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Substitutional doping of Cu in diamond: Mott physics with p orbitals

H. Hassanian Arefi,1 S. A. Jafari∗,2, 3 and M. R. Abolhassani1

1Department of Physics, Tarbiat Modarres University, Tehran, Iran2Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran

3School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran(Dated: October 7, 2013)

Discovery of superconductivity in the impurity band formedby heavy doping of boron into diamond (C:B) aswell as doping of boron into silicon (Si:B) has provided a rout for the possibility of new families of supercon-ducting materials. Motivated by the special role played by copper atoms in high temperature superconductingmaterials where essentially Cud orbitals are responsible for a variety of correlation induced phases, in this paperwe investigate the effect of substitutional doping of Cu into diamond. Our extensive first principle calculationsaveraged over various geometries based on density functional theory, indicates the formation of a mid-gap band,which mainly arises from thet2g and4p states of Cu. For impurity concentrations of more than∼ 1%, theeffect of2p bands of neighboring carbon atoms can be ignored. Based on our detailed analysis, we suggest atwo band model for the mid-gap states consisting of a quarter-filled hole liket2g band, and a half-filled band of4p states. Increasing the concentration of the Cu impurity beyond∼ 5%, completely closes the spectral gap ofthe host diamond.

PACS numbers: 71.55.-i, 71.23.-k, 78.30.Am

INTRODUCTION

Diamond is the best thermal conductor [1, 2], and the hard-est known material with semiconducting properties. Bothpristine form as well as doped films of diamond presentunique ground for high-temperature and high-power applica-tions in the industries [2–4]. Recent discovery of supercon-ductivity in heavily boron doped diamond, has revived the in-terest in impurity band formation from a fundamental pointof view [5, 6]. Similar system of boron doped silicon wasalso studied [7–9]. Synthesis of boron doped diamond wasfirst achieved by Ekimov and coworkers [5] and subsequentlyby Takanoet. al. [6]. They used chemical vapor deposition(CVD) for the synthesis of B-doped diamond. At low dopingconcentrations of typically1017−1018 cm−3, boron atoms in-ject acceptor levels in the electronic energy spectrum of hostdiamond, hence producing ap type semiconductor [10]. Forheavier dopings∼[B]/[C]> 5000 ppm in the gas phase, oneachieves metallic conductivity in diamond [11]. Eventuallyincreasing the doping beyondn > 1021 cm−3, i.e. few per-cents, it superconducts at low temperatures [5, 6]. For higherconcentrations, the dopant atoms come closer to each otherand start to overlap. Therefore the impurity levels are broad-ened into authentic impurity bands [12, 13], which are respon-sible for metallic and superconducting properties [8, 13–18].

Various forms of defects play crucial role in the physicalproperties of diamond. For example the fascinating colorsof diamond as gem stone, is due to the∼ppm concentrationof vacancies [19]. Such vacancies at much higher concentra-tion of the order of few percents, can give rise to an impuritybands inside the diamond gap, which can provide larger num-ber of states than some of the elements from the third and

∗Electronic address: akbar.jafari@gmail.com

fifth column of the periodic table of elements [20]. Such mid-gap bands of the vacant diamond can be attributed to a local-ized Wannier wave function composed from the2p states ofneighboring carbon atoms. Therefore, previous studies of thepossible conductive impurity bands in diamond and/or siliconhosts, have been mainly concerned with the substitution of ap-type orbital for the hostp-states.

In this paper, we are interested in the role of substitution bya 3d element. We choose to study the effect of copper sub-stitution in the diamond host. Why is Cu interesting from theimpurity band formation point of view? First of all, the samehigh-temperature, high-pressure techniques employed forthesynthesis of boron doped diamond can also be applied to at-tempt a substitution of Cu atoms into the diamond host [21].From theoretical point of view, although Cu is a3d transitionmetal, it does not have magnetic complications, and hence onecan avoid possible magnetic ground states at low temperatureswhich tend to compete against superconducting correlations.Choice of Cu also avoids the possibility of local moment for-mation. On the other hand3d9 ”atomic” configuration of Cuis hoped to provide a hole which plays an important role inthe high temperature superconducting (HTSC) materials [22].The nominal3d hole of atomic Cu in HTSC systems is sharedbetween neighboring oxygen2p and Cu3d states, giving riseto the so called Zhang-Rice singlet state [23] which carriesthehole.

