Calculated high pressure crystal structure transformationsfor phosphorus

Rajeev Ahuja*

Condensed Matter Theory Group, Department of Physics, Uppsala University, Box 530, S-751 21,Uppsala, Sweden

Received 5 August 2002, revised 4 October 2002, accepted 4 October 2002Published online 4 February 2003

PACS 61.50.Ks, 61.66.Bi, 71.15.Mb, 71.15.Nc, 71.20.Mq

In this paper we have studied the structural stability of the sp bonded element, P, by means of the firstprinciples calculations. The theoretical calculations made use of a full potential linear muffin-tin orbital(FPLMTO) method adopting the local density approximation to the density functional theory. We repro-duce the observed crystallographic phase stability of P as a function of compression. Our results con-firm the recent experimental finding of Akahama et al. We have also proposed a new structure for anexperimentally reported unidentified intermediate phase in between simple cubic and simple hexagonalphase. This new structure is similar to what has been observed for Si. We have explained the stabilityof different phases under pressure using our calculated density of states (DOS).

Introduction During the last 10 years, one has witnessed a dramatic progress in the study of crystal-phase stability in solids. Impressive advances in the high pressure experiments in the Mbar regimehave made it possible to study crystallographic properties at extreme conditions. Due to the peculiarcrystallographic properties observed, we are here going to focus our attention on the group VA ele-ment, P, in the periodic table. At zero pressure, P has unique anisotropic crystal structures consistingof three-fold coordinated buckled layers of atoms. Black phosphorus is one of the four allotropes ofphosphorus and it is stable in the orthorhombic structure at ambient pressure [1, 2]. In this phase,phosphorus is a narrow band gap semiconductor with an energy gap, 0.3 eV [3]. Kikegawa and Iwasa-ki [4] observed a transition from the orthorhombic (A17) to the A7 phase at 5.5 GPa and a phasechange from the A7 to the simple cubic at 10 GPa. Jamieson [5] observed this transition at 11.1 GPa.No further transition was observed up to 70 GPa. Akahama et al. [6] have studied the stability of thesimple cubic phase of phosphorus up to 151 GPa using angle dispersive powder X-ray diffractiontechnique with a synchrotron radiation source. They have observed a structural phase transition fromthe simple cubic phase to the simple hexagonal phase at 137.0 GPa through an intermediate phase.The structure of this intermediate phase is still unknown. Very recently Akahama et al. [7] havereported another high pressure phase beyond the simple hexagonal phase. They have proposed thestructure of this phase to be the bcc; it is stable above 262 GPa. Thus on the experimental side thereis a fair amount of data available for the high pressure phases of phosphorus.

Early pseudopotential calculations by Chang and Cohen [8] show a transition from the semiconduct-ing orthorhombic phase to a semi-metallic rhombohedral phase (which is the stable phase for As, Sb,and Bi at ambient conditions) at 3.0 GPa whereas the experimental data show this transition in be-

* e-mail: rajeev@fysik.uu.se, Phone: +46 18 471 3626, Fax: +46 18 471 3524

# 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0370-1972/03/23502-0282 $ 17.50þ.50/0

phys. stat. sol. (b) 235, No. 2, 282–287 (2003) / DOI 10.1002/pssb.200301569

tween 4 to 8 GPa. The A7 phase is calculated to be stable up to the volume, V=V0 ¼ 0.5 where V0 isthe experimental volume at ambient condition but the experimental studies show that the transitionfrom the A7 to sc phase is around 11 GPa. In order to get the correct transition pressure, Chang andCohen [8] have shifted the total energy of P by a constant shift of 2.3 mRy/atom. They have sug-gested that this shift may be due to zero-point energy or temperature renormalization of the phononfrequencies. This shift gives the transition pressure for the A7 to sc at 10.5 GPa and for the orthor-hombic (A17) to A7 at 5.1 GPa, which are in good agreement with the experimental data. Moreover,Schiferl [9] who has also used the pseudopotential method, calculated the A17 to A7 transition forphosphorus at 5 GPa in agreement with the experimental data and the A7 to the simple cubic transi-tion has been calculated at 11 GPa, which is again in very good agreement with the experiment. Onecan notice that in the calculations of Schiferl no shift has been made to the total energy of the A7phase in order to get correct transition pressures. Sasaki et al. [10] have also studied the A7 tosimple cubic transition using the norm-conserving pseudopotential and obtain a transition pressure of15.8 GPa. Up to now, we are not aware of any ab initio calculations taking into account the twonewly observed high pressure phases reported by Akahama et al. [6, 7]. The main aim of the pre-sent paper is to calculate all structural phase transitions without shifting the total energies as done inpast and including the newly observed phases by Akahama et al. The other aim is to resolve thestructure of the unidentified intermediate phase between the simple cubic and simple hexagonalphase.

