mixed-order
transition shows two steps phase transition. At first,
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