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Sahu, Devendra K

Khare, P K

Yadav, V S

Steady State Conduction Current of Polyvinylidene Fluoride Film Electrets

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Chouqule, B K

Structure and DC Conductivity Studies of Sm<SUP>3+</SUP> Substituted Mg-Zn Ferrities

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Singh, Shiv Kumar

Optogalvanic Detection of NaNe and Na<SUB>2</SUB> in a Hollow Cathode Lamp

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The Significance of the Schott Energy in the Electrodynamics of Charged Particles and their Fields

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Guptaroy, Sima

Fluctuation and Fractal Characteristics of Ring Like and Jet Like Events Produced at SPS Energies

Advisory Board and Editors

Bhakuni, Gaurav

Bisht, Shuchi

Indian J Phys. 82 (11), 1563-1575 (2008) 
•9-JHjfJ 
Interaction of an electric charge in the radial field 
of a heavy dyon 
Gaurav Bhakuni* and Shuchi Bisht 
Department of Physics, Kumaun University, 
Nainital 263 002, Uttarakhand, India 
E mail bhakugaurS ^ gmail com 
Received 29 February 2008, accepted S Julv 200S 
Abstract Considering one scalar potential and one vector potential for the interaction of an electric charge 
in the radial field of a dyon we have undertaken the study of equation of motion through Lagrangian and 
Hamiltonian formulations The relation of magnetic field with the vector potential has been given a new interpretation 
which leads to an additional angular momentum in a direction transverse to radial direction which could be 
understood as spinning top like angular momentum In presence of this additional angular momentum, the 
commutation relations between the components of total angular momentum and other related operators of the 
system take spherically symmetric forms The eigen value problem of total angular momentum and Hamiltonian 
operator has been also analysed 
Keywords . Dyon, vector potential, topspm 
PACSNo. 14 80Hv 
1. Introduction 
The subject of magnetic charge has been of great interest since the ingenious work of 
Dirac [1] In order to make the Maxwell's equations symmetric and to explain the observed 
quantization of electric charge Later Schwinger [2, 3] and Zwanziger [4] extended this 
idea to dually charged particles namely dyons and successfully developed the quantum 
field theory of these particles. Today magnetic monopoles and dyons have become the 
intrinsic parts of all current grand unified theories [5] with enormous potential importance 
in connection with their roles in catalyzing proton decay [6, 7], the quark confinement 
problem of QCD [8,9] and CP violation [10]. 
* Corresponding Author 
©2008IACS 
1564 Gaurav Bhakuni and Shuchi Bisht 
In this paper we have studied the motion of an electric charge in the radial field of a 
fixed dyon Assigning new concept to vector potential we construct the Lagrangian and 
Hamiltonian of the system which yield the correct equations of motion We have also 
undertaken the study of angular momentum operator and canonical quantization of the 
system in consideration The string singularity in vector potential is completely absent in 
this formulation, moreover being a non relativistic treatment the intrinsic spin of the particles 
has not been taken into consideration The fact, that for the electric charge in any kind of 
