Screening in semiconductor nanocrystals: \textit{Ab initio} results and Thomas-Fermi theory

0.46 MB·English

by  F. Trani

Screening in semiconductor nanocrystals: Ab initio results and Thomas-Fermi theory
∗
F. Trani, D. Ninno, G. Cantele, and G. Iadonisi
Coherentia CNR-INFM and Universit`a di Napoli “Federico II” - Dipartimento di Scienze Fisiche,
Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy
K. Hameeuw
TFVS, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium
E. Degoli and S. Ossicini
8
0 CNR-INFN-S3 and Dipartimento di Scienze e Metodi dell’Ingegneria,
0 Universit`a di Modena e Reggio Emilia, via Fogliani, I-42100 Reggio Emilia, Italy
2
Afirst-principlescalculationoftheimpurityscreeninginSiandGenanocrystalsispresented. We
n
showthatisocoric screeninggivesresultsinagreementwithboththelinearresponseandthepoint-
a
charge approximations. Based on the present ab initio results, and by comparison with previous
J
calculations, we propose a physical real-space interpretation of the several contributions to the
5
screening. CombiningtheThomas-Fermitheoryandsimpleelectrostatics,weshowthatitispossible
1
toconstructamodelscreeningfunctionthathasthemeritofbeingofsimplephysicalinterpretation.

Themainpointuponwhichthemodelisbasedisthat,uptodistancesoftheorderofabondlength
]
i from theperturbation,thecharge response does not dependon thenanocrystal size. Weshow in a
c veryclear way that thelink between thescreening at thenanoscale and in thebulkis given bythe
s
surface polarization. A detailed discussion is devoted to the importance of local field effects in the
-
l screening. Ourfirst-principlescalculationsandtheThomas-FermitheoryclearlyshowthatinSiand
r
t Ge nanocrystals, local field effects are dominated by surface polarization, which causes a reduction
m
ofthescreeningingoingfromthebulkdowntothenanoscale. Finally,themodelscreeningfunction
. is compared with recent state-of-the-art ab initio calculations and tested with impurity activation
t
a energies.
m
-
d I. INTRODUCTION rity screening in nanocrystals is described. The charge
n density induced by donors and acceptors in Si and Ge
o nanocrystals is discussed, pointing to the connections to
Screening in semiconductor nanocrystals is a funda-
c boththepoint-chargecaseandthebulklimit. Itisshown
[ mental issue, the importance of which is mostly due to that, just like in bulk semiconductors,2 the response to
the large amount of technological applications inherent
1 isocoric impurities in nanocrystals is similar to that of
in the world of nanostructures. While screening in bulk
v a point-charge. A description of the local field effects
semiconductors is a well known and widely investigated
2 on the induced charge is given showing that the present
1 subject,1,2,3,4,5 the phenomenon of screening in nanos-
real-space analysis gives a way to distinguish between
3 tructures is still not fully understood. The presence of
surface and bond polarization contributions. In particu-
2 many papers recently published on this subject shows
lar, a very interesting comparison with the work of Ref.
. that there is a strong interest in this field.6,7,8,9,10,11,12
1
4allowsustogiveaphysicalinterpretationofthevarious
0 In particular, it has been shown that the response of
contributions to the screening.
8 a nanostructure to an external field is bulklike well in-
0 side the structure.7,9 This is an interesting feature due Inthesecondpartofthepaper,amodelforthescreen-
v: to the local nature of bonding in semiconductors. On ing is proposed. The well known electrostatic model,
i the other hand, such a behavior is quite different from based on the image charge method,12 is not valid in the
X
the trend shown, in both experimental results and the- neighborhood of the impurity, in that wrong boundary
ar oretical calculations, by the nanocrystal macroscopic di- conditions are given for the induced potential. We pro-
electricconstant,12,13,14,15 thatisshowntoincreasequite pose a generalization to semiconductor nanocrystals of
slowly with the size upon going from small nanocrystals the Thomas-Fermi model as originally proposed in Ref.
to the bulk. The connection between the microscopic 3 for bulk semiconductors. In order to better appreciate
local, bulk-like response and the macroscopic dielectric the merits and the shortcomings of the Thomas-Fermi
properties can be understood only through the study of theory, an alternative derivation of the model from the
the surface polarization contribution, which strongly in- Hohenberg and Kohn theorem17 is described. In the
fluences the nanocrystal properties. presentmodel,boththecorrectlimitintheneighborhood
Preliminaryresultsonthissubjecthavebeenpresented oftheimpurityandthesurfacepolarizationcontributions
in Ref. 16. In the present paper, we investigate the na- are taken into account from the beginning as boundary
ture of the screening in semiconductor nanocrystals by conditions of the Poisson equation. Our results fit well
performing a detailed microscopic analysis. In the first thescreeningdiagonalcontribution4 forbulksilicon,and
partofthepaper,afirst-principlesstudyofshallowimpu- excellent agreement with a recent first-principles calcu-
2
lation of the screening function10 is obtained. Moreover, (a) (b)
1.0
the model gives a fair prediction of the impurity binding
1.0
energies.
