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Localized charged states and phase separation near second order phase transition

Localized charged states and phase segregation are described in the framework of the phenomenological Ginzburg-Landau theory of phase transitions. The Coulomb interactions determines the charge distribution and the characteristic length of the phase separated states. The phase separation with charge segregation becomes possible because of the large dielectric constant and the small density of extra charge in the range of charge localization. The phase diagram is calculated and the energy gain of the phase separated state is estimated. The role of the Coulomb interaction is elucidated.

The problem of spatially inhomogeneous states in the charged systems frustrated by the Coulomb interactions is constantly under debate for different systems [3,4,5,6,7,12,13,14,15,16,17,18]. Usually the first order phase transition is considered where the charge density is coupled linearly to the order parameter (as an external field) [17] or to the square of the order parameter (local temperature) [18]. Very often the Coulomb interaction is discussed on the qualitative level or on the last stage of the calculations [6,13,14,19]. The general method of consideration of the long-range Coulomb interaction for the two dimensional case was proposed in ref. [17,18]. In the recent paper [19] the effect of the longrange Coulomb interaction to the second order phase transition was considered. However the Coulomb interaction was considered only for the special cases, where the density distribution was approximated in the attempts to minimize Coulomb contribution to the free energy [19], rather then it was obtained from the general formulation of the problem. In our opinion, The contribution of the Coulomb interaction is most important in the deter-mination of the form of the inhomogeneous states and therefore we consider this interaction exactly.
Phenomenological approach to the theory of phase transition, proposed by V.L. Ginzburg and L.D. Landau indicates that the properties of the system near the phase transition are determined by the closeness to the phase transition point and weakly dependent on the other properties of the system. Therefore our results demonstrates that the properties of the spatially inhomogeneous states are determined by the closeness to the phase transition point, but not by the properties of the interactions in the system. We obtain the contribution of the Coulomb interaction to the thermodynamic potential of the inhomogeneous state and show that this contribution may be relatively small, to allow the phase separation. Note, that the phase separation is possible for the system displaying phase transition of the second order.
Let us consider doped system in the vicinity of the second order phase transition. We assume that the average concentration of the free carriersρ is proportional to the dopant concentrationx. The thermodynamic potential Φ = φ(η, ρ)dV describes the behavior of the order parameter η near second order phase transition and interaction of the order parameter with the charge density.Close to the phase transition it has the form: where φ 0 is the density of the thermodynamic potential in the high-temperature phase, φ η is the density of the thermodynamic potential in the low-temperature phase, α, β, D are coefficients in the expansion of the thermodynamic potentials in powers of the order parameter (α = α ′ (T − T c ), where T c critical temperature in the absence of doping, α ′ = 1/C, where C is the Curie constant), D is proportional to the diffusion coefficient.
Density of the thermodynamic potential φ int describes the interaction of the order parameter with the charge, σ 1 and σ 2 are the constants of interactions (the term σ 2 η 2 ρ 2 is important to provide global stability of the system [14], σ 2 > 0). The density of the thermodynamic potential φ coul describes charging effects due to longrange Coulomb forces,ρ is the average concentration of the charge. We assume that Coulomb contribution is the strongest. The effect of average dopant concentrationρ to the thermodynamic potential in the high-temperature phase is included in φ 0 . The coefficient γ is inversely proportional to static dielectric constant ε, γ = 1/ε. As a result the coefficients in the thermodynamic potential depend on charge density ρ and the term σ 1 ρ(1 + ρσ 2 /σ 1 ) determines the shift of the critical temperature of the phase transition due to local charge density. Therefore the influence of the electronic system on the phase transition manifests itself as a continues shift of the critical temperature T cρ with the change of the local charge density: The local charge density may be different fromρ in some region of a finite size.
