PHYSICAL REVIEW B

VOLUME 54, NUMBER 21

1 DECEMBER 1996-I

Constant-pressure first-principles studies on the transition statesof the graphite-diamond transformation
Y. Tateyama, T. Ogitsu, K. Kusakabe, and S. Tsuneyuki
Institute for Solid State Physics, University of Tokyo, Roppongi, Minato-ku, Tokyo 106, Japan~Received 13 June 1996!
We have investigated the activation barriers and the intermediate paths of the transformation to cubic
diamond and that to hexagonal diamond from graphite under pressure, allowing both atomic geometry andunit-cell shape to vary, in order to clarify the difference of the microscopic mechanisms between them. For thisinvestigation, we have developed a method of finding a saddle point of the potential surface automatically on
the basis of constant-pressure first-principles molecular dynamics. At the transition states, the length of theinterlayer bonding is universal irrespective of the transformations and pressures, while there is a difference inthe lateral displacement of atoms on the paths. It is found that the activation barrier from graphite to cubic
diamond is lower than that to hexagonal diamond by ;70 meV/atom. These results suggest that, whenevercollective slide of graphite planes is allowed, the transformation to cubic diamond is favored, and that hexagonal diamond can be obtained only when such slide is prohibited. @S0163-1829~96!06145-0#

5GPa = 5×10^9 Pa = 5万気圧

触媒をつかって2000 Kでグラファイトからダイヤモンドへ

I. INTRODUCTION

High-pressure synthesis of diamond from graphite ~Gr!
has been intensively studied because of its technological andfundamental importance. Many experiments for synthesis ofcubic diamond ~CD! have been carried out, so that methods
and the P-T condition for the synthesis have been established empirically.1 Industrially, the catalyst-solvent process
under static conditions is usually utilized because thismethod allows us to make a large crystal in the manageable
P-T region (;5 GPa and ;2000 K!. Direct transformation
to cubic diamond without catalysts, which is desired to makepure crystal, is also possible under shock-wave conditions or
static compression at high temperature.
Furthermore, static compression at room temperature
opens the possibility of different syntheses, namely, synthesis of hexagonal diamond ~HD! which has a different stacking sequence of puckered hexagon from CD. ~HD has AB
stacking, while CD has ABC stacking.! Bundy and Kaspersynthesized HD in the laboratory using well-crystallizedgraphite under static compression in the direction of the caxis up to 13 GPa.2 Recently, Yagi et al. carried out an insitu x-ray diffraction study on the transformation of kishgraphite at room temperature.3 They reported that the transformation took place at 14–18 GPa under quasihydrostaticpressure, and that the high-pressure phase was identified by
the hexagonal diamond structure, where the c axis of HD
was perpendicular to that of the initial graphite.
On the other hand, in recent experiments by Endo et al.,4cubic diamond was observed by heating the sample up to;800 °C under ;20 GPa after static compression of polycrystalline graphite at room temperature. Thus, under static
compression at room temperature, there are two kinds oftransformations from graphite, namely, those to cubic diamond and to hexagonal diamond. Although the difference of
these two transformations might be attributed to the qualityof the starting samples,4 the condition and the microscopic
mechanism for each transformation are still unclear.
0163-1829/96/54~21!/14994~8!/$10.00

54

As to theoretical approaches for these transformations,
there are some reports by means of first-principles calculations. Fahy et al. investigated the activation barriers at ambient pressure for the transformations from rhombohedralgraphite (ABC stacking! to cubic diamond5 and fromAA-stacking graphite to hexagonal diamond6 using a
constant-volume scheme which fixes the shape of the unit
cell. In those works, the transformation paths are assumed
such that the orientation of the interlayer bonding is always
parallel to the c axis of the initial graphite during the transformations. This assumption on the path is not the case in theexperiment for the Gr-HD transformation.3 Recently, finitetemperature and pressure simulations of both Gr-CD and
Gr-HD transformations with a constant-pressure scheme
have been carried out by Scandolo et al.7 In that work, all thesimulations were started from hexagonal graphite (AB stacking! without any assumptions on the path, and both transformations to CD and HD were observed. In addition, the orientation relation in the experiment3 was reproduced in thesimulations for the Gr-HD transformation. However, since
we cannot get good statistics from the limited number ofdynamical simulations, characterization of the paths and estimation of the barriers are required to answer what conditionis crucial for each transformation.
In this work, in order to clarify the difference of the transition probability and the microscopic mechanisms between
the Gr-CD and the Gr-HD transformations, we have quantitatively investigated their transition states in the adiabatic
potential surface under pressure. Actually, we calculate theactivation barriers and the intermediate paths in both transformations at 0 and 20 GPa which correspond to the pressures below and above the critical one P c for the transformations, respectively. From these results, we discuss thepossibility and the mechanisms of the two transformations.
For the calculations, we have used the constant-pressure
first-principles molecular dynamics ~CP-FPMD! method
which allows the shape of the unit cell as well as the atomic
geometry to vary without any symmetry constraints. On the14 994© 1996 The American Physical Society

54

CONSTANT-PRESSURE FIRST-PRINCIPLES STUDIES . . .

