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Effect of Quantum Fluctuations on Structural Phase Transitions In PHYSICAL REVIEW B VOLUME 53, NUMBER 9 1 MARCH 1996-I Effect of quantum fluctuations on structural phase transitions in SrTiO 3 and BaTiO 3 W. Zhong and David Vanderbilt Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08855-0849 ~Received 20 September 1995! Using path-integral Monte Carlo simulations and an ab initio effective Hamiltonian, we study the effects of quantum fluctuations on structural phase transitions in the cubic perovskite compounds SrTiO 3 and BaTiO 3 . We find quantum fluctuations affect ferroelectric ~FE! transitions more strongly than antiferrodistor- tive ~AFD! ones, even though the effective mass of a single FE local mode is larger. For SrTiO 3 we find that the quantum fluctuations suppress the FE transition completely, and reduce the AFD transition temperature from 130 to 110 K. For BaTiO 3 , quantum fluctuations do not affect the order of the transition, but do reduce the transition temperature by 35–50 K. The implications of the calculations are discussed. 12,13 Quantum fluctuations typically have a very important ef- BaTiO 3 ~Refs. 10 and 11! and SrTiO 3 , giving good fect on the structural and thermodynamic properties of ma- agreement with experimental observations. Treating atomic terials consisting of light atoms like hydrogen and helium. motion classically, it predicted FE phase transitions for For example, quantum effects introduce large corrections to SrTiO3 at low temperature, thus giving indirect support for the calculated hydrogen density distribution in the Nb:H the notion that quantum fluctuations ~not included in the system.1 For materials with heavier atoms, however, the theory! must be responsible for the observed absence of a quantum fluctuation can have only a small effect on the dis- low-temperature FE phase. tribution of atomic displacements, and thus typically do not In the present work, we have extended the previous treat- have a noticeable effect on the structural and thermodynamic ment of the first-principles based effective Hamiltonian to properties of the material. However, exceptions may occur. include quantum fluctuations. In particular, we use path- As we shall see, the cubic perovskites can exhibit decisive integral ~PI! quantum Monte Carlo simulations to study the quantum-fluctuation effects, despite the fact that the lightest effect of quantum fluctuations on the structural phase transi- constituent is oxygen. This can occur because these materials tions in SrTiO and BaTiO . For SrTiO , we find that the have several competing structures with very small structural 3 3 3 quantum fluctuations have only a modest effect on the AFD and energetic differences.2 transition temperature, while the FE transition is suppressed A good example is SrTiO . While it has the simple cubic 3 entirely. We discuss the relative importance of AFD and FE perovskite structure at high temperature, SrTiO goes 3 quantum fluctuations in some detail, and examine the poten- through an antiferrodistortive ~AFD! transition at 105 K to a tetragonal phase in which the oxygen octahedra have rotated tial implications of our results for understanding the low- in opposite senses in neighboring unit cells. The observed temperature behavior of the material. For BaTiO 3 , in which softening of the ferroelectric ~FE! polar phonons with further the FE transitions occur at higher temperature, we find that reduction of temperature in the range 50–100 K would ap- the quantum effects are less dramatic. pear to extrapolate to a FE transition close to 20 K, but We start by reviewing the effective Hamiltonian and its instead the softening saturates and no such transition is construction. Two approximations are involved. First, since observed.3 The absence of a true FE transition is suggested to both the FE and AFD transitions involve only small struc- be suppressed by quantum fluctuations, giving rise to a tural distortions, we represent the energy surface by a Taylor ‘‘quantum paraelectric’’ phase at very low temperature.4 expansion around the high-symmetry cubic perovskite struc- Some experiments appear to suggest a sharp transition to this ture, including up to fourth-order anharmonic terms. Second, low-temperature phase at about 40 K, perhaps indicating the because only low-energy distortions are important to the formation of some kind quantum coherent state.5,6 However, structural properties, we include only three such distortions until a plausible candidate for the order parameter of the in our expansion: the soft FE mode, the AFD mode, and an low-temperature phase is put forward, these ideas must re- elastic mode. These are represented, respectively, by local- main highly speculative. mode amplitudes fi , ai , and ui , where i is a cell index. The These developments have stimulated many theoretical ef- local modes are constructed in such a way that a uniform ~or, 4,7–9 forts to understand the quantum effects in SrTiO 3 . How- for AFD, a uniformly staggered! arrangement of the mode ever, the