Solid State Communications 129 (2004) 437–441

www.elsevier.com/locate/ssc

Temperature –pressure phase diagram of silicon determined by

Clapeyron equation
C.C. Yang, J.C. Li, Q. Jiang*
The Key Laboratory of Automobile Materials of Ministry of Education, Department of Materials Science and Engineering, Jilin University,
Changchun 130025, China
Received 27 August 2003; received in revised form 16 October 2003; accepted 12 November 2003 by J. Kuhl

Abstract
The temperature – pressure phase diagram ½T – P for silicon (Si) is predicted through the Clapeyron equation where the
pressure-dependent volume difference is modeled and the corresponding thermodynamic amount of solid transition enthalpy is
calculated by introducing the effect of surface stress induced pressure. The model prediction is found to be consistent with the
present experimental results and other theoretical predictions.
q 2003 Elsevier Ltd. All rights reserved.
PACS: 81.30.Dz; 64.70. 2 p
Keywords: A. Semiconductors; D. Phase transitions; D. Surface stress

1. Introduction useful to determine the T – P curve theoretically in the

following form [24]
Crystalline and amorphous forms of silicon as the
essential materials used for solid-state electronic and DHðT; PÞ
dP ¼ dT ð1Þ
photovoltaic technologies are well-studied although its DVðT; PÞT
new structures and properties are still being discovered
[1 –4]. Structural phase transitions in semiconductors under where DHðT; PÞ and DVðT; PÞ show molar transition
pressure are a subject of considerable experimental and enthalpy and volume between two equilibrium phases. Eq.
theoretical research activity in recent years [5]. It is known (1) can describe the joint rate of change dP=dT along the
that the stable diamond structure (Si-I) under the ambient phase equilibrium lines and estimate the derived properties
pressure undergoes a well-characterized sequence of a first of DH and DV: To utilize Eq. (1) for determination of phase
order transition upon pressurization to the b-Sn structure diagram, a TðPÞ function or an integration of Eq. (1) is
phase (Si-II) and the both phases melt as temperature needed. Since both DHðT; PÞ and DVðT; PÞ are functions of
increases [6 – 23]. Although the updated temperature – temperature and pressure, and the necessary separation of
pressure ðT – PÞ phase diagram of Si has been developed variables cannot be accomplished in any direct and known
experimentally and theoretically [7,8,10,11,21,22], further manner, the integration of Eq. (1) has been carried out
theoretical works are still needed due to the worse through approximate methods ever since the equation was
measuring accuracy of high-temperature and high-pressure first established in the 19th century [24]. Although when
experiments. The classic Clapeyron equation governing all DP ¼ P 2 P0 and DT ¼ T 2 T0 are small, DHðT; PÞ <
first-order phase transitions of pure substances may be DHðT0 ; P0 Þ and DVðT; PÞ < DVðT0 ; P0 Þ have minor error
where the subscript 0 denotes the initial points and D
* Corresponding author. Tel.: þ86-431-570-5371; fax: þ 86-431- denotes the difference [24], as DP and DT increase, exact
570-5876. functions of DHðT; PÞ and DVðT; PÞ must be known [24].
E-mail address: jiangq@jlu.edu.cn (Q. Jiang). Recently, a general equation without any free parameter
0038-1098/$ - see front matter q 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ssc.2003.11.020
438 C.C. Yang et al. / Solid State Communications 129 (2004) 437–441