+2
S
+
P
II. AB INITIO RESULTS 0.6 Al-
-2
Mg
0.6
We have performed ab initio calculations using differ-
ent kinds of impurities in Si and Ge nanocrystals. The 0.2
study hasbeen basedonaplane-wavedensity-functional
theory (DFT) framework. The calculations have been 0.2
done with the QUANTUM-ESPRESSO package,18 us-
-0.2
ing the general gradient approximation with ultrasoft
pseudopotentials.19 A vacuum space of at least 6 ˚A has
been left within the supercell, in order to avoid spuri- 0 4 8 12 16 0 4 8 12 16
ous interactions between a nanocrystal and its replicas. r (a.u.) r (a.u.)
The convergence with respect to the plane-wave basis-
set cutoff has been treated with care, and Makov-Payne FIG.1: (Coloronline)Sphericallyaverageddensityn¯ind(r)/Z
corrections20 have been added for the computation of [panel (a)] and integrated density Qind(r)/Z [panel (b)] in-
the charged nanocrystal total energies. The nanocrys- duced in Si35H36 by different doping species: S+2 (dashed,
tals have a nearly spherical shape, they are centered on pink line) and P+ (solid, black line) donor doping, Al− (cir-
a Si (Ge) atom, and the surface dangling bonds are sat- cles, red) and Mg−2 (stars, blue) acceptor doping.
urated with hydrogen atoms. The undoped structures
havebeenrelaxed,andtheoptimizedgeometriesusedfor
the doped structures. In this way we study the screen- pseudopotentialcore,the acceptor curvesshowa nonlin-
ing due to the electronic response to an external pertur- ear peak that is absent in the donor case), the overall
bation, where fixed ionic positions are considered. The structure is similar to the donor case. However, while
resulting screening does not take into account the con- in the region of space after the second main peak from
tribution due to the ionic displacement. This latter is the impurity the curve is very similar to the donor case,
negligible when covalent semiconductors are considered. the first main peak is lower and pushed away from the
The electrondensityinducedby adonorspecies Xz in impurity site (r = 0). This discrepancy between donor
the nanocrystal SilHm is given by and acceptor doping is in agreement with some results
recently published,11 and is related to nonlinearity ef-
nind =n[(Sil−1XHm)z]−n[SilHm], (1) fects. The problem of nonlinear screening in bulk semi-
conductorshasbeenstudiedwithcareinthepastwithin
where Z is the impurity net charge (atomic units are
a Thomas-Fermi formalism. In Refs. 21,22,23,24, it was
used throughout this work). A similar expression holds
shown that there is an asymmetry in the response when
for acceptors and for Ge nanocrystals. We have focused
positive or negative charged impurities are considered.
our study on the spherical average of n , and on its
ind Whilefordonorscreening(Z =+1,+2)nonlineareffects
integrated density defined as
are negligible, in the case of acceptor screening the re-
r sponse is quite far from linearity. As we shall see in the
Qind(r)= n¯ind(x)dx, (2) next section, the screening in semiconductors is mostly
Z0 confined within a screening sphere around the impurity.
where n¯ (r)=4πr2n (r). The deviation of the radius of such a sphere (screening
ind ind
In Fig. 1, the results for donor and acceptor isocoric radius) from the linear case furnishes a realistic check of
dopants in Si H are reported. The spherically aver- the linearity of the response. From the Thomas-Fermi
35 36
agedinduceddensitiesn¯ (r)/Z [Fig. 1(a)]andtheinte- theory, an increase of the screening radius of about 0.39
ind
grated densities Q (r)/Z [Fig. 1(b)] are shown for the and 0.68 a.u. (with respect to the linear case) was ob-
ind
donors P+, S+2 and acceptors Al−, Mg−2. It is worth servedforZ =−1and−2,againstadecreaseofonly0.16
noticing thatthe densities inducedby P+ andS+2 (solid and 0.26 a.u. found for Z = 1 and 2, respectively.22,24
and dashed line, respectively) are almost indistinguish- These results show the same trend as in Fig. 1, confirm-
able. This suggests that, at least for donor isocoric im- ingourfirsthypothesisofagoodlinearityoftheresponse
purities,thescreeningiswithinalinear-responseregime. in the donor case.
This was already argued some years ago for the bulk,2 In order to show that the features discussed above for
and our analysis shows that this is valid for nanocrys- boththe inducedandintegrateddensityarenotpeculiar
tals too. From Fig. 1 it is seen that, in the case of the to silicon, we have also studied the case of Ge nanocrys-
acceptors Al− and Mg−2 (circles and stars), apart from tals. In Fig. 2, our results for the induced [panel (a)]
a narrow region of space near the impurity (inside the andintegrated[panel(b)]electrondensitiesareshownfor
3
1.2 (a) (b)
1.0 (a) (b)
1.0
Ge34PH36+ Si35H36
0.6 GGee34AAssHH36++ 0.8 0.6 SSii8174H7H76100 0.8
190 148 Si H
Bulk Ge 191 148
0.2 0.4
0.2 0.4
-0.2
0.0
-0.2
0 5 10 15 20 250 4 8 12 16 20 0.0
r (a.u.) r (a.u.)
