Table Of ContentBoron Spectral Density and Disorder Broadening in B-doped Diamond
K.-W. Lee and W. E. Pickett
6 Department of Physics, University of California, Davis, CA 95616
0 (Dated: February 2, 2008)
0
2 ComparisonofperiodicBdopantswitharandomalloyofsubstitionalboronindiamondiscarried
out using several supercells and the coherent potential approximation (CPA) for the random alloy
n
case. Themain peakintheBlocal densityofstatesisshifted tolowerbindingenergycompared to
a
thecorrespondingCpeakinintrinsicdiamond. Insupercells, thisshowsupasstronglyB-character
J
bandssplit from bulk C bands away from the zone center, in an energy region around -1 eV. Even
5 for a 4×4×4 supercell (BC127), effects of the dopant order are evident in the form of primarily
B-character bands just below the Fermi level at the supercell zone boundary. The bands resulting
]
n from the CPA are of continuous mixed C-B character. They resemble virtual crystal bands, but
o broadenedsomewhatreflectingthedisorder-inducedlifetime,andareconsistentwithangle-resolved
c photoemission band maps. The B character is 1.7 times larger than for C (per atom) near the top
- of the valence bands for CPA, and roughly the same for supercells. CPA results are particularly
r
useful since they characterize thewavevector and energy dependenceof disorder broadening.
p
u
PACSnumbers: 71.20.Be,71.55.Cn,74.70.Ad
s
.
t
a
m I. INTRODUCTION atoms.
- Theexistingtheoreticalstudies aimedatgleaning
d
informationabout the B acceptor state arenotvery
n The discovery and confirmation of superconduc-
conclusive. Finite clusters (up to 59 atoms) were
o tivity in diamond heavily doped with boron[1, 2,
used by Mainwood,[17] who obtained for substitu-
c 3, 4, 5, 6] has provoked interest in the mechanism
[ and character of the pairing that is responsible. tional B a threefold degenerate, mainly B 2p, state
2 Thereareseveralconfirmationsofsuperconductivity 0.36eV above the uppermost occupied diamond va-
lence state. The cluster was allowed to relax, with
v around 4 K for 2-3% doping, and a recent report[5]
9 indicates Tc ≈ 11.5 K for 4.7% boron concentra- the B-C bonds increasing by 10% (0.15 ˚A), a much
5 tioninhomoepitaxialfilms. Whilethepictureof3% larger amount than has been obtained in more re-
3 centcalculations. Theverylimitedclustersize,with
hole-dopeddiamondasadegeneratemetalhasbeen
9 its surface effects, does not allow for clear conclu-
the most common assumption,[7, 8, 9, 10, 11, 12]
0 sions. Barnard et al. reported some results from 64
there has also been the possibility that the boron
5
atoms(periodic) supercells,[18]but aswe show here
0 acceptor level is too deep for itinerant conductivity
for a larger supercell, the local density approxima-
/ to arise, leading to a picture of a non-degenerate
t tion(LDA)doesnotproduceagapacceptorstateat
a doped semiconductor[13] with possible correlated
m behavior.[14]Anangle-resolvedphotoemissionspec- such densities (∼ 1%). A tight-binding total energy
method was used by Sitch et al., who reported only
- troscopy(ARPES)study[15]onheavilyB-dopeddi-
d amond has found the Fermi level to lie a few tenths atomic relaxation using a 216 atom supercell.[19] C
n atomsneighboringthe substitutionalB relaxedout-
of an eV within the valence bands, establishing the
o wardby4%(bondlengthof1.60˚A).Theirstudywas
degenerate-metal picture as the correct one for this
c
focused instead on doping of amorphous carbon.