In this paper, we study the formation of impurity bands inCu doped diamond in detail and we find that4p states of Cualong with itst2g orbitals are responsible for the formation ofimpurity band at Cu concentrations1− 5%. At lower concen-trations2p orbitals of neighboring carbon atoms also partic-ipate in the formation of mid-gap metallic band. Dependingon the experimental conditions, either a combination of mostprobable geometries is realized, or by slow cooling the systemwill be given enough time to perform a self-averaging over thephase space of possible geometries. Therefore we undertake

2

extensive averaging over distinct geometries. The main role ofaveraging is to restore the symmetries broken by the specificrealization, e.g. most probable geometry (MPG). For com-parison, we will provide MPG and geometry averaged (GA)results together.

METHOD OF CALCULATION

The calculations of electronic structure are performed withthe plane-wave pseudopotential [24] of DFT using the PBEfunctional [25] for exchange and correlation as implementedin the QUANTUM-ESPRESSO package. Ultra-soft pseu-dopotentials are used with a30 Ry (230 Ry) cutoff for theexpansion of the wave-functions (charge density). We haveconstructed2× 2× 2 supercell containing64 atoms with1, 2and3 Cu impurity, leading to doping concentrations of1.6%,3.1% and4.7%, respectively. The sampling of Brillouin zone(BZ) for structural relaxation and DOS calculations are per-formed on2×2×2 and4×4×4 grid ofk-points, respectively.We first minimized the energy of undoped diamond with re-spect to cell size to calculate the lattice constant. The obtainedvalue was very close to the experimental result. All structureswere relaxed by BFGS algorithm [26].

Two types of doping can be considered: substitutional andinterstitial. In the case of B-doped diamond, interstitialdop-ing is unstable [27]. Since our study is motivated by B-doped diamond, we only consider substitutional doping ofcopper. We randomly substituted Cu impurity into the dia-mond structure. Since there are many different ways of in-troducing impurity into the host carbon lattice, one needs toaverage over sufficient number of realizations of randomness.With m impurities in a host ofn lattice points, there areC(n,m) = n!

(n−m)!m! different ways of substitution. Withn = 64 carbon atoms considered here, the most trivial casecorresponds tom = 1 impurity. By translational symmetry(periodic boundary conditions), it does not matter where weplace the single impurity. Therefore the average DOS cor-responding to1 impurity, is identical to any of the randomrealizations. Next in the case ofm = 2 impurities, there arein principleC(64, 2) = 2016 ways of placing the Cu impuri-ties. However, since therelative distance between the impuri-ties is the physically discriminating factor, many of the aboveconfigurations will be essentially identical by translation androtation symmetries. Therefore the electronic structure calcu-lations must be run for geometrically different configurations.We have used a computer program to count the number of ge-ometrically independent configurations which in the case ofm = 2 impurities gives23. Therefore one has to perform theelectronic structure for all23 distinct geometries, taking intoaccount the multiplicitywg of each geometryg. In the caseof m = 3 impurities, we haveC(64, 3) = 41664 nominallydifferent configurations. Again our counting algorithm gives430 distinct geometries, where each geometry is meant to rep-resents a distinct set of3 mutual distances between the impu-rity atoms. We used60 most probable geometries to calculate

0

30

60

12 15 18

Energy (eV)

1.6%

30

60

Den

sity

of S

tate

s(ar

b U

nit)

3.1%

30

604.7%

GA

12 15 18

Energy (eV)

MPG

Doped DiamondPerfect Diamond

EF

FIG. 1: (Color online) Total density of states for three consideredconcentrations contrasted against pure diamond in background. Leftand right panels correspond to geometry averaged and most probablegeometries, respectively.

the averages. Note that the above60 geometries cover morethan half of the total41664 choices. We have checked that,the average DOS obtained for60 most probable geometriescan be more or less achieved even with10 most likely geome-tries. The geometry averaged values of various quantitiesXsuch asρ(ω), EF can be calculated as follows:

XGA =∑

g

wgXg, (1)

whereg runs over geometries, andwg is its relative weight.