14 997

B. Initial and final states

FIG. 2. Hexagonal diamond, graphite and cubic diamond in themonoclinic unit cell used in this work. The orientation in the conventional cell and the structural parameters u1, u2, R, and l arealso displayed.

symmetry constraint during the simulations. The structuresof the three phases in the unit cell are schematically displayed in Fig. 2. As to the structure of cubic diamond, deformation of the cell is necessary to express it within thisunit cell of two layers because it has ABC stacking of puckered hexagon. From among some possible unit cells, wechoose the one shown in Fig. 2, which has the smallest deformation.
The parameters to characterize the atomic geometry aswell as cell parameters are also shown in Fig. 2. The parameters u1 and u2 indicate the displacement of the atoms perpendicular to planes. By the set of the parameters(u1, u2), one can distinguish the three phases. The set (0,0) indicates graphite, while the sets (1/12, 1/12) and (1/16,21/16) correspond to HD and CD, respectively. These twoparameters are expressed with the internal coordinates in theunit cell. In addition, R is defined as the distance between
atoms in adjacent layers which are connected by formationof sp 3 bonding, and l indicates the average of the interlayer
distance defined as csina.

We first calculated the structures and the electronic properties of graphite, cubic diamond, and hexagonal diamond at
0 and 20 GPa which correspond to the pressures below andabove the critical pressure for the transformations, respectively. The optimized structural parameters are comparedwith the experimental values at ambient condition in Table I.
The results are in good agreement with experiments withinerror of 1%.
The total energies of HD and CD at 0 GPa are higher thanthat of Gr by 0.072 and 0.043 eV/atom, respectively. At 20
GPa, the relative enthalpies to graphite DH5H2H Gr of HD
and CD are found to be 20.208 and 20.234 eV/atom, respectively. Hence, our calculations lead to an energy relationconsistent with the phase diagram of carbon at each pressure.
These energy relations should not change even if zero-point
motion energy is taken into consideration.16 Band gaps ofGr, HD, and CD are estimated ;0, 3.12, and 4.11 eV, respectively. These results are consistent with the previous
ones calculated with the LDA scheme.18
We also compare the total energy obtained here withthose of the previous calculations.18,19 The difference of thetotal energy (0 GPa! between HD and CD is in good agreement with the previous results. On the other hand, the totalenergy of graphite might be underestimated due to the deficiency of k points in spite of sufficient cutoff energy. However, since our results show a consistent energy difference
between HD and CD, we consider that this does not significantly affect the later discussion comparing the transformations to HD and CD.
C. Saddle-point search using the FI technique

Next we describe the search for the saddle points of thepotential surface by means of the FI technique described inTABLE I. Optimized structural parameters at the ~meta!stable state of each phase and at the activation
barriers between them under external pressure P ext50 and 20 GPa in this work, compared with the experimental values indicated in brackets. The cell parameters a, c, and a in this table are in the monoclinic celland definitions of other structural parameters are explained in the text.

P ext50 ~GPa!
a ~Å!
c ~Å!
a ~deg!
u1
u2
R ~Å!
l ~Å!
P ext520 ~GPa!
a ~Å!
c ~Å!
a ~deg!
u1
u2
R ~Å!
l ~Å!

HD

HD-Gr
barrier

GrGr-CD
barrierCD

2.507 ~2.51!
4.348 ~4.36!
90 ~90!
0.0833 ~1/12!
0.0833 ~1/12!
1.542
2.174

2.470
4.895
90
0.04820.04822.073
2.447

2.468 ~2.46!
6.759 ~6.70!
90 ~90!
0.0 ~0!
0.0 ~0!
3.453
3.380

2.457
4.966
107.830.029520.0295
2.085
2.364

2.510 ~2.51!
4.369 ~4.36!
109.42 ~109.42!
0.0624 ~1/16!
20.0624 ~21/16!
1.545
2.060

2.466
4.290
90
0.08340.0834
1.522
2.145

2.444
4.795
90
0.04380.0438
2.077
2.398

2.427
5.614
90
0.0
0.0
2.895
2.822

2.442
4.904
106.520.027720.0277
2.091
2.351

2.472
4.301
109.21
0.062520.0625
1.523
2.031

Δ̲f H = 0.043 eV atom^-1 = +4.1 kJ mol^-1, -0.234 eV atom^-1 = -22.6 kJ mol^-1barier = 0.347 eV atom^-1 = 4030 K, 0.148 eV atom^-1 = 1720 K
14 998TATEYAMA, OGITSU, KUSAKABE, AND TSUNEYUKI

