Molecular dynamics simulation of the ferroelectric phase transition in GeTe displacive or order-disorder Ðorđe Dangić12Stephen Fahy12 and Ivana Savić2y

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Molecular dynamics simulation of the ferroelectric phase transition in GeTe:

displacive or order-disorder?

Ðorđe Dangić1,2,∗Stephen Fahy1,2, and Ivana Savić2†

1Department of Physics, University College Cork, College Road, Cork, Ireland and

2Tyndall National Institute, Dyke Parade, Cork, Ireland

(Dated: October 13, 2022)

Experimental investigations of the phase transition in GeTe provide contradictory conclusions re-

garding the nature of the phase transition. Considering growing interest in technological applications

of GeTe, settling these disputes is of great importance. To that end, we present a molecular dynam-

ics study of the structural phase transition in GeTe using a machine-learned interatomic potential

with ab-initio accuracy. First, we calculate the asymmetric shape of the radial distribution function

of the nearest-neighbor bonds above the critical temperature, in agreement with previous studies.

However, we show that this eﬀect is not necessarily linked with the order-disorder phase transition

and can occur as a result of large anharmonicity. Next, we study in detail the static and dynamic

properties of the order parameter in the vicinity of the phase transition and ﬁnd ﬁngerprints of both

order-disorder and displacive phase transition.

I. INTRODUCTION

The phase transition in ferroelectric materials is usu-

ally discussed in terms of two distinct mechanisms,

which determine whether the phase transition has order-

disorder or displacive character [1–6]. The distinction be-

tween these two mechanisms comes from the analysis of a

simpliﬁed Landau model of ferroelectric materials [7]. In

the displacive limit of the phase transition, the frequency

of a soft phonon mode becomes zero in the higher sym-

metry structure at the critical temperature. The soft

phonon mode freezes in the lower symmetry structure

driving the structural phase transition [1–4]. On the

other hand, in the order-disorder limit of the phase tran-

sition, the local ferroelectric distortion persists above the

critical temperature. In this case the paraelectric nature

of the high-symmetry phase stems from the lack of the

long-range spatial correlation of the polarization [5, 6].

Germanium telluride, GeTe, is an important thermo-

electric material that is also ferroelectric below 600 - 700

K [8–12]. The Landau model of ferroelectric phase tran-

sitions places GeTe at the boundary between materials

exhibiting order-disorder and displacive characters of the

ferroelectric phase transition [13]. This is further con-

ﬁrmed by a number of experimental studies with con-

tradictory conclusions [1–6]. Depending on the spatial

resolution of the experimental method, the phase transi-

tion in GeTe is found to be either order-disorder or dis-

placive. This ambiguity suggests that a computational,

ﬁrst-principles based study of the phase transition would

provide useful insights.

Our recent works have been able to explain a num-

ber of interesting properties of GeTe at the ferroelectric

phase transition, primarily negative thermal expansion

[14] and an increase of the lattice thermal conductivity

[15]. However, both of these studies relied heavily on a

phonon picture of GeTe, implying a displacive character

of the phase transition. There is an open question of

whether the inclusion of order-disorder character in the

calculations would lead to diﬀerent results and conclu-

sions.

Molecular dynamics (MD) simulations are probably

the most direct tool for classical simulations of materi-

als [16–18]. In principle, they can capture all relevant

physical eﬀects at high temperatures, where quantum

corrections are negligible. However, MD simulations for

systems containing many atoms when forces are deter-

mined by density functional theory (DFT) are extremely

computationally expensive [19, 20]. To circumvent this

issue, researchers usually rely on a simple analytic form of

interatomic potentials which have limited accuracy and

transferability [21–23]. Recent works on machine learn-

ing interatomic potentials aim to correct this and provide

interaction models of similar quality to DFT, at a much

more modest computational price [24–28]. These inter-

atomic potentials have been recently used to describe

phase transitions in a variety of materials [29–33].