: system.
v
Adopting the ARPES demonstration[15] that the
i ItisusefultoreviewbrieflythehistoryofBdoping
X Fermi level lies within the itinerant valence bands,
of diamond. Although it has been clear for decades
r that B is responsible for the colorofblue diamonds, there are a few choices to be made in describing the
a electronic structure, and the subsequent evaluation
and the desire for p-type semiconducting diamond
ofthecharacterandstrengthoftheelectron-phonon
hasbeenintenseforsometime,theexistinginforma-
(EP) coupling is determined to some extent by this
tion about the isolated B acceptor is sketchy. The
choice. The published studies of superconductivity
energyoftheacceptorlevelisagreedtobeEB=0.37
in B-doped diamond concur that EP coupling pro-
eV.[16] This binding energy is too large for this im-
videthepairingmechanism,inspite ofapplyingdif-
purity to be treatedaccurately atthe effective mass
ferentapproximationsto achievetheir separateesti-
level, hence its wave function is not as greatly ex-
mates. We briefly describe the various approaches.
tended as that of a shallow level would be. On the
otherhand,thisisnottrulyadeeplevel(sinceEB is Threegroupshaveadoptedthevirtualcrystalap-
only 7% of the band gap) in which the defect state proximation (VCA), making use of the fact that
extends over only a shell or two of neighboring C B is chemically and physically similar to C, espe-
cially when they share a common environment; in ble in principle, because linear response theory for
this case each is surrounded by C atoms on a dia- phonons has not yet been incorporated into CPA
mond lattice. In such cases, the itinerant wavefunc- codes. It is straightforward however to calculate
tions average over the B and C atoms and replac- the CPA electronic structure of B-doped diamond
ing each nucleus by an average charge constitutes andcompareresultswiththeVCAandsupercellap-
the VCA. Adopting the VCA, Boeri, Kortus, and proaches. Inthispaperwestudy inSec. IIIselected
Andersen[7] then applied linear response theory for supercells, and then provide CPA results (Sec. IV)
harmonic phonons and calculated the phonon fre- fordiluteBindiamondandmakesomecomparisons
quencies at ∼50 q points. This allowed the evalu- bothtosupercellresultsandthe VCAbands(which
ation of the EP spectral function and strength of are hardly distinguishable from the diamond bands
coupling. Lee and Pickett[8] also adopted the VCA forsmallBconcentrations). Usingsupercellswecan
fortheunderlyingelectronicstructure. Assuggested study effects due to B-B interaction, and quantify
by the analogy with MgB2 where holes are doped specific effects that arise because of the periodicity
into strong covalent bonds, they found a very large that is imposed. It is also easy when using the su-
EP matrix element for the bond-stretch mode and, percell approach to evaluate the effect of B on the
making use of the small Fermi surfaces, assumed variousshellsofCneighbors. TheCPAaccountsfor
it varied slowly with wavevector q up to 2kF (kF random positioning of the dopants and also allows
is the Fermi wavevector). They calculated by the the evaluation of disorder broadening of the bands
frozen phonon method the phonon renormalization (Sec. IV). We make a few observations about re-
ofthebond-stretchmodebythedopedcarriers,and laxation around a substitutional B atoms (Sec. V)
from two related but separate directions arrived at which is not included in either the VCA or CPA
the EP coupling strength. Also adopting the VCA method, and in Sec. VI provide a summary of the
viewpoint, Ma et al.[11] used methods very similar implications of our results.
tothose ofBoeriet al. (butdifferentcodes)andthe
same phonon mesh, but studied electron doping as
well as hole doping.
II. STRUCTURE AND CALCULATION
Blase, Adessi, and Conn´etable[9] and Xiang et METHOD
al.[10],ontheotherhand,usedasupercellapproach
in which the B dopants keep their chemical identity The bulk diamond bond length is dC−C= 1.54 ˚A.
