First principles study of structural, elastic, mechanical ...

First principles study of structural, elastic, mechanicaland electronic properties of nitrogen-doped cubic diamond

E GULER1,* , S UGUR2, M GULER1 and G UGUR2

1 Department of Physics, Ankara Hacı Bayram Veli University, 06900 Ankara, Turkey2 Department of Physics, Gazi University, 06500 Ankara, Turkey

*Author for correspondence ([email protected])

MS received 15 June 2020; accepted 20 July 2020

Abstract. We report the structural, elastic, mechanical and electronic properties of nitrogen (N)-doped cubic diamond

up to 25%N doping concentrations in the steps of 5%N dopant. Our calculations were performed with the generalized

gradient approximation functional of density functional theory with the Perdew–Burke–Ernzerhof exchange–correlation

energy through virtual crystal approximation. Cubic diamond shows a structural stability up to 15%N doping and it

becomes instable above this dopant concentration. The changes for the typical cubic elastic constants, bulk, shear and

Young’s moduli, Poisson ratio, anisotropy, Pugh ratio, Kleinman parameter and electronic band structures of cubic

diamond vs. applied doping percentages were also evaluated. The obtained results for these parameters were found to be

strictly dependent on the dopant concentration. Although cubic diamond is a well-known insulator, it displays a metallic

character even under the doping of 5%N and keeps this trend for higher doping concentrations.

Keywords. Diamond; doping; DFT; electronic; elastic; mechanical.

1. Introduction

As pointed out in our recent papers [1,2], cubic diamond has

outstanding physical properties like high thermal conduc-

tivity, low thermal expansion, high optical transparency,

ultra-hardness and good insulating capacity. So diamond is

one of the key materials for today’s technology as it covers

wide range of uses changing from high pressure Anvil Cell

experiments to several medical applications [3–6]. On the

other hand, when foreign atoms are introduced in a material,

the physical properties of the related material change

depending on the added impurities [7]. Besides, nitrogen

atom (N) is the simplest impurity, which dominates in most

of natural and artificial diamonds, mainly in the substitu-

tional position of carbon atom at concentrations\1021 cm3

[8,9]. In 1998, Kang [10] performed electronic band struc-

ture calculations within the framework of extended Huckel

tight binding for the properties of n-type impurities for

nitrogen and phosphorus in diamond. He concluded that the

calculated density of states shows the impurity level deep in

the bandgap. Further, Ivanova and Mavrina [8,9] analysed

the physical properties of nitrogen-doped cubic diamond

and evaluated their findings with those of pure diamond

through density functional theory (DFT) within supercell

method. They reported that nitrogen in substitution position

produces a sharp lattice deformation near the impurity atom

and hence lowers the hardness, elastic moduli and

anisotropy of the cubic diamond. As there is still a very

scant theoretical literature on the effect of N doping,

especially for the structural, elastic, mechanical and elec-

tronic properties of cubic diamond, this scarcity motivated

us to perform this research for aforementioned properties of

nitrogen (N)-doped cubic diamond with virtual crystal

approximation (VCA) method [11,12].

2. Computational details

In this study, we performed all the calculations with

CASTEP code [13,14], which allows the self-consistent

DFT calculations within a plane-wave pseudopotential

approach. During present calculations, we applied Perdew–

Burke–Ernzerhof (PBE) exchange–correlation procedure of

the generalized gradient approximation (GGA) functional

for the terms of electron–electron interaction after geometry

optimization of the surveyed crystal structures [15,16].