In the future we perform analysis for 3D case. The analysis for 2D case is similar. Equilibrium state of the system is determined by two equations: ∂φ(η, ρ)/∂η = 0, ∂φ(η, ρ)/∂ρ = 0. The second equation determines the dependence of the charge density ρ 0 on η: Substituting Eq.(3) to Eq.(1) we obtain the expression of the thermodynamic potential (1) φ ρ0 for the equilibrium distribution of the charge density ρ 0 (η): The negative sign in the third term of Eq.(4) (−(∇η 2 ) 2 ) indicates that uniform state may be unstable towards inhomogeneous fluctuations. Similar instabilities were discussed in the case of first order phase transition in Refs. [17,18]. This instability leads to spatially inhomogeneous solutions. Most simple solution is spherically symmetric. This equilibrium inhomogeneous state, obtained by computer minimization of the thermodynamic potential Eq.(1), has spherical form and plotted in Fig.(1). This results are obtained by applying the numerical technique described in Ref. [20]. This solution has characteristic size R 0 and characteristic charge density inside of this region ρ 0 under condition that average charge density in the system is equal toρ. It is clear that the charge is concentrated near the surface of the sphere. It allows to use approximation of the double electrical layer for calculation of the Coulomb energy. At large distances from the sphere the order parameter and the charge density are equal to their equilibrium values ρ =ρ and η = η s (ρ) (η s = 0 or η s = −α(ρ, T )/β). As a result the density of the thermodynamic potential φ s for this solution we can write in the form: Here we use dimensionless concentration x instead of charge density ρ (x = ρa 3 /e, a is the lattice constant, and e is elementary charge) andx corresponds to the level of doping. The coefficients in this formula are defined as follows: In that case it is proportional to dR 4 0 . If the screening is absent this energy is proportional to R 5 0 [7]. The param-eters A(x 0 ) and B(x 0 ) depend on temperature T , and parameter K(x 0 ) depends on the average charge densitȳ x. Parameters (x, T ) are external and we find the phase diagram in the space of these parameters. Sincex is determined by the level of doping, we study the change of the properties of the system with doping. The conditions for the minimum of the potential φ s (R 0 , x 0 ) as a function of R 0 and x 0 and the condition that minimum of the potential is lower then the potential of the uniform state φ s < φ 0 define the following set of equations: These equations define the upper boundary of the existence of the inhomogeneous state. Eq.(6) defines the equilibrium size of the charged domain R 0 : On the upper boundary of the region of the inhomogeneous states (Eq. (8)) we obtain the localization radius R 0 = 2A(x 0 )/3K(x 0 ). As a result Eq. (8) gives the following equation for x 0 as a function of temperature T : Using Eqs. (10) and (7) we write equation for upper boundary of the inhomogeneous states in the space of external parameters (x, T ) (the sign ± is determined by the sign of σ 1 ): x = x 0 (T ) ± ( πa 6 36γe 2 βF D ) 1/2α3/2 (x 0 (T ), T ) = 0 (11) These equations define the temperature T ps1 of the transition to the region of stable inhomogeneous states in the space of external parameters (x, T ) (T ps1 = f 1 (x)) where the energy of this inhomogeneous state is lower then the energy of homogeneous state. Similar consideration leads to the equation which defines lower boundary of the inhomogeneous states T ps2 = f 2 (x). Note that lower boundary T ps2 will be always close to the temperature T cρ (x), because we do not gain, but contrary loos in energy φ η when the bubble of the high-temperature phase is formed (φ η (η = 0) = 0, and φ η (η 0 ) < 0). It leads to the considerable difference between the formation of the bubble with the low-temperature phase in the high-temperature region and the bubble of the hightemperature phase in the low-temperature region. The first one is energetically favorable and therefore the region of the existence of this bubbles is considerably larger. It is important to note that Eqs. (10,11) which determine the phase diagram do not depend on d. This fact allows us to avoid optimization of our analysis with respect to d. III II FIG. 2: Phase diagram of the system with Tc = 350K, σ1/α ′ = 50K, σ2/α ′ = 510 5 K, γe 2 βF D/πa 6 α ′ = 2.33T 3 c : I is is the region of the stable high-temperature phase; II is the region of the stable high-temperature phase with metastable bubbles of the low-temperature phase; III is the region of the stable inhomogeneous phase and metastable homogeneous high-temperature phase; IV is the region of the stable inhomogeneous phase and metastable uniform low-temperature phase; V is the region of the stable uniform low-temperature phase.
