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On the possibility of second order phase transition with nonvanishing order parameter
Volume 144, number 8,9
PHYSICS LETFERS A
19 March 1990
ON THE POSSIBILITY OF SECOND ORDER PHASE TRANSITION WITH NONVANISHING ORDER PARAMETER E.L. NAGAEV and A.!. PODEL’SHCHIKOV NPO “KVANT”, Moscow 129626, USSR Received 15 December 1989; accepted for publication 11 January 1990 Communicated by V.M. Agranovich
In systems with temperature dependent nonuniform charge distribution a second order phase transition may occur with a nonzero order parameter both above and below the transition point.
The Landau theory of second order phase transitions describes a situation when the disordered state continuously transforms into an ordered one with lowering ofthe system symmetry. The order parameter ~j in the ordered state is the measure of symmetry lowering, being equal to zero in the disordered phase at all temperatures. The fact that the symmetry group of the ordered phase is a subgroup of that of the disordered phase leads to the absence of the term linear in ‘i in the free energy expansion in powers of The aim of this short note is to draw attention to the possibility of continuous phase transitions without system symmetry lowering, i.e. transitions occurring without destroying the ordered state of the system. Correspondingly the meaning of the order parameter for such transitions basically differs from that for standard order—disorder phase transitions. Thus, the singularities of thermodynamic quantities for the phase transitions under consideration should be quite different from those in the Landau theory. In particular such phase transitions may be realized in systems with nonuniform charge distribution if that is temperature dependent. The degenerate antiferromagnetic semiconductors may be pointed out as an example. According to refs. [1,2] at T= 0 they may spontaneously get into the state in which the crystal is divided into ferromagnetic and antiferromagnetic parts with all the conduction electrons concentrated in the former (thus, the antiferromagnetic part is insulating). If the conduction electron density n is not very high, the ferromagnetic part consists of spherical droplets ofradii 10—100 A inside the antiferromagnetic host. The droplets are isolated from each other and should form a periodical structure when neglecting the random impurity potential (fig. 1 a). But, with increasing electron density, the ferromagnetic part of the crystal goes over from the multiply-connected state into the singly-connected one with antiferromagnetic droplets inside it (fig. lb). With the structure presented in fig. la the crystal as a whole is insulating and with the structure of fig. lb it ~.
(a)
______________________
(b)
______________________
Fig. 1. Two-phase states of degenerate antiferromagnetic semiconductor: (a) insulating; (b) conducting. The ferromagnetic part of the crystal is shaded, the antiferromagnetic part is not. 0375-9601/90/s 03.50 © Elsevier Science Publishers B.V. (North-Holland)
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Volume 144, number 8,9
PHYSICS LETTERS A
19 March 1990
is highly conducting. Experimentally such nonuniform ferro—antiferromagnetic states were observed in EuSe and EuTe (see an analysis of the experiment in ref. [21). As follows from a detailed calculation, the portion of the ferromagnetic phase in the insulating state (fig. 1 a) increases with temperature. For this reason at some T~ferromagnetic regions begin to make contact with each other, and electron percolation occurs, if at T=0 it was absent. The hypothesis seems quite plausible, that the percolation temperature T~,at which the topology of the conducting part of the crystal changes, is simultaneously the second order phase transition point. Singularities of thermodynamical quantities at T~may be connected, in particular, with the difference in the temperature dependence of the boundary area between antiferromagnetic and ferromagnetic phases at T> T~and T< T~.This difference should lead to a singularity of the surface energy. If so, at T~a second order phase transition should occur from the insulating to the conducting state. One could call the transition topological (one should not confuse such a topological transition in real space at finite temperatures with the well known Lifshitz topological transitions in the impulse space at T=0). A specific feature of the phase transition under consideration is the fact that the symmetry of the state at T< T~is not lower than at T> T~.This allows a linear term in ~j in the expansion of the free energy F. The role of the order parameter may be put as ‘1= ( VAFM/