(actual B atoms), but to apply standard solid state For a substitutional B atom, the B−C bond length
methods the dopant atoms had to be repeated peri-
is increased by 2% and the neighboring C-C bond
odically. This supercell approach restricts dopant
length is decreased by 1.3% in a LDA calculation
concentrations to certain commensurate fractions with 3×3×3 fcc primitive cell (BC53).[9]
but this is no real limitation because B concentra-
We have carried out calculations of “ordered im-
tions are only known approximately. Blase et al.
purities” using several supercells and of “randomly
calculated phonons and EP coupling only at q=0 in
disorderedimpurities” using the CPAmethod, both
a 54 atom (BC53) supercell and estimated the cou- of them with the full-potential nonorthogonallocal-
plingstrengthaccordingly. Xiangetal. treatedboth
orbital minimum-basis scheme (FPLO).[20] The
16 atom and 36 atoms supercells, using phonons at
CPA[21] method implemented in FPLO is based
a few q points.
on the Blackman-Esterling-Berk theory[22] that in-
Naturally the various calculations contain differ- cludes random off-diagonal matrix elements in the
ences in detail and in general it is not possible to local orbital representation. Koepernik et al. pro-
compare several of the intermediate results. Both vide a discussion of how randomness is treated in
groups using the supercell approach have suggested this atomic-orbitalbased procedure.[21]
that there are aspects of their results that make We have used three types of supercells: a large
the supercell approach superior, or perhaps neces- symmetric cell, a doubled cubic cell only in a (001)
sary. On the other hand, the ARPES data[15] lead plane and a set of layered cells. A 4 × 4 × 4 fcc
to band dispersions very much like the diamond (or primitive cell (BC127) was used as a symmetric cell
VCA) bands, perhaps broadened by disorder. and 2× 2 × 1 cubic cell (BC31) also was studied.
Thereisanotherapproachtothetreatmentofran- The latter (≈ 3% doping) is very close to the ini-
domly positioned dopant atoms, the coherent po- tally reported 2.5% doping superconductor. In ad-
tential approximation (CPA). The CPA allows the dition, we used severalsupercells [(1×1×2 (BC15)
dopant to retain its identity, but produces an effec- to1×1×5(BC39)]layeredinthec-direction,which
tivemediumthataccountsforrandomdisorder. The illustratedsomesystematicvariations. Thelowlevel
CPA method has not been applied to evaluate the ofBdopingrequiresasufficientlyfinekpointgridto
EP coupling in B-doped diamond, which is possi- samplethe fewunoccupiedvalencestatesthatarise.
2
Because of the different size of each Brillouin zone 1
(BZ), k points used are different from each other, B1C310.7
C
but we have used from 245 to 1000 k points in the 32
irreducible wedge of each BZ, except for 4×4×4 0
)
supercell. For the 4×4×4 supercell, 3 irreducible V
e
k points should be enough to understand the band y ( −1 0.6 Γ
g
structure. However, CPA requires a much finer k r
e
mesh to sample the wavevector and energy depen- En
dentspectralfunction,so2045k pointswereusedin −2
the irreducible wedge. For all calculations, valence
orbitals for the basis set contained 1s2s2p3d for B
and C. −3Γ X M Γ Z R A Z
III. ORDERED IMPURITIES 1
(SUPERCELLS)
0.5
A. 2×2×1 (32 atom) supercell
0
)
V
e
First we address the supercell having a B concen- (F-0.5
tration(cB ∼3%)veryclosetotheinitiallyreported - E
superconducting concentration (∼ 2.5%). The up- E -1
per portion of the valence band structures of the
2×2×1supercelliscomparedwiththatofintrinsic -1.5
diamondinthe toppanelofFig. 1. Forbetter com-
-2
parison,thebandsofintrinsicdiamondareraisedby Γ X M Γ Z R A Z
theFermienergy(EF)differencefromthedopeddi-
amond so that the bands are aligned at the valence FIG. 1: (Color online) Top panel: Upper portion of the
bandmaximum(VBM).Bdopinghasthreeseparate valencebandstructuresofthe2×2×1supercellforthe
effects that are common to all of the supercells: (1) Bdoped(BC31)andintrinsicdiamond(C32). Thebands
loweringEF owingtoholedoping(oppositedirection arealignedatthetopofthevalenceband. inset: Blowup
to the effect of C doping in Mg(B1−xCx)2,[23]) (2) of the band splitting by 42 meV along the M-Γ-Z lines
band splittings at the high symmetry points in the due to the asymmetry of the supercell. Bottom panel:
“Fatband” representation of the B 2p character. The
BZreflectingsymmetrybreaking,and(3)bandshifts
main character is located in the range of −1.7 eV and
reflecting the difference in the B and C potentials.