0 10 20 0 10 20
r (a.u.) r (a.u.)
FIG.2: (Coloronline)Sphericallyaverageddensityn¯ind(r)/Z
[panel (a)] and integrated density Qind(r)/Z [panel (b)], in- FIG. 3: (Color online) Spherical averaged induced density
dshuocwedn aGree35foHr36P+an(ddaGshee1d91,Hb1l4u8e nlianne)ocarnydstaAlss.+ (Tsohleid,rebslualctks n[¯piandn(erl)([bp)a],necla(lcau)]lataendd fionrtegPr+ateidmpinudriuticeesdidnenSsiitynaQnoincdry(rs)-
line) in Ge35H36, As+ (circles, green) in Ge191H148. In panel tals with increasing size: Si35H36 (solid-black line), Si87H76
(b) we also report the linear response to a point-charge im-
(dashed-red line), Si147H100 (circles-blue) and Si191H148
purity in bulk Ge from Ref. 4 (stars, red).
(stars-green). The full arrows point to the nanocrystal ra-
dius(see text for definition).
P+ and As+ in Ge H (dashed and solid line, respec-
35 36
tively) and As+ in Ge191H148 (circles). For comparison, the induced charge density [stars in Fig 2(b)] receives
in Fig. 2(b) we also report (stars) a previous ab initio contributions from both diagonaland off-diagonalterms
result obtained using a truly point-charge impurity for of the dielectric matrix in reciprocal space. The diago-
computing the linear-response screening in bulk Ge.4 It nal contribution to the integrated density consists in a
is evident that the screening near the impurity is much monotonic increase of Q (r)/Z, followed by constant
ind
more pronounced for the nonisocoric P+ impurity than value. Instead, the off-diagonal contribution, related to
for the isocoric As+. The different core of the dopant the so called local field effects, gives a wavy structure to
atomwithrespecttothehostatomgivesrise,inthefirst the integrated induced density. This undulating behav-
case,tothesocalledcentralcellcorrections,whichgivea ior has been discussed by several authors for bulk sys-
nonlinear contribution. On the other hand, the response tems, andarisesfrom the polarizationofthe bonds.4,5,25
to (isocoric) As+ impurity does not induce central cell Both the diagonal and off-diagonal contributions are re-
corrections. It should be pointed out that these correc- trieved for the nanocrystals, as is clearly seen in Fig. 2,
tions have a finite range, being especially pronounced in but an additional effect due to the polarization of the
the region of space before the second main peak in the surface emerges. It consists of an electron density ac-
Qind(r)/Z curve. Instead, the electron charge localized cumulation around the nanocrystal surface due to the
onthe surface is quite independent of the chemicalshift. dielectric nature of the structure, as is well known from
Basedon Fig. 2, an interesting analysis of the size de- classical electrostatics. The surface polarization is re-
pendenceofthescreeningcanbededucedaswell. Indeed, trieved from the present ab initio formulation and can
from a comparison between our results for Ge35H36 and be seen as a very special kind oflocal field effect causing
Ge191H148,wearguethat,uponincreasingthenanocrys- theannihilationofQind(r)/Z incorrespondencewiththe
tal size, the induced chargeclose to the impurity rapidly nanocrystalboundary[Figs. 1(a)and2(a)]. Weconclude
converges to its bulk value. This can be better inferred that in a nanocrystal, local field effects (i.e., not diago-
from panel (b), showing results at small r very close to nal contributions) show up as both the bond and the
those calculated in Ref. 4 for bulk Ge. surface polarization. Nonetheless, while the bond polar-
From the results illustrated so far, we can reasonably ization, which is a bulk effect, gives a negligible contri-
concludethat(i)theisocoricdopingiswellapproximated bution to the screening and the optical properties, the
by a point-charge screening, (ii) the screening is in the surface polarization is extremely important. Indeed, it
rangeofthelinearresponseregime,(iii)theinducedden- is closely related to the depolarization effects that have
sity rapidly converges to the bulk in the region of space been shown to strongly modify the optical properties of
close to the impurity. low-dimensional structures.26,27,28
Further considerations can be driven from panel (b) The size dependence of the screening can be inferred
of Fig. 2. In Ref. 4, it was shown for bulk Ge that from Fig. 3, where the spherical average of the induced
4
density [panel a]andthe integrateddensity [panel b]are uniformly distributed on the nanocrystal surface, must
shownforasetofP+-dopedsiliconnanocrystalswithin- be introduced. The resulting potential energy is there-
creasingradii. Itcomesoutthatuptoadistanceofafew fore due to a total charge Z −Q localized around the
a.u. from the impurity, both the induced and the inte- impurity, and to a charge Q distributed on the surface.