Further, the ion and electron interactions were carried out

by employing the Ultra soft Vanderbilt pseudopotential

scheme [17] and the electronic wavefunctions were treated

as plane waves with 500 eV cut-off energy. For Brillouin

zone sampling, we applied 12 9 12 9 12 Monkhorst-Pack

grids [18]. The electronic valence configurations for dia-

mond and nitrogen were C: 1s2 2s2 2p2 and N: 1s2 2s2 2p3,

respectively, through the VCA method [19]. This method is

Bull Mater Sci (2021) 44:1 � Indian Academy of Scienceshttps://doi.org/10.1007/s12034-020-02288-z Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)

much simpler and computationally less expensive approach

regarding the commonly used supercell method of crystal

structures. As well, VCA method allows the investigations

of mixed-type crystals by conserving the unit cell of the

initial material [19]. After checking the crystal structure of

the undoped diamond, we achieved various runs by

changing the concentration percentage of N starting from 5

to 25% in steps of 5%.

3. Results and discussion

C11, C12 and C44 referring the typical cubic elastic constants

describe the mechanical hardness of the cubic crystal and

they are required to determine the stability of a given

material. C11, C12 and C44 elastic constants obtained from

the total energy calculations signify the single-crystal

elastic characters, whereas Voigt–Reuss–Hill approach is a

convinced scheme for the elastic constants of polycrys-

talline materials [20–22].

To obtain the accurate values of elastic constants and

other related mechanical parameters of nitrogen-doped

cubic diamond, we considered the Voigt–Reuss–Hill values

for the typical cubic elastic constants during present cal-

culations. Table 1 lists the presently obtained elastic con-

stants of doped and undoped cubic diamond with former

experimental and theoretical data for the sake of compari-

son. Our results are both consistent with experimental data

of undoped cubic diamond and earlier theoretical data of

nitrogen-doped cubic diamond of refs. [8,9]. As well, C11

elastic constant exhibits a clear decrement under 5, 15 and

25% nitrogen doping and a significant increment under 10

and 25% nitrogen doping, as shown in figure 1. Thus, it is

easy to say that C11 elastic constant of cubic diamond,

which represents the longitudinal elastic behaviour of the

related crystal, displays different trends (decrement/incre-

ment) under different nitrogen doping percentages. Further,

the off diagonal elastic constant C12 of cubic diamond

indicates similar trend to C11 constant, as shown in figure 1,

since it increases under 5, 15 and 20% nitrogen doping and

decreases under 10 and 25% nitrogen doping. In addition,

the shear elastic constant C44 of cubic diamond never shows

an increment under nitrogen doping. From figure 1, it is

clear that C44 constants either decrease (5, 15 and 20%) or

keep a constant-like behaviour (as in between 5–10% and

20–25% nitrogen doping) under nitrogen doping.

According to well-known Born stability, cubic elastic

constants C11, C12 and C44 must prove the conditions: C11–

C12 [ 0, C11 [ 0, C44 [ 0, C11 ? 2C12[ 0 for structural

stability and C12\B\C11 for cubic stability [23,24]. As

another result, our calculated elastic constant values well

confirm both structural and cubic stability of diamond up to

15% nitrogen doping, where these conditions break-down

and structural instability begins after 15% nitrogen doping

as in 20 and 25% nitrogen-doped cubic diamond. So, we

can introduce that cubic diamond is found to be stable up to

Table 1. A comparison for some elastic, mechanical and other parameters of undoped and doped cubic diamond with former available

data.

Parameter

This study

Undoped [1] Exp* Undoped 5%N 10%N 15%N 20%N 25%N

C11 (GPa) 1074.8 1079 1046.2 587.1 918 507.4 138.6 420.1

C12 (GPa) 125.6 124 119.4 352.1 188.9 396.1 570.5 417.1

C44 (GPa) 720.6 578 560.2 346.4 362 107.2 -179.1 -187.6

B (GPa) 442 442 428.3 430.4 431.9 433.2 426.5 418.1

G (GPa) 609.5 535 519.2 224.8 363 82.3 -193 -54.1

E (GPa) 1048.5 — 1109.4 574.4 850.8 232.4 -61.8 -169.7

Eg (eV) — 5.5 [31] 4.1 Metallic Metallic Metallic Metallic Metallic

m 0.1 0.1 0.06 0.2 0.1 0.4 0.7 0.5

f 0.26 0.26 0.26 0.69 0.35 0.84 2.22 0.99

*All experimental values were retrieved from ref. [1] and references there in.