−0.7 eV, a regime largely shifted by the B-doping, on
The largesteffect ofthe B doping occurs1−2 eV
the kz =π/a plane. The symbol size is proportional to
below the Fermi level, which is easily seen on the
theB2pcharacter. Thedashedhorizontallineindicates
kz = π/a plane (Z-R-A-Z lines in Fig. 1). Three the Fermi energy of BC31. The symmetry points are
bandsalongthese lines areraisedinenergybymore given such as (0,0,ξ) for Γ(Z), (0,1/2,ξ) for X(R) and
than 1 eV by the addition of boron; the strong B (1/2,1/2,ξ) for M(A).ξ is zero for the first symbols and
character of these displaced bands is illustrated in 1 for the symbols in parentheses; units are πa.
the bottom panel of Fig. 1, where the “fatband”
representationemphasizingBcharacterispresented.
Theseratherflatbandsarestronglyhybridizedwith
thenearestneighbor(nn)C2pstates,whichprovide
aboutone-thirdofthe bandcharacter. Atthe VBM mixing,theBDOSresemblesthatofthennCDOS.
which is threefold degenerate in intrinsic diamond, More specifically, however, the B and nn C are en-
the B character and the nn C 2p character are also hanced at the VBM. Their spectral densities show
strongly mixed. Besides the mixing, the top of the differences: the B DOS has much sharper peaks
valencebandsattheΓpointintheB-dopeddiamond around−1eVwhilethatofthennCcontinuesmore
shows a tiny band splitting of 42 meV, as shown in strongly to -4 to -6 eV. To summarize: the weights
inset of the top panel of Fig. 1. The origin of the of the DOSs are arranged in the order of B, nn C,
splittingisnotclearbutitsvalueisprobablyspecific second neighbor (2nd n) C, and bulk C, in order of
to this supercell. increasing binding energy.[24] This trend is consis-
Thecorrespondingatom-projectedDOSaregiven tent with the interpretationof x-rayabsorptionand
in Fig. 2. As expected from the strong local B−C emission spectroscopy by Nakamura et al.[25, 26]
3
B
0.4
0.5
B doping
0 intrinsic
nn C
0.4 0
)
V
E) (e
( 0 y -0.5
N 2nd n C g
0.4 r
e
n
E
-1
0
bulk C
0.4
-1.5
Γ X W K Γ L W U X
0
−20 −16 E−1 2− E −(8eV) −4 0 FIG.3: (Coloronline)Upperportionofthevalenceband
F structuresof4×4×4supercellfortheBdoped(BC127)
and intrinsic diamond (C128). At the symmetry points,
FIG. 2: (Color online) Atom-projected DOS for B and the band splittings are obvious, especially at the Γ, X,
andthreeCneighborsin the2×2×1supercell(BC31). and L points. The symmetry points follow the conven-
TheBDOSisverysimilartothatofthennC.Notethe tionalnotationforfccBZ.(Ldenotesthezoneboundary
sharp peaks confined to B and the nn C atom between point in <111> direction.) The dashed horizontal line
−2 eV and −1 eV, arising from strong B 2p character indicates EF of the B-doped diamond.
on the plane of kz = π/a. For C atoms progressively
further from the B atom, the peak at −11 eV becomes 1
sharper. The B DOS is clearly shifted higher (to lower BC
23
bindingenergy) compared to that of carbon. 0.5 C
24
V) 0
e
B. 4×4×4 (128 atom) supercell y ( −0.5
g
r
As large a supercell calculation as possible is de- ne −1
sirable to extract information about the isolated B E
impurity. We first study the change in the band −1.5
structure, keeping in mind that it reflects the pe-
−2
riodic arrangement of the 0.8% concentration of B
Γ X M Γ Z R A Z
substitutionals in this supercell. The symmetry is
suchthatthethreefolddegeneracyoftheVBMisnot
FIG.4: (Coloronline)Upperportionofthevalenceband
split. The effective masses are changed very little, structuresofthe1×1×3supercellfortheBdoped(BC23)
butwehavenottriedtoidentify possiblechangesin and intrinsic diamond (C24). This figure shows clearly
masses at the 1% level. Splitting of degeneracies by thedifferentoriginsofthebandsplittingsatΓandatZ.