magnetic field and vice versa the force must be Lorentz force, has been made the central 
point in this treatment 
2. Behaviour of vector potential in radial magnetic field 
The usual electrodynamics, in absence of magnetic charge and corresponding current 
density for all situations of electromagnetic fields uses the relations, 
A = - H x r (21) 
and VxA^=H , (2 2) 
while in the case of radial magnetic field the relation (2 1) looses meaning and so happens 
with the relation (2 2) Therefore for a non vanishing vector potential we use the following 
option to define the vector potential as, 
A = ±HxrT (23) 
where rT is a vector transverse to the vector r / e 
rr ='(rj-rk)*J(rk r / ) + % - r , ) . (24) 
so that, 
r rT = 0 , (2 5) 
and the field can be obtained as, 
V r x A = H (26) 
The realtion (2 6) could be easily obtained using, eq (2 3) for A and identity for vector 
triple product In the relation (2 6) A is a function of rr rjt rk while V r is a transformed 
operator in the new relative coordinates 4, , 4} , 4k defined as, 
h £ = ' • , - ' * . 4,=rk-rlt 4k=r,~'rI (2 7) 
Interaction of an electric charge in the radial field of a heavy dyon 1565 
In this new coordinate system using the derivative transformation, 
= JLL1L+JL^ /> ,v. 
Prt f\i, Pr, P£, Pr, Pfk Pr, etc (2 8) 
we have the following transformations for derivatives 
(X 
Pr> 
P£ Pc 
(2 9) 
so that the components of v r x A, where A is retained as a vector function of old 
coordinates rt, rJt rk are obtained as, 
( V T x A ) | = -
( V 7 x 4 ) ; = 
{*T*A)k = 
fie A-
v'% 
^A / 
( 
r 
(210) 
3. Lagrangian and force law 
We consider a system of an electric charge e1 in the radial field of a dyon having electric 
charge e2 and magnetic charge g2 To describe the interaction between the two we consider 
a scalar potential f{r) and a vector potential A(r) describing the electric-electric (ev e2) 
charge coupling and electric-magnetic (ev g2) charge coupling respectively In terms of 
these potentials the Lagrangian for the relative motion of electric charge in the field of 
heavy dyon could be written as, 
L = llmr?-erf(r) + ^v A(r), 
2 c (31) 
1566 Gaurav Bhakuni and Shuchi Btsht 
where m is the reduced mass of electric charge and </>{r) and A(r) are the functions of 
spatial coordinates r{rt, rrrk) only and not having explicit dependence on time This is 
the usual way of writing the lagrangian in present electrodynamics where all situations of 
electromagnetic fields are attributable to electric charge source and corresponding current 
density where the magnetic charges and corresponding currents are absent, but in our 
case in presence of magnetic charge and an electric charge interacting with it, the behaviour 
of vector potential will be different as demonstrated in section-2 The Lagrangian given by 
eq (3 1) yields, 
(3 2) 
and, 
^ - - e ^ , + ^ - ( r , A , *rkAkl) (3 3) 
where the symbols <fi, and Afl will imply the derivatives (<>)/(<">;) and [PA^^Pr) etc 
Using the derivative transformation eqn (2 9) for A and Akl in (3 3) and substituting we 
get, 
Pr, c 
I PA, PA, 
P * A* 
"J ) 
PA 
(3 4) 
The total time derivative of eq (3 2) is, 
d PL e, r
 A A -i 
JZ mrt + — r.A,, 4 rkA, k 
dt Pr, ' clJ ,J k ' A J 
(3 5) 
Again transforming the derivatives Afj and Alk in eq (3 5) to £ coordinates using eq 
(2 9) we get, 
d oL e< 
off Pr ' c 
(PA PA) 
v */ 
(
 ** J 
+ rL 
y ^ 
PA,) 
1
 s* 
(3 6) 
Substituting eq (3 6) and (3 4) in Euler Lagrange equation and combining the terms 