METALIZATION OF DIAMOND

In Fig. 1 we have plotted the DOS of doped diamond latticewith (left) and without (right) averaging over geometries.Thedoping rates in the figure correspond to1, 2, 3 Cu impuritiesin a host of64 carbon atoms, from bottom to top, respectively.The presence of Cu impurities, affects the entire spectrum (notshown here). In this figure we have focused on the region ofspectrum corresponding to gap in the pristine diamond. Byincreasing the doping rate from1.6 to 4.7 percent, the devia-tion in the DOS of Cu-substituted diamond from the DOS ofperfect diamond becomes more and more significant.

The most spectacular effect of substituting Cu in the in-sulating host of diamond, is the formation of mid-gap impu-rity band. In the right panel of Fig. 1, DOS corresponding to

3

0

2

4

5 10 15 20

Energy (eV)

1.6%

2

4

Loca

l Den

sity

of S

tate

s(ar

b U

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3.1%

2

4

4.7%

GA

5 10 15 20

Energy (eV)

MPG

d LDOSp LDOSs LDOS

EF

FIG. 2: (Color online) LDOS fors, p andd orbitals of Cu corre-sponding to three concentrations considered. Fermi levelEF is in-dicated by vertical dashed line. MPG values on the right panel aregiven for comparison. As can be seen the contribution of4s orbitalsto the impurity band is negligible.

MPG is shown. By averaging over different geometries, thecorresponding geometry averaged DOS in the left panel willbe obtained. As can be seen in the figure, for both GA andMPG, the width of the impurity band increases with increas-ing the Cu concentration. At4.7% Cu substitution, the gapcompletely disappears. The averaged Fermi energy,EF is in-dicated with vertical dashed line. As it is evident, althoughthe most probable geometry in the case of2/64 impurity con-centration gives a little gap in the spectrum, other geometrieswith high probability will fill in that small gap and the result-ing GA density of states will be a metallic band. This indicatesthat the best known insulator, namely diamond, can be metal-ized by few percent Cu substitution. The question we wouldlike to investigate in detail in this paper is, what is the originof the metallic band?

Role of Cu atoms

To consider this question, we first examine the occupationof various orbitals. We find that unlike the atomic configura-tion 4s23d9 of Cu, in the presence of carbon neighbors, theoccupancy of4s orbitals is changed as4s2 → 4s0.5. Themissing1.5 electron changes the3d9 → 3d9.5, 4p0 → 4p1.Hence4p orbitals are pushed down to acquire occupancy of

0

0.5

1

1.5

9 12 15

Energy (eV)

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1.5

Par

tial D

OS

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Cu

d-O

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ls (

arb

Uni

t)

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0.5

1

1.5

4.7%

GA

9 12 15

Energy (eV)

MPG

dz2

dxzdyzdxy

dx2-y2

EF

FIG. 3: (Color online) Copperd-orbital splitting for three consideredconcentrations in diamond crystalline field. The major contributionto the impurity band comes fromt2g subspace ofd-orbitals.

∼ 1. We find that the above Lowdin charges are more or lessindependent of the geometry, as well as concentration of Cuimpurities. This can be qualitatively seen in Fig. 2, where lo-cal density of states (LDOS) fors, p andd orbitals of Cu areplotted. Therefore the relevant orbitals of Cu are both4p, aswell as a subspace of3d states.