0.3−0.4eVcanbeseenatvariousplacesinthebands The dashed horizontal line indicates EF of theB-doped
in Fig. 3: along Γ-X and at L near −0.2 eV. There diamond.
are larger splittings (at L, for example) at energies
below −1 eV in Fig. 3. At the Γ point, the cluster
ofdegeneratebandsat−1.3eVinintrinsicdiamond 0.8% of this supercell could lead to inaccuracies in
properties determined by the Fermi surface, such as
(comingfromzonefolding)aresplitandshiftedcon-
the electron-phonon spectral function.
siderably by B addition. The threefold degenerate
bands at -0.9 eV, having strong B character, have
been raised by 0.5 eV, while others are shifted lit-
tle. Thus even this large supercell shows effects of C. Layered supercells
the periodicity of B atoms that will not be present
if B atoms are distributed randomly. These effects Asetoflayeredsupercellswerestudiedtoobserve
are most evident when they produce new pieces of closelytheasymmetry,changesindopingconcentra-
Fermisurfacerelatedspecificallytothe artificialpe- tion,andeffect ofincreasingthe separationofthe B
riodicity, as is beginning to be the case in Fig. 3 atoms. Although the doped B concentration (2.5%
along the L-W line. A higher doping level than the to 6.25 %) varies in the each calculation, the band
4
(1,1,2)
(1,1,3)
(1,1,4)
(1,1,5)
)
E
(
N
0
−3 −2 −1 0 1
E − E (eV)
F FIG. 6: CPA spectral density, as a “smeared” band
structure,in thefull valence-conductionbandregion for
FIG. 5: (Color online) Total DOS of the (1,1,m) cube x=0.025. The conduction bandsaround 17 eV aremost
supercells (m=2,3,4,5); in each case there is one hole strongly affected bythechemicaldisorder, whiletheva-
persupercellunoccupied. Thearrowsdenotepeakswith lence bandsare less affected.
strongBcharacter. TheFermienergiesofeveryplotare
set to zero and spacing of each plot in y-axis is chosen
as 2 states/eV.
structuresshowconsiderablesimilarities. Theupper
portionofthevalencebandstructuresofthe1×1×3
supercell is shown in Fig. 4. Bands of strong B 2p
character are pushed upward into the -1.5 to -0.5
eV range along the X-M and R-A lines. This is the
same energy range where B character is ‘pushed’ in
the other supercells.
The total DOS of the (1,1,m) cubic supercells
(m=2,3,4,5), corresponding to x=0.0625, 0.0417,
0.0313 and 0.025, are presented in Fig. 5 and show
an interesting trend. As indicated by the arrows,
decreasing x leads to a sharp peak at −1.6 eV and
enhancementaround0.4eV.Withdecreasingx, the
lowest of the bands shown in Fig. 4 along the X-M
line becomes flatter, resulting in enlargement of the
peak at −1.6 eV. As suggested by Xiang et al.,[10] FIG.7: Blowup CPAspectral density,with Fermilevels
we find that the width of the acceptor states (un- aligned, near EF along the K-Γ-L line for x=0.005 and
occupied portion of the valence bands), containing x=0.025. As x increases, EF is lowered and the disor-
one hole per supercell, depends on x up to a cer- dered broadening increases. The spectral density shows
tain value. Note that the unoccupied part of N(E) thebroadening most clearly at theΓ and L points.
is essentially the same in the (1,1,4) and (1,1,5) su-
percells.