we get the equation of motion as, 
mr^eJ.+^iVrxA^-rtiVrxA),]. (3 7) 
interaction of an electric charge in the radial field of a heavy dyon 1567 
Similarly finding mr} and mrk and adding vectonly we get the equation of motion as 
(3 8) mr = - e 1 V ^ - i v x ( v r , A) c 
In our case 
V T x / \ = H=r 92 
and 
r 
(3 9) 
Finally the eq of motion is obtained as 
r v x r 
mr - a —r- + // -— 
r3 r3 
where <7 = e ^ and // - e,g2 
(310) 
(3 11) 
are electric and magnetic coupling parameters respectively The magnetic coupling 
parameter is constrained by the Dirac [1] quantization condition (e,g2)/c ft (integer or 
half odd integer) 
4. Hamiltonian and force law 
Using the general definition 
Jf = Zp,r, (4 1) 
and replacing the canonical momenta p by x-(p-(eJc)A{r)) we get the gauge 
invariant Hamiltonian of the system in consideration as 
* - i a ; ( * - * * ) • * (4 2) 
where the components of kinetic momenta n are 
x. 2m 
D -^- A P,
 C A etc (4 3) 
Now the Hamilton's equatiosn yield, 
r, = 
m 1 
Pi
 C *, Dp, m 
and similar equations for ry and rk (4 4) 
1568 Gaurav Bhakuni and Shuchi Bisht 
as well as, A - ~~y~~ "c'^-' ; / + r* * ' ) ~ e ^ ' | 
and similar equations for py and pk (4 5) 
Taking the total time derivative of eq (4 4) and substituting the value of p, in that 
equation and rearranging the terms we get, 
mrt =-e^,/ + ^ [ r y (A ; i ->A / 7 ) - r f c (A fc" -^ i ) ] (46) 
Transforming the derivatives in eqn (4 6) to (<f,,£r<f*) system with the help of eq 
(2 9) and interpreting the brackets in terms of the components of V r x A with the help of 
eqs (2 10), eq (4 6) takes the form, 
mr, = - e / / + ^ - [ r y (V r x / \ ) f c - r , (V r x / l ) J (47) 
Interpreting the components of V r x A as the component of radial magnetic field and 
obtaining similar eqs for r and rk coordinates and adding the equations vectoniy, we get 
the final equation of motion as, 
r vxr 
mr
 =
 aj3+t<— (48) 
where a and // are given by eqn (3 11) The purpose of demonstrating eq (3 10) and 
(4 8) is to show the consistency of the relations (2 3) and (2 6) with the Lorentz force 
equations These equations of motion have been used by Schwinger et a/ [11] in case of 
classical non relativists description of charge-monopole and dyon-dyon scattering using 
the concept of trajectory 
5. Angular momentum operator 
In consistence with the Lagrangian and Hamiltonian formulations, the gauge invariant 
angular momentum of the system of electric charge in the field of dyon could be written 
as, 
J = rxx (51) 
where n is the kinetic momentum given by, 
n~p-^A (52) 
c 
Interaction of an electric charge in the radial field of a heavy dyon 1569 
Therefore, 
j = rxp—1 r*A. (5 3) 
Using the value of A as A = 1/2 H y rT from eq (2 3) and simplifying the vector triple 
product of second factor in eq. (5 3) as, 
lrxH*rT=l[(r rT)H-(r-/f)rT] , 
with the conditions, 
r rT = 0 and H = — r 
the total angular momentum is obtained as, 
J = £. + £ £ 
2 r 
(5 4) 
where // is the magnetic coupling parameter defined in eq (3 11) The plus sign in the 
mid of eq (5 4) will change to negative for the bound state of the system 
It may be concluded from eq (5 4) that the total angular momentum possesses an 
extra angular momentum (///2) fT besides the orbital and intrinsic spin angular momentum 
which arises due to coupling of the field of magnetic charge in dyon and the field of 
electric charge in motion The direction of the extra angular momentum is transverse to 