Figure 1. Variation of the typical cubic elastic constants (Cij) of

cubic diamond under different nitrogen dopants.

1 of 7 Bull Mater Sci (2021) 44:1

15% nitrogen doping concentration. In other words, nitrogen

doping of cubic diamond above 15% will promote the

instability of cubic diamond.

Bulk modulus (B) implies much information about the

bonding strength in materials and can be explained as the

resistance of a given material to external deformations

[25–27]. Figure 2 displays three elastic moduli (B, G and

E) of nitrogen-doped cubic diamond for the investigated

entire % nitrogen dopant. Meanwhile, the shear modulus

(G) describes the resistance of a given material subjected to

a shearing force [28]. Further, Young’s modulus (E), can be

described as the resistance of materials under uniaxial ten-

sions and provides the materials stiffness degree, i.e., the

higher the value of E, the stiffer is the material [29]. Again,

from table 1, B, G and E values of undoped cubic diamond

obtained during this study are satisfactory when compared

with the experimental values. On the other hand, in figure 2,

under nitrogen doping, B shows a constant-like behaviour

where both G and E moduli increase or decrease depending

on the applied nitrogen concentration.

Since ductility and brittleness are two important adjec-

tives for materials production, we also surveyed the ductile

(brittle) behaviour of cubic diamond under nitrogen doping.

Brittle and ductile adjectives symbolize the two distinct

mechanical performances of solid materials when they are

exposed to external stress. Mostly, deformation does not

much affect the brittle materials and they are found to be

less deformable before fracture. Oppositely, ductile mate-

rials are accepted to be much deformable before fracture

[1,26]. At this decision point, Pugh ratio analysis is a sig-

nificant threshold for determining the ductile/brittle nature

of materials. According to Pugh, if B/G ratio is about 1.75

and higher, the material is considered to be ductile, other-

wise the material is accepted to be as brittle [30]. Figure 3

shows the B/G ratios of nitrogen-doped cubic diamond. As

seen in figure 3, cubic diamond shows a ductile character

after 5 and 15% nitrogen doping, where it behaves as a

Figure 2. Bulk modulus (B), shear modulus (G) and Young’s

modulus (E) variation of cubic diamond up to 25% nitrogen

doping.

Figure 3. Pugh ratio (B/G) analysis of cubic diamond under

nitrogen doping.

Figure 4. Poisson ratio alteration of cubic diamond vs. the

nitrogen doping.

Figure 5. Kleinman parameter of cubic diamond against the

nitrogen doping.

Bull Mater Sci (2021) 44:1 of 7 1

brittle material for other nitrogen dopant percentages (i.e.,

10%N, 20%N and 25%N).

Poisson ratio (m) represents the ratio between the trans-

verse strain (et) and longitudinal strain (el) in the elastic

loading direction of the regarding material. It also imparts

comprehensive knowledge about the bonding force beha-

viour in solids [1,20–24]. For Poisson ratio (m), the values

m = 0.25 and 0.5 denote the lower upper limits of central

forces of solids, respectively. Poisson ratio of cubic dia-

mond begins with 0.06 and increases to 0.27 at 5% nitrogen

doping, as shown in figure 4. It shows a decrease to 0.17 at

10% nitrogen doping and again increase to 0.41 at 15%

nitrogen dopant and reaches 0.76 at 20% nitrogen dopant,

where these results strongly indicate that the interatomic

forces in nitrogen-doped cubic diamond are mainly central

forces.