grateddensities arealmostindependentofthe nanocrys- From electrostatics, the potential energy for an electron
tal size. In order to be more quantitative, it is necessary is
to define the nanocrystal radius. We do not follow the
−Z−Q − Q, 0<r ≤R
standard route, that is, the radius of sphere whose vol- v (r)= r R (3)
ume equals the product of the volume per atom in the c −Z, r≥R,
(cid:26) r
bulktimesthe numberof,say,siliconatoms. Indeed,the
where R is the nanocrystalradius, as defined in the pre-
shortcomingofthisdefinitionisthatthehydrogenspassi-
vious section.
vating the surface are left out. Moreover, in a screening
This electrostatic model is supported by the DFT re-
problem the fundamental objects are the electrons, in-
sults described in the previous section. Indeed, Fig.3
cludingthoseparticipatinginthebondsnearthesurface.
(a) shows that the induced electron density has two
Onthebasisofthesearguments,wedefinethenanocrys-
tal radius through the equation n 4πR3/3 = N, where main contributions concentrated around the impurity
0
andacrossthe surface. A basic hypothesis for the model
n is the bulk valence electron density and N = 4l+m
0
isthatthetotalchargeQisindependentofthenanocrys-
is the total number of valence electrons in Si H . The
l m
tal size. This is supported by the fact that, as shown in
nanocrystalradiicalculatedinthiswayareshowninFig.
Fig. 3, the induced density near the impurity rapidly
3 as arrows. They have slightly higher values than the
converges to the bulk value. As is shown below, the
conventional radii. For instance, in the case of Si H ,
35 36
use of the bulk value for Q is a good approximation for
we retrieve 11.3 a.u. against the conventional value of
nanocrystals.
10.4 a.u.. But, as it is seen from Fig. 3(a), with the
However, this simple model is not accurate in the de-
present definition all the radii lie almost exactly in the
scription of the screened potential close to the impurity
middle of the surface charge density. As a final obser-
site. Indeed, one can easily see that unphysical poten-
vation, we stress that despite the undulation in both
tial values are predicted in the limit r → 0. In order to
the induced and integrated densities, the total induced
correctthe model, we follow Ref. 3, in which a Thomas-
chargearoundthe impurity is almostindependent ofthe
Fermi description for bulk semiconductors is given. The
nanocrystal radius. Although, at this level, this concept
basic point is the concept of incomplete screening occur-
may not appear to be well-defined, we shall see in the
ring in a semiconductor. This means that the charge is
nextsectionhowitallowsthe constructionofa Thomas-
inducedonlyinsideascreeningsphereofradiusR . Out-
Fermi model for the screening. s
sidethe screeningsphere,the systembehavesasaclassi-
cal dielectric medium having a static dielectric constant
ε . In a semiconductor, the total induced charge Q is a
III. THE THOMAS-FERMI MODEL s
finite fraction of the whole external chargeZ introduced
with the impurity. It is well known that the ratio Q/Z
As we have seen in the previous section, the density
closely depends on the static dielectric constant through
induced by a point-charge impurity in semiconductor the relation3
nanocrystals consists of several contributions. The first
contribution is due to the reciprocal-spacediagonal part Q 1
=1− . (4)
of the dielectric matrix, the second contribution comes Z ε
s
from the Si-Si bond polarization, and the third is from
The larger the static dielectric constant is, the closer to
the nanocrystal surface polarization. In this section, we
unitistheratioQ/Z. Theinducedpotentialandthespa-
illustrateaThomas-Fermimodel,whichreproduceswith
tial dielectric function are obtained solving the Thomas-
greataccuracyboththediagonalandsurfacepolarization
Fermi equations with appropriate boundary conditions,
contributions.