the radial direction therefore could be conceived like the angular momentum of spinning 
top instead of helicity like as reported by Schwinger et al [11] and Zwanziger [12] 
Reading the components of rT from eq (2 4) and using the basic commutators, 
[Lnrj] = irwIJkrk , 
[L,,(rr)y] = -/f,r,, 
[L y ,K) , ]—'X. 
(5 5) 
we can easily obtain the following spherically symmetric commutation relation between 
the components of total angular momentum, 
[j„ J^^ihe^J, , (5 6) 
1570 Gaurav Bhakunt and Shucht Btsht 
which follows, 
J2,J,] = [J2,JJ] = [J2IJ,] = 0 (5 7) 
If we use the basic commutators between the particle coordinates and the 
components of canonical momenta as well as with the potential components of the field 
as, 
[r ; lr ;]-[p,,p,] = 0 , (i) 
[ r , ,p j =/;/<*,, , (ii) 
[4 .P/J-0 , (in) | (5 8) 
[>„Py>0 , (iv) 
we get the following gauge invariant and rotationally symmetric commutation relations 
[rn,rJ] = ir,dIJ , (i) 
[*„*,] = () (.,) 
[ y „ * , ] = '*>4*'T* , (m) (5 9) 
[J^r^ihe^r, , (iv) 
and Jr J,]=[{rxx\, (r x * ) J = iti£l}kJk (v) 
In deriving the commutation relation (5 6) and same as (5 9 v) we have adopted two 
different approaches The relation (5 6) depend on the particular representation eq (2 3) of 
the vector potential A in terms of the transverse vector rT while (5 9 v) depend on the 
basic commutation (5 8 m) followed by the components of canonical momentum and that 
of vector potential The two approaches seem to be complementary to each other and 
hence justifies the basic commutator (5 8 m) in one side and the presence of additional 
angular momentum as topspin angular momentum (5 4), on the other 
Expressing the angular momentum (5 4) as the sum of two angular momenta / e 
J - J, + J2, (510) 
nteraction of an electric charge in the radial field of a heavy dyon 1571 
we find the following for the projections of J , and J2 along f and rr directions, 
r J)= r J2 = 0 
fT J,*0 (511) 
and rT J, = / ' (in) 
Now expressing the space quantization of two angular momenta along the reference z-
ixis as ( ^ ) z = m,ti and (J2)z = mji respectively where mf ranges from / to - / in the 
step of one and m/ ranges from ///2 to -(///2) in the step of one, the resulting m, 4 mfl 
values would predict the total angular momentum quantum number / as 
/ f £ / - - (5 12) 
6 Bound state and energy eigen values 
We now introduce the conditions of a - -e,e2 and // - -e,g2 in our treatment for the 
bound states to exist in case of electric charge-dyon system We extend the Pauh method 
[13] applicable to the systems of one scalar potential to the system of one scalar potential 
and one vector potential 
Writing the Runze-lenz vector as, 
1 
2mv ' r 
which could be written in the following simplified form, 
1 r R = — (Jxx-ihft) + a -
mv r 
(61) 
(6 2) 
Using the commutators (5 9 i) and (5 9 it) as well as the following relations 
[(Jx*) | t(Jxir)y] = - i^2J f c , 
(•>x*),.f \ ( J x * ) y : 2,T, ^ 
r 
00 
(in) 
(6 3) 
1572 
we obtain the commutation relation, 
2yf 
Gaurav Bhakuni and Shuchi Bisht 
[R, Rj] = it,£llkJk[ m 
Where #"is the Hamiltonian operator described by equation 
Ti2m r 
with 
The commutation relations (5 9) also yield, 
[J,w]-[R,i/] = o 
Further considering the identities, 
( J y / r ) (JXX) = JZTT2 
— ( J x ^ ) + ( J x ^ ) — =r +2///— n 
r r r r 
r r
 n ti 
n — =- — n - 2/ — 
r r r 
n J = J n = 0 
n (Jyn) + (J*n) n-2\hn2 
we get, 