To further identify which subspace of Cud−orbitals con-tribute to impurity band, in Fig. 3 we resolve the contributionof t2g andeg sub-bands. In the case of one Cu atom, the impu-rity atom feels a perfect diamond environment; thereby givingthree-fold degeneratet2g band and two-foldeg bands. Notethateg bands are located at lower energies with respect tot2gbands. By increasing the number of Cu atoms, the above crys-talline degeneracies start to be lifted, as the crystallineenvi-ronment at higher Cu concentrations will deviate from perfectdiamond symmetry. At the same time the average distancebetween Cu atoms decreases. Especially in the case of threeCu atoms, diamond crystalline environment is totally lost inthe DOS of MPG. Even after averaging over geometries, sub-stantial degeneracy lifting in eacht2g andeg sub-bands can beobserved. The important result from Fig. 3 is that in all threeconsidered concentrations, and for both MPG and GA situ-ations, onlyt2g sub-band of thed-orbitals contribute to theimpurity band, while theeg sub-band will give energy levelsinside the valence band.

4

0

2

4

2 6 10 14 18 22

Energy (eV)

1.6%

2

4

Loca

l Den

sity

of S

tate

s(ar

b U

nit)

3.1%

2

4

4.7%

GA

6 10 14 18 22

Energy (eV)

MPG

Cu LDOSFirst

SecondEF

FIG. 4: (Color online) Spatially resolved DOS at the Cu site.Rightpanel corresponds to MPG, and the left panel includes averaging overgeometries. At high concentrations the dominant contribution comesfrom Cu atom itself, while at lower concentration nearest neighborcarbon atoms have substantial contribution to the impurityband.

Role of neighboring carbon atoms

Now let us concentrate on the effect of neighboring carbonatoms on the formation of the impurity band. In Fig. 4 weplot DOS corresponding to three concentrations for MPG andGA. In terms of the relative weights of neighbors, both MPGand GA give similar results. Note that for one Cu impurity,MPG and GA quantities must always be the same. By movingfrom bottom to top (increasing the concentration) in Fig. 4 onenotices that only at low concentration nearest neighbor (NN)carbon atoms contribute significantly. By increasing the con-centration, contribution of NN carbon atoms compared to theCu atoms is reduced and becomes as low as the next nearestneighbor (NNN) contributions. This can be attributed to thefact that at higher Cu concentration, average Cu-Cu distanceis reduced and the Cu-Cu hopping matrix elements becomedominant. At lower concentrations in addition to Cu-Cu hop-pings, there is also substantial hopping amplitude betweenthedangling orbitals of neighboring carbon atoms. This can bethought of hopping between appropriate Wannier orbitals lo-calized around Cu atoms which is composed of the dangling2p orbitals of neighboring carbon atoms.

Based on a naivesp3 picture of bonding in perfect diamond,one expects2s and2p orbitals of neighboring carbon atomsto have comparable contribution to the formation of impuritybands. However, Cu substitution drastically changes this pic-

0

0.5

1

12 15 18

Energy (eV)

1.6%

0.25

0.5

Loca

l Den

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tate

s(ar

b U

nit)

3.1%

0.25

0.5

4.7%

GA

12 15 18

Energy (eV)

MPG

s-NNp-NN

s-NNNp-NNN

EF

FIG. 5: (Color online) Contribution of first and second neighbors(NN and NNN) to the DOS of the impurity band. Right panel corre-spond to MPG and the left panel includes averaging over geometries.As can be seens orbitals of neighboring carbon atoms do not con-tribute to the impurity band. The relative contribution of neighboringp orbitals to the Cu atomic orbitals become significant in low concen-tration limit.