In this paper, we present a molecular dynamics study

of the ferroelectric phase transition in germanium tel-

luride. To calculate atomic forces and energies along

MD trajectories, we used our recently developed inter-

atomic potential for GeTe using the Gaussian Approxi-

mation Potential (GAP) framework [24, 25]. Our model

of interatomic interactions in GeTe, based on DFT en-

ergies and atomic forces, reproduces the experimental

structural parameters and negative thermal expansion

at the phase transition. The radial distribution function

of the nearest-neighbor bonds in GeTe was found to be

strongly non-Gaussian even at temperatures above the

phase transition. We show that this does not necessar-

ily mean that the phase transition has an order-disorder

character and that this eﬀect could arise as a consequence

of strong anharmonicity. Furthermore, we present a de-

tailed investigation of the order parameter behavior at

the ferroelectric phase transition, which is found to ex-

hibit ﬁngerprints of both order-disorder and displacive

arXiv:2210.06174v1 [cond-mat.mtrl-sci] 12 Oct 2022

character.

COMPUTATIONAL DETAILS

Molecular dynamics simulations were performed us-

ing the LAMMPS code [34]. To calculate the atomic

forces and energies, we used our previously ﬁtted Gaus-

sian Approximation Potential (GAP) for GeTe [15, 35].

This interatomic potential was ﬁtted to DFT energies

and atomic forces. More details about the ﬁtting proce-

dure and the potential are given in Ref. [15]. To obtain

the equilibrium values of structural parameters at diﬀer-

ent temperatures, we ﬁrst run a 10 ps simulation using

the NVT ensemble to equilibrate velocities at the given

temperature, followed with a 20 ps NPT simulation to

equilibrate the structure [36, 37]. We then run a 200 ps

NPT simulation, while collecting data every 0.1 ps. The

timestep is taken to be 1 fs. For convergence study with

the size of the simulation region, see Supplementary ma-

terial [39]. At each temperature we start from the zero

temperature equilibrium structure of GeTe and set ran-

dom initial atomic velocities sampled from the normal

distribution with the variance corresponding to the tar-

get temperature.

To compute the order parameter at diﬀerent temper-

atures, we perform 300 ps NVT simulations on a 512

atoms cell while collecting atomic positions every second

time step. A time step of 1 fs was used in all simulations.

Prior to data collection, we equilibrate the system for 50

ps in the NVT ensemble.

STRUCTURAL PARAMETERS AND THERMAL

EXPANSION OF GETE

Germanium telluride crystalizes in a rhombohedral

structure below 600 K (see Fig. 1), which is described

by the following lattice vectors:

R1=a(b, 0, c),

R2=a(−b

2,b√3

2, c),(1)

R3=a(−b

2,−b√3

2, c).

Here ais the lattice constant of the primitive unit cell

of GeTe, b=q2

3(1 −cos θ)and c=q1

2(1 + 2 cos θ).

θis the angle between the lattice vectors and can be re-

garded as a secondary order parameter, since in the cubic

phase it has a ﬁxed value of 60◦, and a lower, tempera-

ture dependent, value in the rhombohedral phase. The

atomic positions in reduced coordinates are taken to be:

Ge (0,0,0) and Te (0.5 + τ, 0.5 + τ, 0.5 + τ).

We calculate the structural parameters of GeTe (the

lattice constant aand the rhombohedral angle θ) at sev-

FIG. 1. Primitive unit cell of GeTe. Red/blue spheres are

germanium/tellurium atoms. ais the lattice constant, θis

the rhombohedral angle, and τis the order parameter (the

vector from the center of the unit cell, black point, to the

tellurium atom). The side image shows the simulation cell in

molecular dynamics simulations. For presentation purposes

we show the 4×4×4 supercell, instead of the 10×10×10 used

in our calculations. The image was made using the Vesta

software [38].

eral temperatures. At each time step we calculate the

instantaneous values of the lattice constant and rhombo-

hedral angle from the geometry and volume of the simu-

lation region. Following that, we ﬁnd the structural pa-

rameters at a given temperature as a simple arithmetic

mean of the instantaneous values along the MD trajec-

tory. The results are given in Figure 2 and compared

with a number of available experiments [2, 43, 44]. In

this ﬁgure, we show the relative change of the structural

parameters compared to their 300 K values (Vis the vol-

ume of the primitive cell):

αu=uT−u300K

u300K

,for u=a, V,

αθ=θT−θ300K

60◦−θ300K

.(2)