where the screening radius is derived self-consistently.3
The Thomas-Fermi theory establishes the linear rela-
tion between the induced electron density n and the
A. Classical electrostatics ind
screened potential v = v +v inside the screening
scr ext ind
sphere,
As a starting step, we briefly describe a simple elec-
trostatic model that can be used as a first approxima- q2
tioninstudyingpoint-chargescreeninginsemiconductor nind(r)=− [vscr(r)−δµ], (5)
4π
nanocrystals. Thebasicassumptionisthat,asaresponse
to a point-charge potential v (r) = −Z/r, an electron where δµ (the chemical potential) is a constant to be
ext
polarization charge −Q is induced around the impurity. determinedwiththeboundaryconditionsandq isamul-
On the other hand, since the total induced charge must tiplicativeconstantrelatedtothenanocrystalaverageva-
integrate to zero, a compensating charge Q, assumed as lenceelectrondensity.3 Outsidethescreeningsphere,the
5
inducedchargeiszeroandthescreenedpotentialmatches density n = n −n with the screened impurity po-
ind 1 0
the classical expression v given in Eq. (3). The conti- tential v . This potential is defined as the sum of the
c scr
nuity of the induced density, which inside the screening externalperturbationv =v −v ,theinducedelectro-
ext 1 0
sphereisgivenbyEq. (5),atr =R implies[n (r)=0 staticpotentialδv =v [δn],andtheinducedexchange-
s ind H H
if r ≥R ] that correlationpotential
s
δµ=v (R ). (6) δv
c S δv = dr xc n (r). (14)
xc δn(r) ind
Z (cid:12)n0
(cid:12)
B. Derivation of the Thomas-Fermi theory The final result is (cid:12)(cid:12)
Before going to the application of the Thomas-Fermi dr′ δ2Ts[n] n (r′) +v (r)=δµ. (15)
theory to a nanocrystal,it is of some interest to rederive δn(r)δn(r′) ind scr
Eq. (5)withintheDFTframework. FromtheHohenberg Z ( (cid:12)(cid:12)n0 )
(cid:12)
and Kohn theorem, it is known that the total energy of Before showing how an(cid:12)d under what conditions Eq. (5)
anelectronsystem,writtenasafunctionaloftheelectron can be derived from Eq. (15), it is worth mentioning
density, can be decomposed into several contributions, that this derivation is equivalent to the standard linear
response theory, in which the perturbation theory is ap-
E[n]=Ts[n]+J[n]+Exc[n]+V [n], (7) plied to the self-consistent Kohn-Sham equations.31
Equation (15) is an integral equation in which the
where T is the noninteracting kinetic energy, V is the
s exact form of the kernel is unknown. The Thomas-
total one-particle potential energy
Fermi approximation consists in approximating the ki-
netic functional to the free-electron gas case. Therefore,
V [n]= drn(r)v(r), (8) we have32
Z
J is the classical Coulomb energy δ2Ts[n] = α10n (r)−1/3δ(r−r′), (16)
δn(r)δn(r′) 9 0
J[n]= 12 drdr′n|(rr)−n(rr′|′), (9) so that the integra(cid:12)(cid:12)(cid:12)(cid:12)lnc0ondition Eq. (15) reduces to the
Z
following algebraic expression:
and E is the exchange-correlation energy density
xc
functional.17,29 From the variational principle, with the q(r)2
n (r)= [δµ−v (r)], (17)
constraint that the integrated density gives the total ind scr
4π
number of electronsN, we have the stationarycondition
for the ground-state density n (r), where
gs
δE 9 1/2
=µ. (10) q(r)= n (r)1/3 (18)
0
δn 10α
(cid:12)ngs (cid:20) (cid:21)
(cid:12)
Thevariableµhasbeensh(cid:12)(cid:12)owntobethesystemchemical and α = 35/3π4/3/10. In the standard Thomas-Fermi
potential.17,29,30 Inordertostudythescreening,wewrite model n (r) is approximated as the constant spatial av-
0
down the Hohenberg and Kohn equations, for both the eraged valence electron density, which is the number of
unperturbed(labeledwith0)andthe perturbed(labeled valence electrons in a unit cell divided by the unit cell
with 1) systems volume. However, anticipating the results for Si and Ge
to be discussed below, we have found that this approxi-
δTs[n] +v (r)+v [n (r)]+v [n (r)] = µ (,11) mationisnotsoimportant. Whatturnsouttobecrucial
δn(r) 0 H 0 xc 0 0 isthechoiceinEq. (15)forthesecondfunctionalderiva-
(cid:12)n0
(cid:12) tiveofthekineticenergy. Indeed,writingitasafunction
δδTns([rn)](cid:12)(cid:12) +v1(r)+vH[n1(r)]+vxc[n1(r)] = µ1(.12) ofonlythedifferencer−r′ correspondstoneglectingthe
(cid:12)n1 local field effects, in that the response is independent of
(cid:12)
(cid:12) the coordinates in which an external point-charge impu-
Here, v is(cid:12)the one-electron potential, vH is the Hartree
rity is located. This is equivalent, in a bulk system, to
potential
only considering the diagonal component of the recipro-
n(r′) cal space dielectric matrix. This argument is about the
v [n(r)]= dr′ , (13)
H |r′−r| kineticfunctional,anddoesnotregardthesurfacepolar-
Z izationeffects,whicharetakenintoaccountinthemodel
and v is the exchange-correlation potential. By sub- through the use of suitable boundary conditions for the
xc
tracting Eq. (11) from Eq. (12), and assuming linear Poisson equation. We know that local field effects enter
response, we obtain an expression relating the induced the theory through the kinetic functional derivative, as
6
well as through the boundary conditions of the Poisson C. Model description and results
equation. The Thomas-Fermi theory neglects the first,
but retains the second of such effects. In the bulk, only
ThePoissonequation,togetherwithEq. (5),givesthe
the first effect occurs, due to the polarization of bonds,
Thomas-Fermi equation for an external point-charge Z
and a link with the diagonal screening of the analysis of
located at the nanocrystal center,
Ref. 4 can be traced. The Thomas-Fermi theory gives
resultsinfairagreementwiththelinear-responseresults,
∇2v =−4πZδ(r)+q2[v (r)−δµ]. (20)
when the off-diagonal elements of the dielectric matrix scr scr
are neglected.