R J = J R = 0 
and R 2 = R * = a2 + ^ - ( j 2 + A2) 
Now we define the new operators, 
1
 2ft 2ft l 2JfJ 
(64) 
(6 5) 
(66) 
(6 7) 
(6 8) 
(6 9) 
(610) 
Interaction of an electric charge in the radial field of a heavy dyon 1573 
and f \ > = • 2» ' 2ti 
m 
1/2 
(6 11) 
The operators r, and r2 could be identified as new independent angular momentum 
operators following the Lie algebra 
TA y r, =• /f/r, 
r 2 x r 2 = ITIT2 (6 12) 
and because of the eqs (6 9) and (6 10) we obtain, 
1 
To = • 
1 mer 
7f 2ti2 
+ 1 (613) 
The Lie algebra (6 12) and the relation (6 13) suggest a set of simultaneous eigen 
states for rf, rf, r1/f, r2* as, 
\</>}^\rvr2,r,ktr2k), 
which is also the eigen state of Hamiltonian operator j{ because of, 
[ r 1 , ; / ] = [ r 2 > « ] = 0 
The resulting eigen value equations are, 
(6 14) 
(6 15) 
r?W-r | | #>3( |
 + 1 
where 
and 
r2* *7 = m 2 
— - < m, < -1 
2 2 2 
(i) 
(") 
(in) 
(616) 
(617) 
1574 Gaurav Bhakuni and Shuchi Bisht 
Because of the construction of T, and r2 the eigen values will be simply numbers not 
in /> Using the relations (6 13) and eigen value eq (6 16) i) we get the eigen value 
equation of Hamiltonian as, 
tf|*> = E„|*), (618) 
where 
a2 1 
En
=~
m2tf r 2 ' ( 6 1 9 ) 
in which we have set k + 1 = ny and a is the electric coupling parameter The quantum 
numbers are n mv m2 and energy eigen values are hydrogen atom like as expected 
because coupling between electric charge in motion and magnetic charge of dyon does 
not add and extra energy in the system because of the nature of Lorentz force 
7. Conclusions 
Usual electrodynamics assumes the absence of magnetic charge, where the presence of 
any sort of magnetic field is expressed as Curl of a vector potential (eq 2 2) which is 
reciprocally related with the field as eq (2 1) Considering the problem of an electric 
charge in the radial field of dyon we have come to the conclusion that the current of 
electric charge couples with the radial field of magnetic charge of dyon through a vector 
potential given by eq (2 3) which is related to the field as eq (2 6) This notion is consistent 
with the Lagrangian and Hamiltonian formulations of equation of motion as eqn (3 10) and 
eq (4 8), as well as gives rise to an additional angular momentum of electric charge 
transverse to radial direction besides the orbital and intrinsic spin angular momentum and 
hence could be conceived like top spin angular momentum In presence of this spinning 
top like angular momentum the total angular momentum becomes spherically symmetric 
and leads to the possibility of formation of bound state of the system The analysis of 
eigen value problem of Hamiltonian yields hydrogen atom like energies 
References 
[1] P A M Dirac Proc Roy Soc (London) A133 60 (1931) 
[2] J Schwinger Phys Rev 144 1087 (1966) 
[3] J Schwinger Science 165 757 (1969) 
[4] D Zwanziger Phys Rev 176 1489 (1968) 
[5] C Dokos and T Tomaras Phys Rev D21 2940 (1980) 
[6] C G Callan (Jr) Phys Rev D26 2058 (1982), D25 2141 (1982) 
[7] V A Rubakov J E T P Lett 33 645 (1981), Nucl Phys B203 311 (1982) 
[8] S Mandelstam Phys Rep C23 245 (1976). Phys Rev D19 249 (1979) 
Interaction of an electric charge in the radial field of a heavy dyon 
[9] G't Hooft Nucl Phys B138 1 (1978) 
[10] E Witten Phys. Lett. B86 283 (1979) 
[11] J Schwinger et al, Annals of Physics. 101 451-495 (1976) 
[12) D Zwanziger Phys. Rev. 176 1480 (1968) 
[13J W Pault Z Physik. 36 336 (1926) 


