Atomistic simulation techniques have been employed in order to investigate key issues related to intrinsic defects and a variety of dopants from trivalent and tetravalent ions. The most favorable intrinsic defect is determined to be a scheme involving calcium and hydroxyl vacancies. It is found that trivalent ions have an energetic preference for the Ca site, while tetravalent ions can enter P sites. Charge compensation is predicted to occur basically via three schemes. In general, the charge compensation via the formation of calcium vacancies is more favorable. Trivalent dopant ions are more stable than tetravalent dopants.
Hydroxyapatite (
Many works have investigated the substitution of extrinsic dopant in hydroxyapatite (HAP) in order to improve the material properties [
A systematic investigation of cation substitution in HAP may provide a better understanding of their properties, which is still lacking. Some theoretical studies based on first principle calculations for HAP materials have been realized in order to elucidate some basic physics [
For Eudoped HAP, for example, an investigation of the structural response is essential due to the intimate relationship between its properties and lattice distortions caused by the Eu substitution and the charge compensation involved in the Eu emission, which are associated with the crystal field [
In the present work, atomistic simulation based on the lattice energy minimization is used to provide useful information on the energetically favoured intrinsic defects and a variety of dopants from trivalent and tetravalent ions. Specifically, trivalent rare earth ions and some trivalent and tetravalent transition metals were considered due to their influence on various properties [
The simulation techniques used in this work are based on energy minimization, with interactions represented by interatomic potentials. Interactions between ions in the solid are represented in terms of a longrange Coulomb term plus a shortrange term, as described by Buckingham potential, that accounts for electron cloud overlap Pauli repulsion and dispersion (Van der Waals) interactions. Polarizability of the oxygen ions is incorporated by means of the DickOverhauser shell model [
Suitable potential parameters are highly necessary in order to give a good description of the material by atomistic simulation. The refitted potential parameters used in this work are listed in Table
Interatomic potentials parameters for hydroxyapatite.
Buckingham 




H_{2 core}–O_{1 shell}  311.97  0.2500  0.00 

850.00  0.3316  0.00 

1288.00  0.3334  0.00 
P_{core}–O_{2 core}  518.47  0.3510  0.00 
P_{core}–O_{1 shell}  914.00  0.3380  0.00 
O_{2 core}–O_{2 core}  22764.0  0.1490  0.00 
O_{1 shell}–O_{2 core}  22764.0  0.1490  13.94 
O_{1 shell}–O_{1 shell}  22764.0  0.1490  32.58 


Morse 





O_{2 core}–H_{2 core}  7.0525  3.1749  0.9485 


Spring 




O_{1 core}–O_{1 core}  98.67  


Cargas ( 



O_{1 core}  0.86 

2.0 
O_{1 shell}  −2.86  P_{core}  5.0 
O_{2 core}  −1.426  H_{core}  0.426 
Lattice parameters (
Lattice parameters  






 
Present work  9.419  9.419  6.874  90.0  90.0  120.0 
Tanaka et al. (Exp.) [ 
9.419  9.419  6.881  90.0  90.0  120.0 
Mostafa and Brown (Theo.) [ 
9.412  9.412  6.853  90.0  90.0  120.0 
de Leeuw (Theo.) [ 
9.563  9.563  6.832  90.0  90.0  120.0 
Lee et al. (Theo.) [ 
9.528  9.528  6.607  90.0  90.0  120.0 
Rabone and de Leeuw (Theo.) [ 
9.350  9.350  6.860  90.0  90.0  120.0 
Hauptmann et al. (Theo.) [ 
9.455  9.455  6.901  90.0  90.0  120.0 
Calculated and experimental elastic constants of hydroxyapatite.
Elastic constants (GPa)  






Bulk modulus  
Present work  158.58  57.10  58.90  142.11  43.70  89.72 
Hughes et al. (Exp.) [ 
166.7  13.96  66.3  139.6  —  84.6 
Katz and Ukraincik (Exp.) [ 
137.0  42.50  54.90  172.0  39.60  82.60 
Pedone et al. (Theo.) [ 
157.5  —  59.7  147.3  43.9  90.66 
de Leeuw et al. (Theo.) [ 
134.4  48.9  68.5  184.7  51.4  90.0 
Snyders et al. (Theo.) [ 
117.1  26.2  55.6  231.8  56.4  76.0 
To determine Frenkel and Schottky type defect formation energies, isolated point defect (vacancy and interstitial) energies and relevant lattice energies were first calculated. Several possible positions were tested to confirm the optimal position of the interstitial site for defect occupancy, and the positions which have the lowest energy were taken for interstitial cations and oxygen, respectively. We also calculated antisite pair defects, which involve the exchange of a cation with another cation of different species. In addition, two other intrinsic defect schemes are considered. Scheme (i) involves one calcium vacancy and two hydroxyl vacancies, and scheme (ii) involves five calcium vacancies and two phosphorus interstitials. These defects are expressed by KrögerVink notation and are shown in Table
Solution energy of intrinsic disorder in hydroxyapatite (eV/defect).
Type  Site  Defect equation  Solution energy 