It can be written as
In order to give a first indication on the validity of
the Thomas-Fermi approximation in nanostructures, we
compare the chemical potential δµ calculated from DFT ∇2vscr =q2[vscr(r)−A] (21)
with the Thomas-Fermi prediction. Within DFT, the
chemical potential of a given structure is calculated as with the condition
(I+A) lim[rv (r)]=−Z. (22)
µ=− 2 , (19) r→0 scr
where I =E+−E and A=E −E− are, respectively, The leading boundary condition is given by the continu-
S S S S
thestructureionizationpotentialandtheelectronaffinity ity of the potential on the screening sphere,
(E isthetotalenergy).29 Wehavecalculatedthechemi-
S
calpotentialvariationδµforSi35H36,fortheisocoricP+ vscr(Rs)=vc(Rs). (23)
impurity, obtaining δµ = −2.79 eV from Eq. (19) and
δµ=−2.62 eV from the Thomas-Fermi Eq. (6) (for the Solving the Poisson equation, we find that the Thomas-
determination of the parameters, see below). Consider- Fermi expression for the effective spatial dielectric func-
ingalltheapproximationsinvolved,theagreementofless tion, defined as the ratio between external and screened
than 0.2 eV can be considered very good. potential, is
1−Q/Z {sinh[q(R −r)]+rq}+ r Q −1, 0<r ≤R
ε˜(r)= n qRs 1− QZs+ Rr QZ −1, RZo Rs ≤r ≤Rs . (24)
n o

Moreover,from the continuity of the electric field on the that
screeningsphere,arelationshipbetweentheproductqR
s
and Q can be obtained, 1−Q/Z {sinh[q(R −r)]+rq} −1, 0<r ≤R
ε˜bulk(r)= n qRs 1− QZs −1, o r ≥Rs. s
sinh(qR ) 1
s
= . (25) n o (26)
qRs 1−Q/Z This is theequation first derived in Ref. 3. Since the
material behaves like a dielectric medium beyond the
screeningradius,wegetε =1/(1−Q/Z)fromtheabove
In the following, we shall use Eq. (25) for determining s
equation, in agreement with Eq. (4).
thevariableqforeachnanocrystal,oncethetotaldensity
From this equation we have Q = 0.91 and 0.93 for Si
Q and the screening radius R are known.
s andGe,respectively,correspondingtothe bulkstaticdi-
The determination of the Thomas-Fermi parameters electric constants ε = 11.4 and 14.3. The choice for Q
s
Q and R is not obvious. We propose to use the bulk in the case of Ge is motivated by the fact that it is the
s
Q, and calculate the screening radius from the ab initio diagonal contribution to the integrated density, in the
nanocrystal-induced density. As we shall see below, the limit r → ∞.4 Also, the corresponding ε is very close
s
R dependenceonthenanocrystalsizeisveryweak,and, to the Walter andCohen pseudopotentialcalculationsof
s
a posteriori, we can say that the use of a constant value dielectric function (they used ε = 14 for bulk Ge).33
s
forthe screeningradiusdoesnotsignificantlychangethe Thisallowsustoshowthatthepresentmodelfitswellto
results. For determining Q in the bulk, it is necessary to a standard RPA calculation. Finally, we want to point
consider Eq. (24) in the limit R → ∞. It is easy to see out here that the experimental value ε = 16 is not far
s
7
1.2 1.2
TABLEI:Thomas-FermiparameterscalculatedforseveralSi (a) (b)
andGenanocrystals andfor bulkSi. Thenanocrystalradius
R,thescreeningradiusRs,andtheparameterq,ascomputed
from Eq. (25), are reported. All the quantities are in atomic
0.8 0.8
units.
R Rs q
Si35H36 11.31 5.58 0.836
0.4 0.4
Si87H76 15.17 5.39 0.866
Si147H100 17.82 5.35 0.873
Si191H148 19.58 5.36 0.871
Sibulk ∞ 5.33 0.875 0 0
Ge35H36 11.92 5.90 0.84 0 10 200 10 20
Ge191H148 20.62 5.61 0.882 r (a.u.) r (a.u.)
FIG. 4: (Color online) Integrated induced density calculated
fromthevalueweuse. Wehavecheckedthatsmalldiffer- for bulk Si [panel (a)] and Ge [panel (b)] using both DFT
encesinthestaticdielectricconstantdonotsignificantly and the Thomas-Fermi model. In panel (a), Thomas-Fermi
change our results. (full,blackline)andDFTresults(circles,green)forSi191H148
ItremainsnowtofixthescreeningradiusR . Thiscan nanocrystal. Inpanel(b),Thomas-Fermi(full,blackline)and
s
linear response results (stars, red) of Ref. 4. The diagonal
bedonestartingfromtheDFT-inducedchargeillustrated
component of screening is also shown (dashed, blueline).
in the previous section. One can define R as the radius
s
corresponding to an integrated induced charge equal to
Q. Namely, the screening radius is given by the solution
of the equation R =4.28andq =1.10forsiliconandR =4.71andq =
s s
1.08forgermanium. Althoughthepresentestimationsof
Qind(Rs)=Q. (27) Rs (q)arebigger(smaller)thanthose ofRef. 3, we shall
see below that the dielectric functions for both the bulk
However,because of the wavy behaviorshownby the in-
and nanocrystals are in excellent agreement with DFT
duced charge,there can be severalsolutions to Eq. (27).
and empirical pseudopotential results.