Details of the calculations We have performed calculations using the full potential linear muffin-tinorbital (FPLMTO) method [11], in which basis functions, electron densities, and potentials were calcu-lated without any geometrical approximations. These quantities were expanded in spherical waves(with a cut-off lmax ¼ 8) inside non-overlapping spheres surrounding the atomic sites (muffin-tinspheres) and in a Fourier series in the interstitial region, between the spheres. We have used theHedin–Lundqvist parameterization of the exchange-correlation potential [12]. The muffin-tin spheresoccupied approximately 50% of the unit cell. The radial parts of the basis functions inside the muffin-tin spheres were calculated from a wave equation for the l ¼ 0 component of the potential inside thespheres that included mass velocity, Darwin term, and higher order corrections, but not the spin–orbitcoupling. Spin–orbit coupling and the higher l components of the potential in the muffin-tin spheresand all of the Fourier components of the potential in the interstitial region were included in the crystalHamiltonian [13]. In the calculations reported here, two sets of energy parameters were used, oneappropriate for the semi-core states, and the other appropriate for the valence states. The resultingbasis formed a single, fully hybridizing set. The special k-point method [14] was used for samplingthe irreducible wedge of the Brillouin-zone (BZ).

Results and discussion We have calculated the total energies within the local density approximationfor different phases, namely, the A17, A7, simple cubic (sc), simple hexagonal (sh), bcc, and our newidentified phase Imma. In Figure 1a, we show total energies for different structures as a function ofvolume (V=V0, where V0 is the experimental volume at ambient condition) for phosphorus relative tothe bcc structure. For simplicity we have not shown the A17 structure in Fig. 1a. In Figure 1b, weshow the stability of our newly identified phase, Imma, sc phase, and sh phase relative to the bccstructure in a small volume range. Our calculation shows a transition from the A7 to the sc phase at avolume compression V=V0 ¼ 0.75, the sc phase to Imma at V=V0 ¼ 0.48, the Imma to sh phase atV=V0 ¼ 0.42, and the sh to bcc phase at a volume compression, V=V0 ¼ 0.39. This (bcc) phase re-mains stable up to the highest studied volume compression. Our calculated transition pressures andvolume changes are shown in Table 1. One can see from Table 1, that the agreement between calcula-tions and experiment is very good. For calculating transition pressures for the A17 to A7 or A7 to sc,no shift is included in the total energy. One can also notice that there is almost no volume changefrom the Imma to the sh phase transition. In Table 2, we have given the calculated volume, bulkmodulus, and its pressure derivative along with experimental data. Again the agreement between theo-ry and experiment is good. In Table 3, we have shown our calculated (fully relaxed) axial ratio and

phys. stat. sol. (b) 235, No. 2 (2003) 283

internal positional parameters. Our results compare very well with the existing data. For the Immastructure, our values are predictions.

In Figures 2a and 2b, we have plotted the density of states (DOS) of black phosphorus for differentstructures at different pressures. The DOS in Fig. 2a, lower panel, for the A17 phase at ambient pres-sure shows semiconducting nature. The calculated band gap is 0.25 eV and this compares very wellwith the experimental value of 0.3 eV [3]. The states just below and above the Fermi level (EF) arisefrom the p-states, while the states between 16 eV to 8 eV below the (EF) are mainly due to the s-states. The DOS for the A7 phase shows the semi-metallic nature, which is in agreement with experi-

284 Rajeev Ahuja: High pressure crystal structure transformations for phosphorus

0.35 0.45 0.55 0.65 0.75 0.85V/V0

−70.0

−50.0

−30.0

−10.0

10.0

30.0

50.0

70.0E

nerg

y D

iffe

renc

e (m

Ry)

A7−bccsc−bccsh−bcc

A7

sc

shbcc

P

a)

0.35 0.40 0.45 0.50 0.55V/V0

10.00

5.00

0.00

5.00

10.00

Ene

rgy

Dif

fere

nce

(mR

y.)

scbccImmabccshbcc

Imma

shbcc

sc

P

b)

Fig. 1 Total energy of phosphorus as a function of volume (V=V0, where V0 is experimental volume)for different structures; (a) wide volume range and (b) small volume range. The total energy of the bccstructure is taken as a reference level.

phys. stat. sol. (b) 235, No. 2 (2003) 285

Table 3 Calculated and experimental b=a, c=a, angle (a) and internal positionalparameters for A17, A7, Imma, and sh phases.

calculated expt.