Kleinman parameter (f), formulated with f = (C11 ? 8C12)/

(7C11 ? 2C12) explains the relative ease of bond bending to

the bond stretching in cubic materials [1,23,28]. Minimizing

bond bending leads to f = 0, where minimizing bond

stretching leads to f = 1. Figure 5 shows the Kleinman

parameter of nitrogen-doped cubic diamond vs. nitrogen

Figure 6. Electronic band structure and corresponding TDOS curve of undoped cubic diamond with typical insulating nature

with the bandgap energy of Eg = 4.118 eV.

Figure 7. Electronic band structure and corresponding TDOS curve of cubic diamond under 5% nitrogen doping showing a

metallic bandgap.


concentration. Since it is naturally dependent on the cubic

elastic constants C11 and C12, Kleinman parameter shows a

similar behaviour like C11 and C12 under nitrogen doping.

Depending on the amount of applied nitrogen dopant,

Kleinman parameter may either increase (5, 15 and 20%) or

decrease (10 and 25%). As listed in table 1, obtained results

of Kleinman parameter for the undoped diamond correspond

well both experimental and theoretical data of former studies.

In addition, before the structural instability (at 15%) its value

is found to be as 0.84, which implies that the bond stretching

takes place in the nitrogen-doped cubic diamond before it

becomes instable.

Figure 6 shows the electronic band structure of the

undoped cubic diamond with its corresponding total density

of states (TDOS) curve. As is obvious from figure 6, the

undoped cubic diamond exhibits an evident insulating

character with a typical direct band having gap energy of

Eg = 4.1 eV. Of course, this value is slightly lower than the

experimental value of bandgap energy of cubic diamond

with Eg = 5.5 eV [31] due to well-known underestimating

limitations of presently applied GGA approach of DFT [32].

However, it is also interesting to note here that the insu-

lating undoped cubic diamond becomes metallic even under

5%N doping, as in figure 7, and this attitude on the

Figure 8. Electronic band structure and corresponding TDOS curve of cubic diamond under 15% nitrogen doping when it

becomes instable.

Figure 9. Electronic band structure and corresponding TDOS curve of cubic diamond under 25% nitrogen doping with

metallic character.


electronic band structure of cubic diamond keeps going on

also under 15%N doping (figure 8) and 25%N doping

(figure 9). This behaviour of cubic diamond can be attrib-

uted to nitrogen doping, where Fermi energy (EF) shows a

clear increment under increasing dopant concentration

unlike the boron doping, in which EF shows a decreasing

tendency via rising boron dopants in carbon materials like

graphene [33–35]. Figure 10 indicates the presently

obtained increasing Fermi energies of nitrogen-doped cubic

diamond vs. increasing nitrogen dopant, contributing to the

electrical conductivity (r). To get more information about

the electrical conductivity of nitrogen-doped cubic dia-

mond, we also plotted the electrical conductivity of cubic

diamond at 300 K under rising nitrogen concentrations by

using the BoltzTrap code [36] during our computations. As

in the ultra-nano crystalline diamond thin films [37], we

observed that the electrical conductivity of cubic diamond

clearly increases with increase in nitrogen-dopant concen-

trations, as in figure 11, due to effects of aforementioned

nitrogen dopant, where s represents the electron relaxation

time [38–40].

4. Conclusion

The structural, elastic, mechanical and electronic properties

of nitrogen (N)-doped cubic diamond up to 25%N doping

concentrations were investigated within the GGA functional

of DFT with PBE exchange–correlation energy through

VCA. Cubic diamond shows a structural stability up to

15%N doping and it becomes instable above this dopant

concentration. The changes for the typical cubic elastic

constants, bulk, shear and Young’s moduli, Poisson ratio,

Pugh ratio, Kleinman parameter of cubic diamond vs. applied

doping percentages were also considered. The obtained

results for these parameters were found to be strictly

dependent on the nitrogen-dopant concentration. Although

undoped cubic diamond is a well-known insulator, it displays

a metallic character even under the doping of 5%N and

sustains this trend for more doping concentrations.

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