for surface stress f has been established as follows [25] of Si-II, and are denoted as I– L, I– II and II– L by
subscripts, respectively. Note that this definition differs
f ¼ ðh=2Þ½3DSvib DHm ðTÞ=ðkVRÞ1=2 ð2Þ
from Eqs. (2) – (4) where a subscript m denotes the melting.
where h is atomic diameter, DS ; the vibrational part of the
vib According to Eq. (5), three TðPÞ curves are obtained by
overall melting entropy DSm ; k ¼ 2DV=ðVPÞ; compressi- integrating Eq. (5) after suitable considerations on initial
bility of the crystal, V; molar volume of the crystal, R shows points and DHðTÞ and DVðPÞ functions. Thus, in the
the ideal gas constant and DHm ðTÞ is temperature dependent following, three deduced DHðTÞ and DVðPÞ functions and
molar melting enthalpy. DHm ðTÞ function can be deter- the corresponding initial points are considered one by one.
mined by Helmholtz function, DHm ðTÞ ¼ DGm ðTÞ 2 T The first one in consideration is I– L transition. For this
dDGm ðTÞ=dT where DGm ðTÞ denotes the temperature transition, DHI – L ðTÞ function can be simply taken as Eq. (3).
dependent solid– liquid Gibbs free energy difference. For DVI – L ðPÞ ¼ ðVL 2 VI Þ þ ðDVL 2 DVI Þ where DV ¼ 2VPk:
semiconductors, DGm ðTÞ ¼ DHm TðTm 2 TÞ=Tm2 where DHm To find a solution of the equation, a relationship between
is the melting enthalpy at the melting temperature Tm [26]. two pressures of PL and PI must be found. To do that, a
Thus spherical particle with a diameter D is considered. In light of
the Laplace – Young equation, PI ¼ 4fI =D and PL ¼ 4g=D
DHm ðTÞ ¼ DHm ðT=Tm Þ2 ð3Þ where g is the surface energy of the liquid [25]. Thus,
When T , Tk with Tk ¼ Tm =2 being the Kauzmann DVL ¼ 2VL PI ðg=fI ÞkL because PI =PL ¼ fI =g: Substituting
temperature or isentropic temperature where ›DGm ðTÞ= this relationship into DVI – L ðPI Þ function
›T ¼ 0 [27], DHm ðTÞ ¼ DHm ðTk Þ: As result, at T , Tk DVI – L ðPI Þ ¼ VL 2 VI þ ½VI kI 2 VL ðg=fI ÞkL PI ð6Þ
DHm ðTÞ ¼ DHm =4 ð4Þ
When the initial point of ðP0 ; T0 Þ is selected as ð0; TmI Þ
The predicted f values of various materials in terms of Eq. where TmI is the melting temperature of Si-I under
(2) are in agreement with the known experimental and ambient pressure, integrating Eq. (5) with DHI – L ðTI – L Þ
theoretical results obtained from the first principle and the and DVI – L ðPI Þ functions in terms of Eqs. (3) and (6)
classic mechanics calculations [25]. Since the measured ðP I
thermodynamic amounts in Eq. (2) has reflected usually {VL 2 VI þ ½VI kI 2 VL ðg=fI ÞkL PI }dPI
0
unknown surface states of materials [25], Eq. (2) supplies an
easy way to establish a relationship between the surface ðT
stress induced internal pressure Pi for small particles and T; ¼ ðDHI – L =TmI
2
Þ T dT
TmI
which brings out a possibility to determine DVðT; PÞ
function. In terms of Clapeyron equation associated with or
Eq. (2), the melting temperature – pressure curve of silicon qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
TðPI Þ ¼ TmI 1 þ {2ðVL 2 VI ÞPI þ ½VI kI 2 VL ðg=fI ÞkL P2I }=DHI – L
in diamond structure is predicted and the model prediction is
consistent with other experimental and theoretical results ð7Þ
[28,29]. 0
For I – II transition, the subscript I is used to substitute I
In this contribution, through assuming that Eq. (3) may for distinguishing this transition pressure from that of
be utilized to estimate DHðT; PÞ function and DVðT; PÞ melting of the Si-I phase since the initial point as the
function may be established by considering Eq. (2), TðPÞ boundary condition is selected as (PI – II ; 273Þ; which has
curves are obtained with an integration of Clapeyron a much larger pressure value than zero and leads to the
equation when suitable original points for each integration related parameters being affected by the pressure. Since
are selected. It is found that the model prediction of the T – P PI0 varies a little in the full transition temperature range
phase diagram of Si is consistent with the experimental [8,10,11,16,21,22], as a first order approximation,
results and other theoretical predictions. DVII < DVI0 is assumed
DVI – II ðPI 0 Þ < VII 2 VI0 ð8Þ
2. Model Substituting Eq. (8) into Eq. (6), VII fII kII < VI0 fI0 kI0 : As a
result
To find a solution of Clapeyron equation, as a first order
approximation, DHðT; PÞ < DHðTÞ and DVðT; PÞ < DVðPÞ fII < ðVI0 =VII ÞðkI0 =kII ÞfI0 ð9Þ
are assumed [24], which lead to a simplification of Eq. (1) The corresponding thermodynamic amounts of Si-II are
DHðTÞ different from Si-I due to their different structures.
dP ¼ dT ð5Þ Since the specific heat difference between different
DVðPÞT
polymorphous solid phases DCp is small [30], it is
In the T – P phase diagram of Si, there are together three assumed
transitions with the corresponding TðPÞ functions, which are
the melting of Si-I, the Si-I to Si-II transition and the melting DSvib vib vib
II – L < DSI – L þ DSI – II ð10Þ
 C.C. Yang et al. / Solid State Communications 129 (2004) 437–441 439

where DSvib
I – II is the vibrational part of the overall solid dependent transition of particles. When Pi < 0 with D !
transition entropy of DSI – II with DSI – II ¼ DHI – II =TI – II ; 1; P ¼ Pe ; which is the usual situation of the pressure-
which may be determined by dependent transition for bulk materials. Since any pressure
source should have the same effect on materials properties,
I – II ¼ ðDHI – II =TI – II ÞðDSI – L =DSmI Þ
DSvib vib
ð11Þ
Pi can be substituted by Pe : Thus, although P denotes Pi in
where DSvib I – II =DSI – II is supposed to be equal to DSI – L =
vib all the above three TðPÞ equations of Eqs. (7), (15), and (16),
DSmI as a first order approximation. it has been considered as Pe and can be simplified as P:
With neglecting of DCp ; DHI – II < DHII – L 2 DHI – L : As
the transition occurs at T , Tk ; in terms of Eq. (4)
DHI – II ¼ ðDHII – L 2 DHI – L Þ=4 ð12Þ 3. Results and discussion

