Superconductivity in heavily vacant diamond
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Superconductivity in heavily vacant diamond
M. Alaei , S. Akbar Jafari and H. Akbarzadeh Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran (Dated: November 7, 2018)Using first principle electronic structure calculations we investigated the role of substitutionaldoping of B,N,P,Al and vacancies ( V ) in diamond (X α C − α ). In the heavy doping regime, at about ∼ −
6% doping an impurity band appears in the mid gap. Increasing further the concentrationof the impurity substitution fills in the gap of the diamond host. Our first principle calculationindicates that in the case of vacancies, a clear single-band picture can be employed to write down aneffective one band microscopic Hamiltonian, which can be used to further study various many-bodyand disorder effects in impurity band (super)conductors.
I. INTRODUCTION
Diamond has a number of unique attributes that makeit highly suited as a gem stone. It is the hardest knownmaterial which can only be scratched by another dia-mond. The thermal conductivity of diamond is the high-est among all materials [1, 2]. Irradiation of diamond byvarious particles (e.g. electrons, neutrons, α particles)followed by annealing to repair damaged sp bonds givesrise to fascinating colours of diamond, which are due tothe so called colour centers [3].Diamond is also a material with semiconductor prop-erties that are superior to Si, Ge, or GaAs, which are nowcommonly used. The use of diamond in electronic appli-cations is not a new idea, but limitations in size and con-trol of properties restricted the use of diamond to a fewspecialized applications. The vapor-phase synthesis ofdiamond, however, has facilitated serious interest in thedevelopment of diamond-based electronic devices. Theprocess allows diamond films to be laid down over largeareas. Both intrinsic and doped diamond films have aunique combination of extreme properties for high speed,high power and high temperature applications [2, 4, 5].Ekimov and coworkers [7] and subsequently Takano et.al. [8] used chemical vapor deposition (CVD) to synthe-sized B-doped diamond. Doping diamond by low concen-tration of typically 10 − cm − boron atoms givesrise to acceptor level, rendering it to a p type semiconduc-tor [6]. Increasing the doping level to [B]/[C] > n > cm − , i.e. ∼ few %, makes it supercon-duct at low temperatures [7, 8]. Increasing the dopingrate amounts to bringing the boron atoms closer, and al-low them to overlap more effectively, which broadens theacceptor levels in to decent bands of electrons [10, 11],which are responsible for metallic and superconductingproperties [11, 12, 13, 14, 15, 16, 17].Also some authors have investigated the doping of sili-con with boron, aluminum and phosphorus [18, 19]. Ex-periment has showed that the transition temperature forB-doped Si is T c ∼ .
35 K. Therefore doping C withboron gives more than an order of magnitude larger T c ∼ ∼ − T c . In the example of high temperature cuprate su-perconductors (HTSC) [20], an effective one-band modelfor the so called Zhang-Rice singlet [21] can be writtendown in terms of the hole states of the O2 p and Cu3 d .In X α C − α case also the effective impurity band has amixed X np and C2 p character. Here n = 2 for X=B,Nand n = 3 for X=Al,P. In the case of X= V the pic-ture is even simpler. The metallic band in the middleof the diamond gap well isolated from both bands is al-most entirely due to the nearest neighbor C atoms sur-rounding the vacant site. In this work we present ourpreliminary result on comparison of the impurity bandformation for the above elements for various doping rates α = 1 / , / , / II. METHOD OF CALCULATION
In this study we used the Plane Wave-PseudopotentialQuantum-ESPRESSO code [22]. We used Density Func-tional Theory with General Gradient Approximation(GGA). The GGA exchange-correlation functional whichhas been used is PBE [23]. We employed ultrasoft pseu-dopotential [24] to describe electron-ion interaction. Theenergy cutoff for expansion of wave function in planewave was 25 Ry, and 150 Ry has been used for expan-sion of charge density. We chose 2 × × × × × × α C − α with α = 1 / , / , /
16. B, N,Al, P are chosen to substitutionally replace one carbonatom in a diamond structure. The k-point is sampledaccording to table I. To accelerate electronic structurecalculation, we use Methfessel and Paxtons Fermi-levelsmearing method (width 0.01 Ry) [25]. The impurity hasbeen only substituted by one of carbon atom in diamondstructure without any relaxation.
