Boron-doped diamond
Vassiliki Katsika-Tsigourakou*
Section of Solid State Physics, Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, 157 84 Zografos, Greece Abstract-Boron-doped diamond undergoes an insulator-metal transition at some critical value (around 2.21 at %) of the dopand concentration. Here, we report a simple method for the calculation of its bulk modulus, bad on the thermodynamical model, by Varotsos and Alexopoulos, that has been originally suggested for the interconnection between the defect formation parameters in solids and bulk properties. The results obtained at the doping level of 2.6 at %, which was later improved at the level 0.5 at %, are in agreement with the experimental values.
Keywords- Compressibility, Point defects, Mixed crystals, Elastic properties, Defect volume, Activation energy, Boron-doped diamond
*E-mail- ****************.gr
1 INTRODUCTION
Pandey et al. [1] studied the pudo elastic behavior of liquid alloys using pudo potential model bad on the density functional theory with both the local density approximation and the generalized gradient approximation for the exchange
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correlation function. Very interesting results were obtained which showed that the elastic constants of the elemental cubic model depend primarily on the bonding variance, the density at the cell boundary and the symmetry of the lattice.
In the above paper Pandey et al. applied the model of Varotsos and Alexopoulos using slight modification in volume due to concentration. It is the scope of this short paper to extend the ufulness of the challenging findings of Pandey et al. and show that the u of Varotsos and Alexopoulos model can also rve for treating a problem (e below) of major technological interest.
不等号Diamond has been extensively studied (e.g., e refs [2-12]) in view of its remarkable properties. For example, it has a very large Debye temperature [13] and the largest elastic moduli known for any material and correspondingly the largest sound velocities [13], [14]. Nowdays, the diamond anvil cell (DAC) technique, which is extensively ud as a unique tool for producing high pressure in the laboratory [15], exploits the extreme hardness of diamond.
Diamond is a wide band-gap miconductor. The high interest of studying both doped natural diamonds and high-level doped synthetic diamonds [16] originates from the discovery of the profound influence of dopants on their physical properties. Specifically, doping diamond with boron leads to the insulator-metal transition [17]. Electrical conductivity measurements of diamond revealed that for boron concentrations higher than some critical value estimated as 2.21 at %, the conductivity on the metallic side of the transition at low temperature a T b law. For metallic samples, b was found to be 1/3, approaching 1/2 at higher concentrations [17]. Some uncertainty remains in predicting the boron concentration above which metallic conduction takes place [17-21].
The isothermal bulk modulus B (and the compressibility κ,1/B κ=) can be ud as a quantitative characteristic describing relations between the structure and atomic forces, from one side and the physical properties of solids, from another side. Dubrovinskaia et al. [22] reported the results of high pressure-high temperature synthesis of boron-doped diamond and the results of experimental determination of its bulk modulus. In addition, they proceeded to detailed a theoretical calculation which suggested very little (within computational uncertainty) effect of the doping on the compressibility of diamond for impurity concentrations up to 3 at %. The calculations also confirmed that boron atoms prefer to substitute C-atoms in a diamond structure. It is the aim of this s
hort paper to draw attention to the following point: Instead of the aforementioned tedious theoretical calculation, a simple thermodynamical model can be alternatively ud for the estimation of the boron-concentration dependence of the compressibility of diamond. This thermodynamical model, has been originally suggested for the formation and/or migration process of defects in solids [23-25]. The same model was extended [26] to describe the physical properties of the electric signals that precede rupture [27-29]. In the next ction, we recapitulate this model (termed cB Ω model, e below) and then in ction 3 we apply it to the ca of the compressibility of boron-doped diamond. We note that the success of this model to reproduce the lf-diffusion coefficients of diamond has been already checked in [30].
2 THE MODEL
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简便计算公式大全Let us denote 1V and 2V the corresponding molar volumes, i.e. 11V N υ= and 22V N υ= (where N stands for Avogadro’s number) for diamond (density 3.51
gr/cm 3) and B 4C (density 2.48 gr/cm 3) respectively. Defining a “defect volume”蓬莱仙洞风景区
[31] d υ as the variation of the volume 1V , if one “molecule” of type “1” is replac ed by one “molecule” of type “2”, it is evident that the addition of one “molecule” of type “2” to a crystal containi
ng N “molecules” of type “1” will increa its volume by 1d υυ+. Assuming that d υ is independent of composition,
三位一体自我介绍the volume N n V + of a crystal containing N “molecules” of type (1) and n “molecules” of type “2” can be written as [27,31]: 1[1()]d N n V n N V n υ+=++ (1)
The compressibility κ of the doped diamond can be found by differentiating (1) with respect to pressure which finally gives [27]: ()1111d d N n V V n N N V κκκυκ+⎡⎤=++⎣⎦ (2)
where d κ denotes the compressibility of the volume d υ, defined as [32] (1)()d d d T d dP κυυ≡-⨯.
Within the approximation of the hard-spheres model, the “defect -volume” d υ can be estimated from: 21()d V V N υ=- (3)
Thus, if N n V + can be determined from (1) (upon either considering (3) or other type of measurements and/or method), the compressibility κ can be found from (2) if a procedure for the estimation of d κ will be employed. In this direction, we adopt a thermodynamical model, termed cB Ω model, for the formation and migration of the defects in solids [23-27]. According to this model, the defect Gibbs energy i g is
interconnected with the bulk properties of the solid through the relation i i g c B =Ω where B stands, as mentioned, for the isothermal bulk modulus (1/)κ=, Ω the mean volume per atom and i c is dimensionless quantity. (The superscript i refers to the defect process under consideration, e.g. defect formation, defect migration and lf-diffusion activation). By differentiating this relation in respect to pressure P , we find the defect volume i υ [()]i T dg =. The compressibility ,d i κ defined by ,d i κ[()]i T d n dP υ≡-, is given by:
,22(1)()[()1]d i T B d B dP dB dP κ=-- (4) This relation states that the compressibility ,d i κ does not depend on the type i of the defect process. Thus, it ems reasonable to assume [32], [27] that the validity of (4) holds also for the compressibility d κ involved in (2), i.e.,
22111()[()1]d T d B dP dB dP κκ=-- (5) where the subscript <1> in the quantities at the right side denotes that they refer to the undoped diamond crystal.
3 APPLICATION OF THE MODEL TO THE BORON-DOPED DIAMOND In general, the quantities 1dB dP and 221d B dP , can be roughly estimated from the modified Born model according to [27], [31]:
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1(7)3B dB dP n =+ and 2211()(43)B B d B dP n =-+ (6) where B n is the usual Born exponent. In cas, however, where the Born model does not provide an adequate description, we can solely rely on (4), but not on (6). In other