Frenkel  Ca 

3.75 
P 

16.38  
O 

4.46  
OH 

6.99  


Schottky  Total 

4.77 
Ca 

3.14  
P 

8.01  


AntiSchottky  Total 

7.65 


Schemes (i)  Ca 

−0.71 
Schemes (ii)  Ca 

38.62 
In the doping process, a description of the favorable substitution site and the charge compensation mechanism is very important information. From atomistic simulation it is possible to obtain quantitative estimates of the relative energies of different modes of dopant substitution. For this, trivalent (
Trivalent dopants
Tetravalent dopants
In (
The solution energies for the trivalent defect were calculated by combining the appropriate defect and lattice energy terms and are listed in Tables
Solution energies (eV) for substitution of the transition metals (M^{3+} = Fe, Mn, Cr, and Sc) and rare earth (M^{3+} = Lu and Yb) ions in HAP.
Mechanisms  Dopant  

Fe^{3+}  Mn^{3+}  Cr^{3+}  Sc^{3+}  Lu^{3+}  Yb^{3+}  
Equation ( 

4.04  4.06  4.07  3.28  3.01  2.95 

7.63  7.11  5.20  3.68  1.49  3.65  
Equation ( 





2.12 


6.75  6.23  4.32  2.79 

3.65  
Equation ( 

6.01  6.04  6.05  5.47  4.71  4.64 

10.50  9.84  7.46  5.96  2.79  5.52  
Equation ( 

8.20  8.22  8.23  7.63  2.62  2.57 

10.90  10.50  9.07  7.93  1.46  3.10  
Equation ( 

7.46  7.01  6.40  6.60  7.71  8.77 
Equation ( 

6.85  6.39  5.79  5.98  7.09  8.15 
Equation ( 

16.29  15.74  15.02  15.55  16.58  17.86 

21.34  20.79  20.07  20.60  10.10  11.37 
Solution energies (eV) for substitution of the rare earth (M^{3+} = Tm, Er, Ho, Dy, Tb, and Gd) ions in HAP.
Mechanisms  Dopant  

Tm^{3+}  Er^{3+}  Ho^{3+}  Dy^{3+}  Tb^{3+}  Gd^{3+}  
Equation ( 

3.06  2.91  2.85  2.81  2.60  2.72 

3.31  3.31  3.05  2.99  2.74  1.56  
Equation ( 


2.17 



1.80 

3.31  3.31  3.05  2.99  2.74  1.56  
Equation ( 

4.78  4.59  4.52  4.47  4.20  4.35 

5.09  2.99  4.77  4.70  4.38  2.90  
Equation ( 

2.65  2.54  2.50  2.47  2.31  2.40 

2.84 

2.65  2.61  2.42 


Equation ( 

8.90  8.69  8.98  8.78  8.55  8.90 
Equation ( 

8.30  8.07  8.36  8.17  7.93  8.29 
Equation ( 

18.03  17.76  18.11  17.87  17.60  18.02 

11.55  11.30  11.62  11.39  11.11  11.54 
Solution energies (eV) for substitution of the rare earth (M^{3+} = Eu, Sm, Nd, Pr, Ce, and La) ions in HAP.
Mechanisms  Dopant  

Eu^{3+}  Sm^{3+}  Nd^{3+}  Pr^{3+}  Ce^{3+}  La^{3+}  
Equation ( 

2.67  2.64  2.59  2.55  2.80  2.08 

1.52 


2.32  2.62  2.52  
Equation ( 

1.79  1.75  1.70 




1.52  1.42  1.29  2.32  2.62  2.52  
Equation ( 

4.29  4.25  4.19  4.15  3.93  3.56 

2.85  2.73  2.56  3.86  4.23  4.10  
Equation ( 

2.36  2.34  2.30  2.28  2.14  1.92 


1.43  1.33  2.10  2.32  2.25  
Equation ( 

9.12  9.04  8.93  8.66  3.88  9.34 
Equation ( 

8.51  8.42  8.32  8.05  3.27  8.72 
Equation ( 

18.29  18.18  18.05  17.73  12.00  18.54 

11.80  11.70  11.57  11.25  5.51  12.06 
In the Eudoped HAP, the
The charge compensation tends to occur basically via three schemes. In general, charge compensation via the formation of calcium vacancies is more favorable for charge compensation in the trivalent dopants, except for Er, Gd, and Eu, where the charge compensation is more favorable through interstitial hydroxyl than by calcium vacancies, and for Sm and Nd, where charge compensation by interstitial oxygen is more favorable. The charge compensation for Eu ions is in agreement with that proposed by Martin et al. [
The production of stoichiometric HAP is obtained with a Ca/P ratio of 1.67. Experimental work realized by Mayer et al. [
In Table
Solution energies (eV) for different dopants in HAP.
Mechanisms  Dopant  

Mn^{4+}  Cr^{4+}  
Equation ( 

6.71  6.96 

7.97  9.35  
Equation ( 

5.39  5.64 

6.65  8.02  
Equation ( 

11.22  11.58 

13.02  14.99  
Equation ( 

4.95  5.34 

5.51  6.71  
Equation ( 

4.28  5.75 
Equation ( 



Equation ( 

9.41  11.06 

5.35  7.03 
The basic defect chemistry including intrinsic defects and trivalent and tetravalent extrinsic dopants has been investigated using atomistic simulation. The most favorable intrinsic defect schemes are formed by one calcium vacancy and two hydroxyl vacancies. Trivalent and tetravalent dopant ion substitutions are found to take place preferentially on the Ca and P sites, respectively. In both cases, calcium vacancy defect formation is the more likely charge compensation mechanism for most of the trivalent and tetravalent dopants. In some trivalent ions, charge compensation by interstitial hydroxyl and interstitial oxygen is more favorable. The favorable doping process for these polyvalent dopants should be an
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to acknowledge the financial support by FINEP, CAPES, and CNPq.