Our calculations reproduce the experimental results

very well. All studies show negative expansion of the lat-

tice constant at intermediate temperatures (above 300 K

and below the critical temperature) and positive expan-

sion in the cubic phase. We ﬁnd that the lattice constant

has a positive thermal expansion coeﬃcient for tempera-

tures below 300 K, as measured in the experiment. The

rhombohedral angle tends to the cubic value of 60◦at

high temperatures. From the behavior of the rhombo-

hedral angle, we can infer that the critical temperature

in our study is 634 K (the middle point between the

last rhombohedral structure at 631 K and the ﬁrst cubic

structure at 637 K), which is in the range of experimental

results (600 - 700 K) [8].

We also calculate the volumetric thermal expansion of

GeTe (see Fig. 2 (c)). Again, our results follow closely

200 400 600 800

Temperature (K)

−0.020

−0.015

−0.010

−0.005

0.000

Rel. change in lattice constant

(a)

200 400 600 800

Temperature (K)

0.0

0.2

0.4

0.6

0.8

1.0

Rel. change in angle

(b)

GAP MD

Exp. 1

Exp. 2

Exp. 3

200 400 600 800

Temperature (K)

-5

Rel. change of volume

FIG. 2. Relative change of (a) the lattice constant, (b) the

rhombohedral angle and (c) the volume of GeTe with tem-

perature. The red lines are our MD results, while the points

represent experimental data taken from Refs. [2] (green), [43]

(magenta) and [44] (black).

experimental ﬁndings, both showing negative thermal ex-

pansion (NTE) at the phase transition. In the cubic

phase, GeTe regains positive thermal expansion, in agree-

ment with experiment. We do not see a discontinuity in

the calculated thermal expansion of GeTe. To be precise,

we see a decrease of the volumetric thermal expansion co-

2.50 2.75 3.00 3.25 3.50 3.75 4.00

Distance (˚

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

RDF (arb. units)

(a)

300 K

500 K

631 K

800 K

200 400 600 800

Temperature (K)

2.8

2.9

3.0

3.1

3.2

3.3

3.4

3.5

Bond length (˚

(b)

Calc.

Exp. 1

Exp. 2

Exp. 3

Exp. 4

FIG. 3. (a) Radial distribution function of GeTe for

the nearest-neighbor bonds at diﬀerent temperatures. (b)

Nearest-neighbor bond lengths in GeTe. Our calculations

are in red, Ref. [5] is in blue, Ref. [6] is in green points

(experiments reporting the order-disorder character of the

phase transition). Ref. [3] is in magenta and Ref. [2] is in

black points (experiments reporting the displacive character

of the phase transition). The full symbols represent longer

bond lengths, while the empty symbols represent shorter bond

lengths. The lines are guides to the eye.

eﬃcient as we approach the phase transition from lower

temperatures, which eventually becomes negative ther-

mal expansion at 631 K. For more elaborate discussion

of negative thermal expansion near the phase transition,

please see Supplementary material [39].

Finally, we compute the Ge - Te nearest-neighbor bond

lengths. A number of experimental [5, 6] and theoreti-

cal studies [19, 20] claim that they observe persistence of

unequal bond lengths in the cubic phase. Previous the-

oretical studies infer this eﬀect from the distorted Gaus-

sian shape of the radial distribution function (RDF) for

these bond lengths (≈3 Å). If the interatomic interaction

is perfectly harmonic, one would expect bond lengths to

be normally distributed around some mean value which

is the reported bond length. A distortion of this Gaus-

sian shape is usually attributed to the presence of two

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MoleculardynamicssimulationoftheferroelectricphasetransitioninGeTe:displaciveororder-disorder?ÐoreDangi¢1,2,StephenFahy1,2,andIvanaSavi¢2y1DepartmentofPhysics,UniversityCollegeCork,CollegeRoad,Cork,Irelandand2TyndallNationalInstitute,DykeParade,Cork,Ireland(Dated:October13,2022)Experimentalinvesti...

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Molecular dynamics simulation of the ferroelectric phase transition in GeTe displacive or order-disorder Ðorđe Dangić12Stephen Fahy12 and Ivana Savić2y

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