A17

b=a 3.145 3.162 [2]c=a 1.308 1.321 [2]u 0.0789 0.0806 [2]v 0.1013 0.1017 [2]

A7

a 57.51 57.25 [4]u 0.2313 0.228 [4]

Imma

b=a 0.90 ––c=a 0.54 ––z 0.2484 ––

sh

c=a 0.932 0.948 [7]

Table 1 Calculated and experimental transition pressure (GPa) and volume change (%).

transition pressure volume change

expt. calc. expt. calc.

A17 ! A7 5.5 [7] 4.5 10.2 11.0A7 ! sc 11.1 [7] 10.5 3.7 1.5sc ! Imma 103.0 [7] 110.0 –– 5.5Imma ! sh 137.0 [7] 157.0 –– 0.2sh ! bcc 262.0 [7] 234.0 2.8 2.2

Table 2 Calculated and experimental volume (�A3), bulk modulus (GPa), and its pressure derivative(B0

0).

volume bulk modulus (B0) B00

expt. calc. expt. calc. expt. calc.

A17 19.0 [7] 17.5 36:0� 2:0 32.0 4:5� 0:5 4.5A7 16.5 [7] 15.1 46:0� 4:0 62.0 3:0� 0:6 5.3sc 15.2 [7] 14.4 95:0� 5:0 132.0 2:1� 0:8 3.3Imma –– 14.3 –– 90.0 –– 3.6sh 14.2 [7] 14.2 84:1� 1:0 99.0 4.7 3.5bcc [7] –– 13.7 –– 111.0 –– 3.3

ments. In the upper panel of Fig. 2a, we have shown DOS at 4.5 GPa showing that both the A17 andA7 phases are now semi-metallic. The shape of DOS is not much changed. But as we go to highpressures (Fig. 2b), the DOS for the bcc, sh, and sc structures have become broadened and look free-electron like. We can explain the stability of different structures in terms of bandwidth just by lookingto the DOS. At ambient pressure (Fig. 2a), the DOS for the A17 phase is broader than the A7 phasesuggesting that the A17 is the more stable phase at ambient pressure. As we increase the pressure(upper panel of Fig. 2a), at 4.5 GPa, now the DOS of the A7 phase is broader than the A17 phase,stabilizing the A7 phase in comparison to the A17 phase. For a further increase of pressure (lowerpanel of Fig. 2b), at 10.5 GPa, the DOS of the sc phase is broader than the A7 phase. Similarly at120 GPa (middle panel), the DOS of the sh phase is broader than the sc phase, and finally at 234 GPa(upper panel), the DOS of the bcc phase is broader than the sh phase, showing the stability of the bccphase at this pressure. In other terms one can also understand the stability of the A7 phase at equili-brium is thus to be viewed as Peierls distortion, the symmetric structures are metallic whereas thedistorted structure is not. The distortion is driven by the relatively narrow p-bands. At low volumesthe bands become broadened, this leads to a reduction of the driving force for the distorted structures.Instead the electrostatic energy and the Born–Mayer repulsion are stabilizing the high symmetricstructures.

Summary In summary, we have studied the electronic and structural properties of phosphorus usingthe all electrons FPLMTO method within the local density approximation (LDA). We have resolvedthe unidentified intermediate structure in between the simple cubic and simple hexagonal phase. The

286 Rajeev Ahuja: High pressure crystal structure transformations for phosphorus

−20.0 −15.0 −10.0 −5.0 0.0 5.0Energy (eV)

0.0

1.2

A17A7

0.0

0.5

1.0

1.5

2.0

A17A7

0.0GPa

4.5 GPa P

a)

−28.0 −20.0 −12.0 −4.0 4.0Energy (eV)

0.0

0.7

A7sc

0.0

0.2

scsh

0.0

0.2

shbcc

10.5 GPa

120.0 GPa

234.0 GPa P

Tot

al D

OS

(sta

tes/

eV)

b)

Fig. 2 Total DOS of phosphorus for different structures and pressures; (a) 0.0 GPa (lower panel) and 4.5 GPa(upper panel), (b) 10.5 GPa (lower panel), 120.0 GPa (middle panel), and 234.0 GPa (upper panel). The Fermilevel is set at zero energy level and shown as vertical solid line.

Tot

al D

OS(

Stat

es/e

V)

new structure belongs to the Imma space group. This structure is similar to what has been observedfor Si by McMahon and Nelmes [15] under pressure in between the b-Sn and the simple hexagonalstructure. Calculated band gap, transition pressures, equilibrium volumes and bulk modulus are ingood agreement with the experimental data.

Acknowledgements The author acknowledges financial support from VR and ATOMICS (SSF).

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