VFM)T—
( VAFM/ VFM)T~,
where VAFM and VFM are the volumes of the antiferromagnetic and ferromagnetic phases respectively. Obviously, j is positive at T< T~and negative at T> T~.One can write a simple phenomenologicalexpression for F using only the assumption that the percolation temperature is singled out, i.e. the specific heat C is singular at T~.Then unlike the Landau expansion, in this case the corresponding expansion for the free energy has the form (1) F=a~+bi~4, where a_—ce(T—T~),b(T~)>0.The quadratic term is absent in (1) since otherwise the point T~would not be singled out. The cubic term is excluded in order to ensure real values of ~ at T both higher and lower than T~. It follows from (1), that C—~ T— T~I—2/3, i.e. unlike the Landau theory where the specific heat displays a jump, in the present theory it diverges at T~(similarity to the scaling theory result is only apparent). In any case, the expression (1) for F supports the possibility of second order phase transition without the order parameter vanishing both below and above T~. Unfortunately the accuracy of the calculation based on the microscopic model [1,2] is unsufficient to confirm eq. (1) or even the fact of second order phase transition at the percolation point. But this calculation makes it possible to prove that with increasing temperature the topology of fig. 1 a should go over into the topology of fig. lb if the parameters are favourable. Thus, the electron percolation may be caused by the increase of temperature. The calculation was carried out within the framework of the s-f-model using a variational principle for the free energy of the system which generalized the variational principle used in refs. [1,21 at T= 0 for finite temperatures. The generalization consists in accounting for the fact that the magnetization is partially destroyed in regions where conduction electrons are concentrated. Correspondingly in insulating regions the antiferromagnetic ordering is completely or partially destroyed, too. The geometry of nonuniform states is represented by figs. la, lb. The ratio x= VAFM/ VFM and the radius R of spherical droplets of another phase inside the host are chosen as variational parameters. Since the conduction electrons are degenerate the contribution of their thermal excitations to their free energy can be neglected. For these reasons the expressions from refs. [1,2] can be used for the electron energy including both bulk and surface parts E~and E 5, as well as for the Coulomb energy EQ. The density of the magnetic subsystem free energy FM is found in the mean field approximation. It is assumed that the undoped 474
Volume 144, number 8,9
PHYSICS LETTERS A
19 March 1990
crystal is antiferromagnetic with exchange interaction only between the first nearest neighbours and that the conduction electrons inside the ferromagnetic part of the crystal are completely spin polarized. Then FM is given by the sum of contributions from antiferro- and ferromagnetic regions respectively, FM= .j~~FM(0~ T)+
.~~LFM(~vOn(l+x)A, T),
I s H2 voFM(H,T)=——-——+~IJoIS~—Tln( ~
exp(mIJoISi/T)), H~2IJ 0IS1,
4l~1oI
\m=—S
J
Is
=—iIJoIS~—Tlfl(\ ~ exP[(m/T)(H_IJoIS2)])~ H~2IJ0IS1, m= —s where H means the effective magnetic field in the corresponding part of the crystal; A is the s-f-exchange integral, v0 is the volume of the unit cell, S is the magnitude of the f-spins, J0 is the exchange integral in the undoped crystal. The quantities S~and S2 obey the following self-consistency equations:
S1=SB5(SIJ0IS1/T), S2=SB5[S(H—1J01S2)/T], where B~is the Brillouin function. The stationary state of the system is determined from the condition of minimum total free energy of the system,
F=Ey+Es+EQ+FM with respect to x and R. The numerical calculation was carried out at the same values of parameters as in refs. [1,21 (they correspond to the rare earth of EuTe It was obtained that,lb) for occurs example, 3 the transition from thecompounds insulating state (fig.or 1EuSe a) totype). the conducting one (fig. at at n =2.51020 cm— values of the variational parameters at the transition point x= 1, R = 31.4 A. T~= K with
References [1] E.L. Nagaev, Pis’ma Zh. Eksp. Teor. Fiz. 16 (1972) 558; V.A. Kashin and E.L. Nagaev, Zh. Eksp. Teor. Fiz. 66 (1974) 2105. [2] E.L. Nagaev, Physics ofmagnetic semiconductors (Mir, Moscow, 1983).