previous work has all been qualitative or empirical vectors represents the desired low-energy excitation.11 Thus, in approach. Although it was shown that quantum zero-point we work with local-mode vectors instead of atomic displace- motion is capable of suppressing phase transitions,9 a de- ments. This reduces the number of degrees of freedom from tailed microscopic approach is needed to gain a quantitative 15 to 9 per cell and greatly reduces the complexity of the and detailed understanding of the quantum effects at finite Taylor expansion. The Hamiltonian is specified by a set of temperature. Recently, an ab initio effective-Hamiltonian expansion parameters determined using highly accurate first- scheme has been developed to study structural phase transi- principles calculations with Vanderbilt ultrasoft tions of cubic perovskites. It has been successfully applied to pseudopotentials.14 The details of the Hamiltonian, the first- 0163-1829/96/53~9!/5047~4!/$10.0053 5047 © 1996 The American Physical Society 5048 BRIEF REPORTS 53 principles calculations, and the values of the expansion pa- Here, i and j are the cell indices, while a and b denote rameters have been reported elsewhere.10–13 Cartesian components. j(ia) is the eigenvector describing In our previous work, we have used this effective Hamil- atomic displacements associated with local mode f ia , and M tonian by applying Monte Carlo simulation techniques to is a ~diagonal! mass matrix in the 15L3-dimensional space of study the thermodynamics of the system in the classical atomic displacements of our L3L3L supercell. Similarly, limit. Assuming the ionic motions are classical is usually a mass matrix elements connecting different kinds of local good approximation for systems such as cubic perovskites vectors, such as those between fi and ai , are also included. containing atoms no less massive than oxygen. However, the The entire mass matrix can be calculated once and for all, structural differences and energy barriers between the cubic and the extension of the PI technique to handle a nondiago- structure and the possible ~rhombohedral or tetragonal! dis- nal mass matrix is straightforward. torted structures are very small. A rough estimate of the im- The study of the thermodynamic properties of the classi- portance of quantum fluctuations can be obtained from the cal system is performed using Monte Carlo ~MC! Heisenberg uncertainty principle Dp•Dq>\/2, or equiva- simulations.18 The original simulation cell is an L3L3L lently, cube, with three vectors fi , ai , and ui at each lattice point i. 2 2 Periodic boundary conditions are used, and homogeneous DE>\ /~8mDq !. ~1! strains of the entire supercell are included. Each local vector 3 Here, Dq denotes the uncertainty in the structural coordinate, is converted to a string of P beads, so that we have 9PL which is related to the structural difference between phases. degrees of freedom per simulation supercell. We use a DE is the energy uncertainty, or zero-point energy, which single-flip algorithm, making trial moves of the vectors at may prevent the occurrence of the distorted phase if it is each site in turn and testing acceptance after each move. We larger than the classical free-energy reduction. So if the say that one Monte Carlo sweep ~MCS! has been completed structural and energetic differences between phases are small when all vectors on all sites have been tried once. Because of enough, quantum suppression may occur even for fairly mas- the 1% lattice-constant error in our local-density approxima- sive ions. For a quantitative understanding, we need to per- tion ~LDA! calculations and the strong sensitivity of the form statistical simulations that treat the ionic motion quan- structural transitions to the lattice constant, all our simula- tum mechanically. tions are performed at a negative pressure to restore the ex- 10–13 Here, we adopt the path-integral ~PI! technique15 of quan- periment lattice constant, as in our previous work. tum simulations, which has proven to be a very successful The Trotter number P should be large enough to ensure method for studying H- and He-related systems.1,16 The that the quantum effects are correctly accounted for. On the method is based on Feynman’s PI formulation of quantum other hand, the computational load increases rapidly with mechanics.17 This formulation states that the partition func- increasing P, because of both larger system size and longer tion of the original quantum-statistical systems of particles correlation time with larger P. In our simulation, the proper can be approximated by the partition function of P sub- Trotter number for each temperature is chosen empirically. systems of classical particles with each quantum particle re- For a certain temperature, we simulate systems with increas- placed by a cyclic chain of P beads coupled by harmonic ing Trotter number P51,2,4,8,16,.... We equilibrate springs. Each subsystem ~comprising one bead from each systems with each P and monitor their order parameters.
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