TABLE I: The k-point sampling for different supercells.Supercell k-point grid number ofatomic sites2 × × × × × × × × × × × × D O S ( s t a t e s / e V / C e ll / nu m be r o f a t o m s ) × × × × × × FIG. 1: The density of state (DOS) for various concentrationsof B, Al, N, P and V impurities in diamond. III. RESULTS AND DISCUSSION
The results of this work has been summarized in Fig. 1where we have plotted the density of state (DOS) forvarious impurities and different impurity concentrations.We have shifted the data such that the Fermi level inall cases is at E F = 0. The impurity band formation forthe vacancy and Aluminum are the most clear among thecases studied here. Their bandwidths are between 1.5 eVto 0.5 eV for different impurity concentration.For the perfect diamond lattice the DOS for all sizesof the unit cell coincide and give the clean diamond gap.However substitutional doping with B and Al starts tocreate acceptor levels on top of the valence band whichbroaden into bands in the regime of few percent impurityconcentration. Also in this regime the relevant orbitals of the impurity atoms will hybridize with the C2 p orbitals,and the impurity band has a mixed character, similarto the case of Zhang-Rice singlet of cuprates [21]. It isclearly seen that the impurity band peak in the case ofAl is stronger than B. Also the value of the DOS at theFermi level ( ρ ) is larger for Al doping than B doping. Nand P similarly create donor levels at the buttom of theconduction band. Again qualitatively one can see that Ptends to give a sharper impurity band peak than the N.Here also the impurity band arises from a combinationof nearest neighbor C2 p and N2 p or P3 p bands.In the case of vacancy, the story is different. First of all,since V is neither acceptor, nor donor level, the resultingband will be in the middle of the gap, well isolated fromthe valence and conduction band. Secondly the impu-rity band arises from the n.n. carbon atoms surroundingthe vacancy. This qualitative picture can be inferred bylooking at the orbital and site resolved partial DOS (notshown here). Therefore for the effective one band modelof the impurities proposed by Baskaran [11], doping by V seems to be more suitable than the other cases studiedin this investigation.Note that in this study we have ignored issues likethe formation of various complexes. For example, ni-trogen doping in diamond usually leads to the formationof nitrogen-vacancy complex [26]. Ignoring such com-plications, one can use the argument of Mott to get arough estimate of the typical density needed to make thethe resulting half-filled impurity band superconduct: Thecritical concentration needed for metalization is given by a B n c = 1 /
4, where the Bohr radius a B of the impuritycan be estimated from the binding energy E B of the im-purity levels as a B = e ε E B . For typically 0 . − . V , the Bohr radius will be 1 − ∼ cm − or 5 −
15 percents. For the concentrationsaffordable in our calculations, at α = 1 / ≈
6% non ofthe impurities studied here fills in the gap. However Cudoping at nearly 6% already metalizes the diamond [27].According to a disordered RVB mechanism suggestedby Baskaran [11], for such an effective single band at halffilling the critical temperature for the superconductivityis given by k B T c ≈ W e − ρ J , where J is superexchange, W the band width, and ρ is the DOS at the Fermi levelobtained from the DFT bands. The effective one bandmodel must have large enough W to escape the Mott-Hubbard splitting in the large U limit, such that thehalf-filled band picture remains valid. Also the disor-dered nature of the impurity centers will start to localizethe states close to the impurity band edges. Again thebandwidth W must be wide enough such that the mobil-ity edge will not cross the Fermi level.If one assumes that the superconducting mechanismremains the same for various impurities X studied here,vacancy and Al doping offer a more clear picture of thehalf-filled band undergoing Anderson-Mott to RVB su-perconducting scenario of Baskaran [11]. The enhance-ment of ρ observed for the case of similar concentrationof Al and V is advantageous in giving larger transitiontemperature than the case of Boron. Note that larger ρ even within the BCS picture is an advantage of V and Aldoping compared to doping by B.In terms of practical fabrication, utilizing the heavilyvacant diamond may offer a new method in addition tohigh temperature high pressure techniques used in pro-duction of CVD diamond doped with various elements.Irradiation by different particles may offer a method ofproducing high concentration of vacancies in diamondwhich can possibly lead to higher transition temperature than the CVD diamond doped with elements. IV. ACKNOWLEDGMENTS
S.A.J. was supproted by Alavi Group Ltd. S.A.J.would likes to thank S. Maekawa for the hospitality dur-ing his visit to the institute for materials research, To-hoku university. We are grateful to G. Baskaran for usefulcomments and discussion. [1] Field, J.E.,
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