Amongthepossiblesolutions,achoicecanbedonestart-
Before presenting the results for the dielectric func-
ing from physical motivations; in particular, we expect
tions,letuslookattheintegrateddensitiesinbulksilicon
that the screening radius would take a value close to
and germanium. The results are shown in Fig. 4, where
(actually slightly greater than) the impurity host bond
thesolidlinecomesfromtheThomas-Fermimodelusing,
length. Indeed, the whole displaced charge has to be in-
forsilicon,thecalculatedbulkparametersandforgerma-
cluded in the model, and it is likely that, after giving off
nium those correspondingto Ge H (see Table I). In
thewavycontribution,theinducedchargeextendstothe 191 148
the case of Si [panel a], we compare the model with our
nearest-neighbor distance.4
nanocrystalDFTresults. ForGe[panelb],acomparison
In Table I, we report the calculated values for
isdonewiththelinear-responsecalculationofRef. 4. We
the Thomas-Fermi parameters, for several Si and Ge
note from Fig. 4 that the characteristic wavy structure
nanocrystals. Wehavealsoincludedinthistabletheval-
thatistypicalofthefullresponse(stars)isabsent. More
uesforbulkSithathavebeenobtainedwithacalculation
importantly, the agreement with our Thomas-Fermi cal-
on a supercell comprising 512 atoms. The important re-
culation is very good. In other words, the model is per-
sult coming from Table I is that R is quite independent
s fectly capable of reproducing the response without local
of the nanocrystal size (apart from the case of Si H ,
35 36 fieldeffects. This isalsotrueforsilicon,asshowninFig.
very small variations are seen, being less than 0.1 a.u.
4(a). HereitisclearlyseenthattheThomas-Fermicurve
goingfrom Si H to the bulk Si). We cansaythat this
87 76 liesinthemiddleoftheDFTcurve,asinthecaseofger-
independence of R of the system dimension is a mani-
s manium. We can therefore reasonably conclude that the
festation of the local nature of the response as originally
present model gives results consistent with an ab initio
suggested in Ref. 9.
calculation in which local field effects due to the bond
It is important to point out that we use a different
polarization are either neglected or play a minor role.
procedurefromthatinRef. 3inthedeterminationofthe
parameters. In that case, q was calculated starting from Within the present model, the wave-vector-dependent
the materialFermienergy,while the screenedradiuswas screening dielectric function of a bulk semiconductor
aderivedvariable,calculatedfromEq. (25). Atvariance takes the form
with that procedure, we directly calculate the screening
radius fromfirst-principlesresults and then use Eq. (25) q2+k2
ε (k)= . (28)
to getq. The values of Rs andq obtained in Ref. 3 were s q2(1−Q/Z)sin(kRs)/kRs+k2
8
(a) 16 (b)
12 3.0
12
2.5
) 8
k )
( 8 r
(
ε ~ε 2.0
4
4
1.5
0 0 1.0
0 0.5 1 0 0.5 1
k (a. u.) k (a. u.) 0 2 4 6 8 10
r (a.u.)
FIG. 5: Wave-vector-dependent screening dielectric function
for bulk Si [panel (a)] and bulk Ge [panel (b)]. The dashed FIG. 6: (Color online) Screening function ε˜(r) for Si35H36.
Solid (black)line: Thomas-Fermimodel with theparameters
lines are the RPA empirical pseudopotential calculation of
of Table I; dotted (blue) line: the same but with the pa-
Ref. 33. The solid line are the Thomas-Fermi results from
rameters of Ref. 3; dashed (red) line: real-space ab initio
Eq. (28).
calculation of Ref. 10.
In Fig. 5, the results are shown for both bulk Si and
Ge. A comparison with the empirical pseudopotential TABLEII:Impurityactivation energiescalculated withboth
calculationofWalterandCohen33(dashedline)indicates the ab initio method (EDFT) and the Thomas-Fermi model
that the model parameters listed in Table I give a very (Em) for several silicon nanocrystals. The DFT data are for
good agreement in the bulk limit. thedonor isocoric impurities P+. All thevalues are in eV.
In order to see how the present model performs in EDFT Em
the case of nanocrystals, a comparison has been done
Si35H36 3.05 2.84
with recent real-space ab initio calculations. In Fig. 6,
theSi H effectivespatialscreeningfunctioncalculated Si87H76 2.27 2.11
35 36
with the Thomas-Fermi model and the ab initio results Si147H100 1.91 1.72
taken from Ref. 10 are shown. For the Thomas-Fermi Si191H148 1.79 1.60
model, we have used both the derived values of R and
s
q given in Table I (full line) and those of Ref. 3 (dotted
line). It is seen that in this last case the agreement with
the ab initio results of Ref. 10 is not perfect, particu- using the perturbation theory as
larly for the height and the position of the main peak.