Note that Eqs. (3) and (4) are also applicable for DHII – L ðTÞ: Fig. 1 shows a comparison between the model predic-
In terms of Eqs. (2), (4), and (9), ðVI0 =VII ÞðkI0 =kII ÞhI0 ½DSvibI–L tions of Eqs. (7), (15), and (16) and experimental and other
DHmI =ðkI0 VI0 Þ1=2 ¼ hII ½DSvib
II – L DHmII =ðkII VII Þ ; it yields
1=2
theoretical results of T – P phase diagram of Si where the
necessary parameters are listed in Table 1.
DHII – L ¼ ðVI0 =VII ÞðkI0 =kII Þðh2I0 =h2II ÞðDSvib
I – L =DSII – L ÞDHI – L ð13Þ
vib
As shown in the figure, Clapeyron equation, without any
In terms of Eqs. (10)– (13), the value of DHI – II is obtained as adjustable parameter, is consistent with the shown exper-
imental results and other theoretical predictions and may
DHI – II ¼ {½ðTI – II DSmI 2 250DHI – L Þ2 þ 1000 predict the full T – P phase diagram accurately due to
DHI – L TI – II DSmI ðVI0 =VII ÞðkI0 =kII Þðh2I0 =h2II Þ1=2 successful assumptions on DHðTÞ and DVðPÞ functions and
the value of DHI – II :
2 250DHI – L 2 TI – II DSmI }=2000 ð14Þ The P – T relationship in Fig. 1 is made by a
generalization where the internal pressure of small particles
Integrating PI0 from PI – II ; to PI0 and T from TI – II to T in is considered to be equivalent to that of the bulk one.
terms of Eqs. (8) and (14) However, the low size limit of nanocrystals for the
ðPI0 ðT 1 application of Pi in Eqs. (7) and (16) must be considered,
ðVII 2 VI0 ÞdPI0 ¼ DHI – II dT which is equal to 6h , 12h [29]. For I– L transition, the
PI – II TI – II T
corresponding values are 6hI ¼ 1:4112 nm and 12hI ¼
it yields 2:8224 nm with Pl ¼ 10:5 GPa and Pl ¼ 5:25 GPa in
TðPI0 Þ ¼ TI – II exp½ðVII 2 VI0 ÞðPI0 2 PI – II Þ=DHI – II  ð15Þ terms of Eq. (6), respectively. Although the model
prediction in this pressure range on the P – T curve has a
Let Eq. (7) be equal to Eq. (15), the Si-I/Si-II/liquid triple little error between the model prediction and the exper-
point ðPt ; Tt Þ is obtained, which is considered as the known imental results, it will bring uncertainty to ascertain
threshold point for the melting curve of Si-II. Assuming the position of the triple point ðPt ; Tt Þ: In order to improve
Eqs. (3) and (6) to be applicable for DHmII ðTÞ and DVII – L ðPÞ
functions through substituting the initial point of ð0; TmI Þ by
ðPt ; Tt Þ; with a similar way dealt above, Eq. (5) is integrated
from Pt to PII for PII and from Tt to T for T
ðPII
{VL 2 VII þ ½VII kII 2 VL ðg=fII ÞkL PII }dPII
Pt

ðT
¼ ðDHII – L =Tt2 Þ T dT
Tt

which yields
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
TðPII Þ ¼ Tt 1 þ {2ðVL 2 VII ÞðPII 2 Pt Þ þ ½VII kII 2 VL ðg=fII ÞkL ðPII 2 Pt Þ2 }=DHII – L

ð16Þ
Although the above discussion on P is related with the
Fig. 1. A comparison for T – P phase diagram of bulk Si between the
surface stress induced internal pressure Pi for a spherical
model predictions of Eqs. (7), (15), and (16) (solid lines) and the
particle, they may be extended to a general case for the
experimental (the symbols W [7], A [8] and K [11]) and other
pressure effect on T; which is illustrated as follows: let P theoretical results (dash [10] and dot [21] lines). Other symbols
denote the sum of Pi and the external pressure Pe ; namely denote I– II transition pressure at room temperature where (X)
P ¼ Pi þ Pe ð17Þ denotes the theoretical result [10], (þ ) denotes the experimental
results under non-hydrostatic pressure [16], (V) [8], (S) [16] and
When Pe < 0; P ¼ Pi : This is the case for the size- (L) [11] denote the experimental results under hydrostatic pressure.
440 C.C. Yang et al. / Solid State Communications 129 (2004) 437–441