Typical phase diagram of the inhomogeneous state is presented in Fig.2 for the case of of positive σ 1 > 0. Phase transition to the nonhomogeneous state represents typical phase transition of the first order. Metastable inhomogeneous phase appears at the temperature T ps0 which is much higher then the temperature of phase transition and it is shown in the phase diagram by dashed line. This line is determined from the conditions Eqs.(6,7) and A(x 0 ) 2 = 4K(x 0 ,x)B(x 0 ,x) instead of Eq. (8). The characteristic size of the charged regions in that case is determined by the condition R 0 = A(x 0 )/2K(x 0 ,x).
Very often the Coulomb energy is estimated as e 2 /a ≃ 4eV , which corresponds to two elementary charges localized in neighboring unite cells. However in our case to make the phase separation possible it is sufficient to have only small variation of the charge density per unite cell in comparison with average charge density a 3 (ρ 0 −ρ) ≃ 0.1 − 0.2e. Therefore we can estimate the Coulomb contribution to the free energy u ρ as well as the energy gain due to formation of the low-temperature phase u η (the second and the forth terms in Eq.(5)) per one unite cell in the nonhomogeneous phase u i = 3|φ i |a 3 V 0 /4πR 3 0 . We assume that d is small and neglect surface term in Eq. (5). For the Coulomb contribution we obtain: The characteristic energy in the brackets is of the order u 0 = 0.0003 − 0.001eV , here we use static dielectric constant ε = 30 − 40, which is consistent with the measurements on manganites and high-temperature superconductors. The increase of the bubble size leads to the increase of the energy u ρ by the order of magnitude or more in comparison with u 0 , but on the other hand geometrical factor F decreases u ρ . (F = 1/18 in the limiting case of the double electrical layer). Therefore u ρ is evaluated as 0.003 − 0.01eV . The corresponding estimate for the energy gain due to formation of the low-temperature phase is u η ≃ 3k B T c /4 (u η ≃ a 3 (α(x 0 , T = 0)) 2 /4β ≃ a 3 T c η 2 0 (x 0 , T = 0)/C, where C = N (η 0 (T = 0)a 3 ) 2 /3k B and N = a −3 . Therefore u η ≃ 0.02 − 0.05eV for T c ≃ 270 − 700K. Therefore in our case the phase separation becomes possible. The typical length scale of the nano-regions is estimated as The analysis of the pair distribution function obtained by neutron scattering shows that the charge density in manganites is localized on the scale of 3 to 4 interatomic distances. This is the case even if holes are localized in the form of polarons [5,21]. It confirms that holes are localized not in the form of well separated polarons but in the form of nanoregions with finite charge density of polarons. The extra charge in that case is not more then 0.1 − 0.2e per unite cell. This is in consistence with our estimates. This state is characterized as the state with nano-dimensional charge and phase separation. Dynamics of these charged nano-regions may lead to high value of the dielectric constant in the low frequency range [11].
Therefore we have shown that the second order phase transition with critical temperature equal to zero at the certain doping (quantum critical point) is unstable with respect to formation of the spatially inhomogeneous charged states. Within the phenomenological Landau theory we have shown that these states appears at some temperature T ps1 , which is substantially higher then the temperature T cρ (Fig.2). As a result the phase transition becomes effectively first order phase transition. Note that the Coulomb interaction determines the charge distribution, the screening and the characteristic length scale of the nonhomogeneous states. The spatially inhomogeneous state becomes possible in the systems with the large dielectric constants (30-40) and with relatively small charge density variations. The nonhomogeneous states were discussed also in the case of the phase transitions of the first order [17,18]. However in that case spatial variations are related to the existence of two minima in the free energy in the region of hysteresis of the first order phase transition. In our case the nonhomogeneous states are related to the spatial charge redistribution.
In conclusion we note that the localized charged states and the phase separation appears even in the case of the second order phase transition. Properties of these states are described within the phenomenological theory of the phase transitions. The Coulomb interaction determines the spatial charge distribution, the screening and the characteristic length of charge localization. The charge and the phase separation becomes possible because of large dielectric constants and relatively small spatial variation of the charge density. We derive phase diagram and estimate different contributions to the free energy.