However, considering that there is no need of doing any
calculationfor getting the parametersofRef. 3, depend- 1
E =−Z ψ ψ , (29)
ing onthe applicationathand, this may surelybe a first m c ε˜(r)r c
approximation. In any case, it is interesting to see that (cid:28) (cid:12) (cid:12) (cid:29)
(cid:12) (cid:12)
althoughthecalculationsofRef. 10includetheexchange (cid:12) (cid:12)
(cid:12) (cid:12)
and correlationterms, the agreementbetween our result where ε˜(r) is the effective spatial dielectric function and
(full line) and the ab initio one (dashed line) is very im- ψ is the first empty state of the undoped nanocrystal.
c
pressive. We want to remark once again that the most Using the undoped nanocrystal DFT wave function ψ
c
importantpartofthescreeningisduetoelectroncharges and the effective dielectric function calculated with the
accumulated both near the surface and close to the im- model, we cancompute the binding energy andcompare
purity. it with the DFT results. In Table II, the binding-energy
We also performed a calculation of the donor impu- results are shown. The interesting thing is that there is
rity binding energy in order to compare the Thomas- a systematic difference between the DFT results andthe
Fermi model and DFT results. Within a DFT frame- values estimated from the model. This size-independent
work, the binding energy can be calculated as the dif- contribution is quite small, about 0.2 eV for Si. It can
ference E =I −A between the doped nanocrystal be due to both the bond polarization effects and the ex-
DFT d u
ionization energy and the undoped nanocrystal electron changeandcorrelationcontributions,whicharenottaken
affinity.34,35 The binding energy can also be calculated into account in the present model.
9
IV. CONCLUSIONS nanoscale.
The first-principlescalculationspresentedinthis work
Screening in covalent semiconductor nanocrystals has have shown that the concept of bulklike response to an
beenstudiedusingbothadvancedab initio methods and external perturbation introduced in Ref. 9 is valid also
a Thomas-Fermi model. Our DFT calculations have in a screening problem. Although the induced electron
shown that isocoric donor dopants essentially behave as density contains some oscillations, we have shown that
a point-charge giving an induced charge that agrees well it is possible to define a screening radius that is prac-
with the linear-response approximation. Comparing the tically independent of the nanocrystal dimension. The
induced integrated densities with those obtained in Ref. progressive reduction of the screening action is due to
4, we have been able to conceptually isolate the features the surface polarization. In this paper we have shown
directlyrelatedtolocalfieldeffects. Itis,atleastinprin- howthiscontributionmaybedescribedwithsimpleelec-
ciple, noteasyto distinguishthe severalcontributionsto trostatics. The important point is that now we have an
the screeningdue to localfields. However,in the specific analytical expression for the effective screening function
cases we haveanalyzed, there is anindication ofthe fact whose validity for both the nanoscale and bulk has been
thatsurfacepolarizationisthedominantlocalfieldeffect proved.
insemiconductornanocrystals. Thiswasprovedcombin- A final note should be devoted to a possible general-
ing the Thomas-Fermi theory including electrostatics of ization of the model to the case of off-center impurities.
surface polarization with the result that the screening This could be done, at least in principle, by considering
function agrees very well with state-of-the-art ab initio the matching of the Thomas-Fermi solution to the full
calculations. image potential of an off-center point-charge. However,
It is worth mentioning that local field effects related beyondthetechnicaldifficultiesindoingthat,oneshould
tosurfacepolarizationsmayhavedramaticconsequences considerthatthemethodfails,particularlyforverysmall
also in the optical response, particularly for anisotropic nanocrystals,whentheimpurityisatadistancefromthe
structure such as wires and ellipsoidal dots. Indeed, it nanocrystalsurfacethatislessthanorcomparabletothe
hasbeenshownforbothsilicon27andgermanium28wires screening radius.
thattheopticalfrequency-dependentabsorptionfunction
isstronglysuppressedforlightpolarizedperpendicularly
to the wire axis. This suppression does not come out
ACKNOWLEDGMENTS
when local fields are neglected. From a classicalpoint of
view, these local fields are again dominated by surface
polarization, as originally recognized in interpreting the We are grateful to Professor Serdar O¨˘gu¨t for a crit-
polarizationanisotropyofporoussilicon.36 Althoughthe ical reading of the manuscript. Financial support
present model cannot be generalized to the case of the by COFIN-PRIN 2005 is acknowledged. Calculations
responsetoanexternalfield,itis ofinterestto notethat havebeenperformedatCINECA-“ProgettiSupercalcolo
inbothcases,thatis,point-chargescreeningandexternal 2006” and “Campus Computational Grid”-Universit`a di
field,surfacepolarizationplaysafundamentalroleatthe Napoli “Federico II” advanced computing facilities.
∗
Electronic address: [email protected] 165311 (2005).
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