Table 1
Necessary parameters for calculating TðPÞ phase diagram of Si in term of Eqs. (7), (15), and (16). T is in K, P in GPa, V in cm3 mol21, k in
10211 Pa21, g and f in J m22 and DH is in kJ mol21

I –L transition I–II transition II –L transition

TmI 1693 [21] TI – II 273a Tt 960b

PI – II 12a Pt 11.6b
VI 12.06c VI0 11.00c VII 8.53c
VL 10.93c VII 8.53c VL 10.93c
kI 1.02d kII 0.885d
kL 10.00 [32] kL 10.00 [32]
fI 3.707e fII 2.797e
g 0.765f g 0.765f
DHI – L 50.55 [31] DHI – II 0.78g DHII – L 53.67g
a
This value is about mean value among the experimental results [8,16,17].
b
The triple point ðPt ; Tt Þ is calculated by letting Eq. (7) be equal to Eq. (15).
c
VI ¼ M=rI and VL ¼ M=rL with M ¼ 28:09 g mol21 being the molar weight [31] and r being the density, rI ¼ 2:33 g cm23 and rL ¼ 2:57
g cm23 [32]. VII ¼ N0 vII and VI0 ¼ N0 vI0 where N0 denotes the Avogadro’s constant and v is the mean atom volume within the corresponding
crystalline structures. vII ¼ a2II cII =4 ¼ 0:014176 nm3 with the lattice constants aII ¼ 0:4690 nm and cII ¼ 0:2578 nm for b-Sn structure [16] and
vI0 ¼ ðaI0 Þ3 =8 ¼ 0:018275 nm3 where the lattice constant aI0 ¼ 0:5268 nm for diamond structure [16].
d
kI ¼ 1=BI and kII ¼ 1=BII with B being the bulk modulus, BI ¼ 97:7 GPa [33] and BII ¼ 113 GPa [19,20]. Note that as the first order
approximation, kI0 < kI is assumed [28].
e
fI is calculated by Eq. (2) with hI ¼ ð31=2 =4ÞaI ¼ 0:2352 nm due to its diamond structure where aI ¼ 0:5431 nm denoting the lattice constant
21 21
[32] and DSvib I – L ¼ 6:7 J mol K [34]. fII is calculated through Eq. (9) where fI0 ¼ 1:882 J m22 determined by Eqs. (2) and (4) with
kI < kI ¼ 1:02 £ 10
0
211
Pa being a weak function of pressure and hI0 ¼ ð31=2 =4ÞaI0 ¼ 0:2281 nm for diamond structure.
21
f
The value 0.765 J m22 cited here [35] differs from 0.865 J m22 cited before [28] since the latest experimental results support the present value [35].
g
DHI – II is calculated by Eq. (14) where DSmI ¼ DHI – L =TmI ¼ 29:86 J mol21 K21 and hII ¼ cII ¼ 0:2578 nm due to its b-Sn structure. The
value of DHII – L is determined by Eq. (12).

the accuracy of the prediction and the consistence with the two reasonable simplifications of DHðT; PÞ < DHðTÞ and
experimental results, P – T curve at P . 5:25 GPa is deter- DVðT; PÞ < DVðPÞ to estimate the T – P phase diagram of Si
mined by a linear extension of the curve along the slope at by introducing the effect of surface stress induced pressure.
P ¼ 5:25 GPa: With a similar consideration, for II – L It is found that the model prediction is consistent with the
transition, at 12hII ¼ 3:09 nm; P2 ¼ 3:62 GPa in terms of present experimental and theoretical results. Since the
Eq. (9) and the largest applicable pressure is 15.22 GPa, which Clapeyron equation may govern all first-order phase
is the sum of Pt and P2 : transitions, the Clapeyron equation supplies a new way to
In the T – P diagram, the I–II phase boundary is very determine the T – P phase diagram of materials.
indefinite with a wide range of the reported transition pressure
[13] where the transition with hysteresis is very sluggish,
especially at room temperature [8]. Although a mean original
value of PI – II at the room temperature among different Acknowledgements
experiments has been selected to ascertain the I–II transition
curve, the model prediction is in good consistent with the The financial support from the NNSFC under Grant No.
experimental results [8] where the P – T relationship of I–II 50071023 and 50025101 is acknowledged.
transition is about a straight line with dP=dT <
20:58 MPa K21 which is in the range of experimental results
of 26.5 to 2.0 MPa K21 [11]. Thus, the agreement of the
calculated transition pressure with experiment is considered to References
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