475
PHYSICS LETFERS A
19 March 1990
ON THE POSSIBILITY OF SECOND ORDER PHASE TRANSITION WITH NONVANISHING ORDER PARAMETER E.L. NAGAEV and A.!. PODEL’SHCHIKOV NPO “KVANT”, Moscow 129626, USSR Received 15 December 1989; accepted for publication 11 January 1990 Communicated by V.M. Agranovich
In systems with temperature dependent nonuniform charge distribution a second order phase transition may occur with a nonzero order parameter both above and below the transition point.
The Landau theory of second order phase transitions describes a situation when the disordered state continuously transforms into an ordered one with lowering ofthe system symmetry. The order parameter ~j in the ordered state is the measure of symmetry lowering, being equal to zero in the disordered phase at all temperatures. The fact that the symmetry group of the ordered phase is a subgroup of that of the disordered phase leads to the absence of the term linear in ‘i in the free energy expansion in powers of The aim of this short note is to draw attention to the possibility of continuous phase transitions without system symmetry lowering, i.e. transitions occurring without destroying the ordered state of the system. Correspondingly the meaning of the order parameter for such transitions basically differs from that for standard order—disorder phase transitions. Thus, the singularities of thermodynamic quantities for the phase transitions under consideration should be quite different from those in the Landau theory. In particular such phase transitions may be realized in systems with nonuniform charge distribution if that is temperature dependent. The degenerate antiferromagnetic semiconductors may be pointed out as an example. According to refs. [1,2] at T= 0 they may spontaneously get into the state in which the crystal is divided into ferromagnetic and antiferromagnetic parts with all the conduction electrons concentrated in the former (thus, the antiferromagnetic part is insulating). If the conduction electron density n is not very high, the ferromagnetic part consists of spherical droplets ofradii 10—100 A inside the antiferromagnetic host. The droplets are isolated from each other and should form a periodical structure when neglecting the random impurity potential (fig. 1 a). But, with increasing electron density, the ferromagnetic part of the crystal goes over from the multiply-connected state into the singly-connected one with antiferromagnetic droplets inside it (fig. lb). With the structure presented in fig. la the crystal as a whole is insulating and with the structure of fig. lb it ~.
(a)
______________________
(b)
______________________
Fig. 1. Two-phase states of degenerate antiferromagnetic semiconductor: (a) insulating; (b) conducting. The ferromagnetic part of the crystal is shaded, the antiferromagnetic part is not. 0375-9601/90/s 03.50 © Elsevier Science Publishers B.V. (North-Holland)
473
Volume 144, number 8,9
PHYSICS LETTERS A
19 March 1990
is highly conducting. Experimentally such nonuniform ferro—antiferromagnetic states were observed in EuSe and EuTe (see an analysis of the experiment in ref. [21). As follows from a detailed calculation, the portion of the ferromagnetic phase in the insulating state (fig. 1 a) increases with temperature. For this reason at some T~ferromagnetic regions begin to make contact with each other, and electron percolation occurs, if at T=0 it was absent. The hypothesis seems quite plausible, that the percolation temperature T~,at which the topology of the conducting part of the crystal changes, is simultaneously the second order phase transition point. Singularities of thermodynamical quantities at T~may be connected, in particular, with the difference in the temperature dependence of the boundary area between antiferromagnetic and ferromagnetic phases at T> T~and T< T~.This difference should lead to a singularity of the surface energy. If so, at T~a second order phase transition should occur from the insulating to the conducting state. One could call the transition topological (one should not confuse such a topological transition in real space at finite temperatures with the well known Lifshitz topological transitions in the impulse space at T=0). A specific feature of the phase transition under consideration is the fact that the symmetry of the state at T< T~is not lower than at T> T~.This allows a linear term in ~j in the expansion of the free energy F. The role of the order parameter may be put as ‘1= ( VAFM/