cB. The CPA spectral density for x=0.025, ren-
dered as a “smeared” band structure, is displayed
IV. DISORDERED BORON (CPA) in the full band regionin Fig. 6. The bands arelike
those from the virtual crystal approximation used
Now we describe CPA calculations which explic- earlier[7,8]exceptthatlifetimeeffects(appearingas
itly treat randomly distributed B substitutionals a broadening) are visible. The effect of disorder is
withconcentrationx. Wecalculatex=0.005=1/200 strongestin the conduction(unoccupied) band near
toapproximateanisolatedboron,andx=0.025very 17 eV. The valence (occupied) band is less affected.
close to the optimal superconducting concentration Evenfor only 2.5% B-doping, the disorderedbroad-
5
1
0.8 B
relaxed
0.5 unrelaxed
0.4
)
E ) 0
V
( 0
e
N0.8 C (F −0.5
E
0.4 − −1
E
0 −1.5
−20 −15 −10 −5 0
E − E (eV) −2
F Γ X M Γ Z R A Z
FIG.8: (Coloronline) Atom-projectedDOSatx=0.025 FIG. 9: (Color online) Relaxed and unrelaxed band
from the CPA calculation. The picture that results is structuresof theB-doped diamond for the2×2×1su-
verysimilartothatfromthesupercellcalculations. The percell. The relaxation shows little effect. As expected,
main peak of B is located at -2 eV, while that of C lies the strongly hybridized bands between the B and nn C
at higher binding energy and is almost identical to that on the kz = π/a plane are most affected by the relax-
of the bulk C. ation. The difference between the Fermi energies is 25
meV.
ing is clearly visible in the doubly degenerate band
nearEF alongΓ-L,andnear−12eValongtheX-W The C DOS is hardly distinguishable from that of
line. TheformercomesfrommixingwithB2pstates the bulk C. The weight of the B DOS is closer to
andthe latterhasmoreadmixtureofB2sstates. It EF than that of the C DOS, as in the supercells.
isofcoursenoaccidentthatthedisorderbroadening Near EF, the DOS of B is 1.7 times larger than the
resulting from the CPA treatment is largest in the DOSofC,whichindicatestheBinfluenceonmetal-
valence band (in the zero to -2 eV region relative lic properties of C1−xBx is greater than that of an
to EF) where the B-derived band shifts are largest. average C atom (but by less than a factor of two),
Thebroadeningarisesbecausethequantummechan- consistent with measurement by soft X-ray absorp-
ical averaging of the wavefunction over the atomic tion and emission spectroscopy.[26]
potentialsbecomes lesssystematicinenergyregions
wheretheatomicpotentialsdiffermore. Inthiscase
a single well-defined band persists but with larger V. RELAXATION
broadening. In strong scattering systems (which it
turnsoutthisisnot),split-bandbehaviorcanresult. Reports of the B-C bond length, when allowedto
Because the behavior near EF is important for relax, differ. The 10% value reported in the cluster
(super)conductivity,the blowupofthe spectralden- calculationofMainwood[17]isclearlyunrealistically
sities nearEF alongthe K-Γ-Lline forx=0.005and large. Sitch et al., using a realistic (but not first
x=0.025 are given in Fig. 7. Increasing the B con- principles) method with large supercell (216 atoms)
tent (x) increases the chemicaldisorder broadening, founda4%lengthening,[19]whileBlaseet al., using
inadditiontoloweringEF. Thechangeofthedisor- first principles methods but only a 54 atom super-
der broadening at the Γ point is different from that cell, reported only a 2% elongation.[9] These values
of the L point, because the main B 2p character is arewithrespecttotheC-Cbondlengthof1.54˚Afor
located just below −2 eV at x=0.025. (See below.) intrinsic diamond. The relaxed and unrelaxed band
As a result, the broadening at the Γ point is half structures of the B-doped diamond for the 2×2×1
of that at the L point at x =0.025, while the width supercell given in Fig. 9 show that the effect of the
at the Γ point is 10% wider than at the L point at 2%elongatedB-Cbondlengthistofurtherraisethe