Sec. II. As to the reference vector u for the transformation toCD ~HD!, we adopted the direction to CD ~HD! from orthorhombic graphite whose stacking manner is slightly different
from hexagonal one within AB stacking. This is because theatomic geometry of orthorhombic graphite is closer to both
CD and HD in the configuration space than that of hexagonal
one so that this choice leads to less computational task. To
check its validity, we compared energies of these two graphite structures. The difference of the total energy between
them is less than 1 meV, which is beyond the accuracy of
our calculations, and other electronic and structural properties are almost the same. In addition, this orthorhombic structure is the same as the intermediate one suggested by Scandolo et al.7 In fact, our simulations using the above reference
vectors certainly converged.
In order to check whether the obtained states are on asaddle point or not, we then performed the usual structure
optimizations starting from states shifted infinitesimally fromthe converged points. Since the structure changes into bothsides of the transformation with a decrease of the enthalpy,
we confirmed that the converged points are true saddlepoints. In these investigations, we carried out some simulations using the FI technique with changing the initial state in
the intermediate possible region of the configuration space,and found that all the simulations give the same saddle point.
D. Transformation paths

The structural parameters at the obtained saddle points inboth transformations under 0 GPa and 20 GPa are listed inTable I. At the saddle points, namely, the transition states,
R has almost similar values, 2.07–2.09 Å, regardless of thetransformations or external pressures. This suggests that a
critical R seems to be universal in the transformations between sp 2 and sp 3 bonding of carbon. Thus, the transition
states would be determined by the distance between atoms
combining the adjacent layers.
As to the transformation paths to the saddle points, there
is little difference in the reduction of the interlayer distancel and c-axis length. They are reduced from that of graphite
similarly in both transformations such as 75% at 0 GPa and
60% at 20 GPa.
However, there exists a difference in the lateral displacement of atoms. Indeed, we can clearly see a difference in thedeformation angle of the unit cell a . In the Gr-HD transformation, a retains 90° because of no change of stacking. On
the other hand, a in the Gr-CD transformation alreadychanges over 106.5° which is almost the same as that of CD.
As to the out-of-layer displacement of atoms, u1 and u2 in
the Gr-HD transformations change over 50% from Gr to HD,
while those in the Gr-CD transformations rather have values
closer to graphite. Thus, the collective sliding of the layers,which leads to the change of stacking, is necessary for reaching the saddle point in the Gr-CD transformation, while, because of no need to change of stacking, the local change ofatomic configurations such as the buckling of planes is more
required in the Gr-HD one.

54

FIG. 3. Schematic view of the potential energy surface at 0 GPa
with solid line and 20 GPa with dashed line. ~The lines are a guidefor the eye.! Relative adiabatic enthalpy H5E1 P extV ~eV/atom!
to that of graphite is shown for each pressure.

hexagonal diamonnd ←graphite→simple cubic
potential energy surface at 0 and 20 GPa is shown in Fig. 3,
where the calculated values of the relative enthalpy to graphite, DH, are displayed. At 0 GPa, the activation barriers fromgraphite are 0.347 and 0.414 eV/atom for the transformations
to CD and HD, respectively, while those at 20 GPa fall into
0.148 and 0.222 eV/atom. Thus, the activation barrier to CD
is slightly lower than that to HD by 0.067 eV/atom at 0 GPa
and 0.074 eV/atom at 20 GPa. In order to confirm the convergence of the barriers on the number of k points, we also
carried out calculations with a larger k-point sampling ~uniform 128 and 162 points! for the obtained barrier structures.
As a result, the difference between the two barriers changed
by less than 0.01 eV/atom in these calculations. This wouldensure that the activation barrier for the Gr-CD transformation is lower than that for the Gr-HD one.
The barrier from Gr to CD at 0 GPa in this work is similar
to the barrier between rhombohedral graphite and CD calculated by Fahy et al.,5 where the path is fixed so that the caxis is always parallel to that of the initial graphite during thetransformation. This is because the transition state between
Gr and CD obtained in this work without any constraints on
the path has the structure with almost ABC stacking.
From the potential surface shown in Fig. 3, all the threephases can survive as metastable phases at ambient condition
due to the existence of the high barriers between them at 0GPa. Under high pressure, on the other hand, both barriers
become sufficiently small for the state to go over at room
temperature. By comparing these two transformations, it isconfirmed that, at high pressure, the transformation to cubic
diamond from graphite takes place under thermal equilibrium condition. Furthermore, from the viewpoint of chemical
kinetics, the transition probability to cubic diamond is higher
than that to hexagonal diamond even under nonequilibrium
conditions because of the difference between the barrierheights. In fact, the transition probability to hexagonal diamond from graphite is approximately estimated as 10% ofthat to cubic diamond at room temperature under 20 GPa.

E. Activation barriers

F. Discussion

We next show the activation barriers in the transformations at the resulting saddle points. A schematic view of theNow we discuss the conditions and the microscopic
mechanisms for both Gr-HD and Gr-CD transformations.