VFM)T—
( VAFM/ VFM)T~,
where VAFM and VFM are the volumes of the antiferromagnetic and ferromagnetic phases respectively. Obviously, j is positive at T< T~and negative at T> T~.One can write a simple phenomenologicalexpression for F using only the assumption that the percolation temperature is singled out, i.e. the specific heat C is singular at T~.Then unlike the Landau expansion, in this case the corresponding expansion for the free energy has the form (1) F=a~+bi~4, where a_—ce(T—T~),b(T~)>0.The quadratic term is absent in (1) since otherwise the point T~would not be singled out. The cubic term is excluded in order to ensure real values of ~ at T both higher and lower than T~. It follows from (1), that C—~ T— T~I—2/3, i.e. unlike the Landau theory where the specific heat displays a jump, in the present theory it diverges at T~(similarity to the scaling theory result is only apparent). In any case, the expression (1) for F supports the possibility of second order phase transition without the order parameter vanishing both below and above T~. Unfortunately the accuracy of the calculation based on the microscopic model [1,2] is unsufficient to confirm eq. (1) or even the fact of second order phase transition at the percolation point. But this calculation makes it possible to prove that with increasing temperature the topology of fig. 1 a should go over into the topology of fig. lb if the parameters are favourable. Thus, the electron percolation may be caused by the increase of temperature. The calculation was carried out within the framework of the s-f-model using a variational principle for the free energy of the system which generalized the variational principle used in refs. [1,21 at T= 0 for finite temperatures. The generalization consists in accounting for the fact that the magnetization is partially destroyed in regions where conduction electrons are concentrated. Correspondingly in insulating regions the antiferromagnetic ordering is completely or partially destroyed, too. The geometry of nonuniform states is represented by figs. la, lb. The ratio x= VAFM/ VFM and the radius R of spherical droplets of another phase inside the host are chosen as variational parameters. Since the conduction electrons are degenerate the contribution of their thermal excitations to their free energy can be neglected. For these reasons the expressions from refs. [1,2] can be used for the electron energy including both bulk and surface parts E~and E 5, as well as for the Coulomb energy EQ. The density of the magnetic subsystem free energy FM is found in the mean field approximation. It is assumed that the undoped 474
Volume 144, number 8,9
PHYSICS LETTERS A
19 March 1990
crystal is antiferromagnetic with exchange interaction only between the first nearest neighbours and that the conduction electrons inside the ferromagnetic part of the crystal are completely spin polarized. Then FM is given by the sum of contributions from antiferro- and ferromagnetic regions respectively, FM= .j~~FM(0~ T)+
.~~LFM(~vOn(l+x)A, T),
I s H2 voFM(H,T)=——-——+~IJoIS~—Tln( ~
exp(mIJoISi/T)), H~2IJ 0IS1,
4l~1oI
\m=—S
J
Is
=—iIJoIS~—Tlfl(\ ~ exP[(m/T)(H_IJoIS2)])~ H~2IJ0IS1, m= —s where H means the effective magnetic field in the corresponding part of the crystal; A is the s-f-exchange integral, v0 is the volume of the unit cell, S is the magnitude of the f-spins, J0 is the exchange integral in the undoped crystal. The quantities S~and S2 obey the following self-consistency equations:
S1=SB5(SIJ0IS1/T), S2=SB5[S(H—1J01S2)/T], where B~is the Brillouin function. The stationary state of the system is determined from the condition of minimum total free energy of the system,
F=Ey+Es+EQ+FM with respect to x and R. The numerical calculation was carried out at the same values of parameters as in refs. [1,21 (they correspond to the rare earth of EuTe It was obtained that,lb) for occurs example, 3 the transition from thecompounds insulating state (fig.or 1EuSe a) totype). the conducting one (fig. at at n =2.51020 cm— values of the variational parameters at the transition point x= 1, R = 31.4 A. T~= K with
References [1] E.L. Nagaev, Pis’ma Zh. Eksp. Teor. Fiz. 16 (1972) 558; V.A. Kashin and E.L. Nagaev, Zh. Eksp. Teor. Fiz. 66 (1974) 2105. [2] E.L. Nagaev, Physics ofmagnetic semiconductors (Mir, Moscow, 1983).
475