x=0.005. Thus disorder broadening effects (such as B-derived bands, i.e. increase the separation from
the residualresistivity)willbenonlinearinxinthis the bulk C bands, although the effect is not major
region. (a maximum of 60 meV). The region between −1.7
The atom-projected B and C DOS from CPA at eV to −0.7 eV on the kz = π/a plane of this su-
x=0.025 is shown in Fig. 8. The DOS is similar to percell is most strongestaffected, as expected, since
thosefromsupercellcalculations,butwithoutsharp that is the region in which B 2p states are strongly
peaksintherangeof−2eVand0eVthatarisedue mixed with nn C 2p states. This 2% relaxation re-
totheperiodicarrayofdopantBatoms. TheBDOS sults in a small (3%) increase in the Fermi energy
has a peak at −2 eV and decreases asymmetrically. (by25meV),whichcanbe consideredasaslightin-
6
creaseintheeffectivemass. Themagnitudeofthese inBxC1−x givenbytheCPA),andissplitawayfrom
shiftssuggeststhatrelaxationwillchangenumerical bands that are primarily C in character. This effect
results by a few percent but will not change signifi- arises in both the smaller and the larger supercells
cantly any of the general features of B doping that (3% doping down to 0.8%). The CPA method, on
are discussed above. the other hand, leads to bands much like those of
intrinsic diamond but disorder broadened to some
extent, and with a B content that varies smoothly
VI. SUMMARY along the band (that is, with energy and with mo-
mentum). The CPA description is buttressed by
In this paper we have studied the boron spec- ARPES data,[15] where mapping of the bands re-
tral density relative to that of C, and quantified to sultsindiamond-likebandtopologyasoccursinthe
someextentthedisorderbroadening,inB-dopeddi- VCA or (accounting for disorder) CPA.
amond, using a combination of supercell and ran-
Anattractivefeatureofthesupercellmethodisits
dom disorder (CPA) approaches. All experimental
ability to investigate the local environment of the
results that we are aware of support the prevailing
dopant atom, including lattice relaxations. While
viewpointthatthissystemisaheavily-dopeddegen-
the cluster CPA method makes it possible within
erate (metallic) semiconductor at the concentration
a disordered system, there is very little experience
range where superconductivity has been seen, and
with this capability. The ability to relax atomic po-
our studies lie only within this scenario. As far as sitions was used by Blase et al.[9], who obtained a
the averagedistributionofB characterinthe itiner-
2% longer B-C bondlength than that of diamond;
antbandsystemisconcerned,theCPAandsupercell
this elongation will lead in turn to a change in the
results are similar. In the regionof interest (around
B-Cforceconstant. Whetherthisfactorcanaccount
the Fermilevelforreportedlevelsofdoping,notex-
for the difference in spectral distributions obtained
ceeding 5%), the CPA result is that the fraction of
by the two supercell methods[9, 10] is yet to be re-
B characteris 1.7times that of a C atom. Supercell
solved.
results are consistent with this value, at least if a
VII. ACKNOWLEDGMENTS
little energy-averagingis considered.
In the energy-momentum distribution of B char-
acter,however,theperiodicityinthesupercellsleads WeacknowledgeK.Koepernikforimportanttech-
to regions where a band has a much larger fraction nical advice. This work was supported by National
ofBcharacterthan1.7x(thefractionofBcharacter Science Foundation grant No. DMR-0421810.
[1] E. A. Ekimov, V. A. Sidorov, E. D. Bauer, N. N. [9] X.Blase,Ch.Adessi,andD.Conn´etable,Phys.Rev.
Mel’nik, N. J. Curro, J. D. Thompson, and S. M. Lett. 93, 237004 (2004).
Stishov,Nature 428, 542 (2004). [10] H.J.Xiang,Z.Li,J.Yang,J.G.Hou,andQ.Zhu,
[2] Y. Takano, M. Nagao, K. Kobayashi, H. Umezawa, Phys. Rev.B 70, 212504 (2004).
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