ture. As can be seen in Fig. 5, in both MPG and GA situationss orbitals are irrelevant and the contribution of NN and NNNcarbon atoms comes from their2p states. These contributionsbecome comparable to local DOS arising from Cu atoms onlyin low Cu concentration limit. When comparing Figs. 4, 5one has to note the scales. Note that 1/64 concentration canalso be reproduced in the simulation with two Cu atoms inhost supercell of128 carbon sites. Therefore the growth inthe relative weight of neighboringp orbitals can be partly dueto finite size artifacts of the simulation, which of course isex-pected to be minimal with the periodic boundary conditionsemployed in the calculations. Therefore we expect the neigh-boring atoms to have significant2p orbital contributions forconcentrations well below∼ 1%. At higher concentrations,the contribution of neighboring atoms becomes less and lesssignificant. A comparison of MPG and GA density of states inFig. 5 indicates that the role of averaging is to slightly increasethe contribution ofp-orbitals which is essentially a statisticaleffect.

5

DISCUSSION AND CONCLUSION

Examining the bond lengths show that Cu-C bond is elon-gated by approximately0.2 relative to C-C bonds in undopedstructure which has been almost compensated by a similar de-crease in the surrounding C-C bond lengths. C-C bonds faraway from Cu undergo negligible changes of the order0.01A.Therefore in terms of structural changes, Cu substitution willgive rise to local effects. In terms of electronic properties, inthe regime of∼ 5% Cu substitution, the most significant pro-cess behind the formation of impurity band, is the hoppingbetween Cu atoms themselves. In this regime a minimal two-band effective model can describe the electronic states nearthe Fermi energy:

H = tdd∑

〈i,j〉σ

(

d†iσdjσ + h.c.)

− tpp∑

〈i,j〉σ

(

p†iσpjσ + h.c.)

+tpd∑

〈i,j〉σ

(

d†iσpjσ + h.c.)

, (2)

where the hopping matrix elementstdd, tpp > 0 are betweenthe t2g hole bands and4p electron bands of the adjacent Cuatoms residing at random sitesi, j. If the inter-band ma-trix elementtpd is negligible, one would have a quarter filledband of3d holes, along with a half-filled4p band constitut-ing the impurity band. In this limit for large enough Coulombcorrelations among the Wannier orbitals derived from parent4p states, the electron-like band undergoes a Mott transition,while the 1/4-filled hole bands derived fromt2g states can besubject to various forms of density waves [28]. Theoreticallyspeaking, in the opposite limit whentpd is large, the origi-nal half-filled electron band and the quarter filled hole band,get mixed giving rise to an effective quarter filled band whoseelectrons have a mixed character,

c†iσ =d†iσ + p†iσ√

2. (3)

This can be thought of as analogue of Zhang-Rice singletstate [23]. Although this limit seems to be hard to realizein the experiment, but from theoretical point of view it mayprove useful in studying transition from two-band situation toeffective one-band model. In the intermediate regime where|tpd| ∼ tpp, tdd, copper doped diamond will offer an op-portunity to study both theoretically and experimentally thecompetition between the instabilities of a half-filled electronicband [29] with that of a quarter-filled hole band [28]. Ourmodel Hamiltonian (2) defines the competition between in-stabilities of two bands in 3D. The randomness in the matrixelements can not be avoided, as there is no control over the lo-cation of Cu impurities. Robustness of Lowdin charges in var-ious geometries imposed the restrictiontdd, tpp > 0 for study-ing Anderson localization effects in such 3D bands. The factthat intpd ≪ tpp, tdd limit electronic bands of4p orbitals arehalf-filled, suggests the possibility of Mott-Anderson physicsin 4p orbitals.

To summarize, from both experimental and theoreticalpoint of view, impurity band formation in Cu/C system pro-vides interesting ground to study the orbital selective phenom-ena and novel forms of Anderson localization properties in acombination of electron and hole like bands with different fill-ings in 3D.

ACKNOWLEDGEMENT

We thank Dr. M. Alaei for fruitful discussions. H.H.F.would likes to thank R. Sarparast, K. Haghighi and M. Hafezfor assistance in computing. S.A.J. was supported by the ViceChancellor for Research Affairs of the Isfahan University ofTechnology (IUT) and the National Elite Foundation (NEF)of Iran.

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