Interaction of an Electric Charge in the Radial Field of a Heavy Dyon

Chaudhuri, Jyotipratim Ray

Barik, Debashis

Self Consistent Microscopic Theory of Frictional Ratchet in a Nonequilibrium Environment

Dubey, J D

Kushwaha, J P

Prasad, Bhimlal

Indian J. Phys. 82(12) 1701-1705 (2008) JJ. 
Surface stress calculation for one-monoiayer adsorption of 
As on the Si(001) and the Ge(001) surfaces 
J D Dubey1*, J P Kushwaha2 and Bhimlal Prasad1 
department of Physics, Vinobha Bhave University, Hazanbag-825 301, Jharkhand, India 
2J P College, Narayanpur, Tilka Manjhi Bhagalpur University, Bhagalpur-853 203, Bihar, India 
E-mail : jd_dube2004@yahoo.co.in 
Received 12 February 2008, accepted 8 May 2008 
Abstract : We report a surface stress calculation for one-monolayer adsorption of As on the Si(001) and the 
Ge(001) surfaces. The density functional theory within local density approximation has been used. A comparison 
of our two results show that the As-adsorbed-Ge surface is more isotropic than the As-adsorbed-Si surface. 
The adsorption of As removes the original Si (Ge) asymmetric dimers and forms symmetric dimers. This is 
attributed to the decrease in stress anisotropy due to the introduction of defects on the surface. This inference 
is in agreement with that of He and Zhang who studied a Sb-adsorbed-Si(001) surface. 
Keywords : Surface stress, adsorption, arsenic, germanium, silicon, density functional theory, stress 
anisotropy. 
PACS Nos. : 73.20.-r, 73.20.At, 73.20.Hb 
The adsorption of group-V elements on the surfaces of group-IV elements has been the 
subject of great interest both theoretically and experimentally in the last few years. The 
interest has grown because of the influence of group-V elements in enhancing the 
abruptness of the interface in crystal growth of heterostructures by acting as surfactants, 
and in modifying the electronic properties by acting as dopants [1]. The adsorption of 
As, under good experimental conditions leads to a well ordered symmetric dimer 
structure on both Si(001) and Ge(001) structure [2,3]. Arsenic lowers the surface free 
energy of both Ge and Si, and it can be used as surfactant in the growth of Si on 
Ge(001) [4]. The surface stress has been recognized to play important role in the 
adsorption process of As on Si(001) and Ge(001) surfaces [5,6]. In this paper we report 
our study of the surface stress calculation of 1-mono-layer adsorption of As on the 
•Corresponding Author © 2008 (ACS 
1702 J D Dubey, J P Kushwaha and Bhimlal Prasad 
Si(001) and Ge(001) surfaces employing an ab initio pseudopotential method. 
In this study we have used the density functional theory of Hohenberg, Kohn and 
Sham (HKS) [7-9] with the utility of the ABINIT code which in based on pseudopotential 
and plane waves [10]. The electron-ion interaction was included in the form of norm-
conserving pseudopotential [11] that are fully separable in the Kleinman-Bylander form 
[12]. The pseudopotentials were generated using the FHI98pp code [13]. The electron-
electron interaction was considered within the local density approximation using the 
correlation scheme of Perdew and Wang [14]. 
A brief outline of the density functional theory is given below. 
This theory puts charge density at the centre stage. The solution of Schrodinger 
equation in an external potential would result in a certain ground state wave function 
& to which is associated a certain charge density 
P ( ' I ) = £ M / > ( * I > * 2 xm\dxudx2 tfxm). (1) 
Further, there is a certain energy functional which is minimized by the unknown ground 
charge density, p. The Kohn-Sham energy functional is formally written in the form 
Hk8 = h2/2mv2+Ve„ (2) 
where the effective potential is defined as for a one-electron potential 
Vetf =VN(p) + VH(p) + Vxc(p) (3) 
The energy term associated with the nuclei-electron interaction \s(VN\p) while that 
associated with the electron-electron interaction \s(VH\p) where VH is Hartree potential. 
V„ = / p ( r t y | r - r ' | - c f r \ (4) 
The Kohn-Sham energy functional is written as 
E(p) = {-hz/2m)J2 J <pUr)*2$i(r)dr + J p(r)VSon(r)dr 
+V2ff
 P(r)p(r')/ \r - r'\ • drdr' + Ex {p(r)} . (5) 
Within local density approximation the exchange energy is expressed as 
f x { ^ ) } = / ^ ) E j / > ( 0 ] c f 3 r (6) 
where Ex [p(t)] is the exchange energy per particle of a uniform gas at a density of 
p. The exchange potential can be determined from the functional derivative of Ex[p(r)) 
LB. VX(P) = SEX(P)/Sp- (7) 
Using Hartree-Fock theory it has been shown that the exchange energy of a free 
Surface stress calculation for one-monolayer adsorption of As on the Si(001) and the Ge(001) surfaces 1703 
electron gas is given by 
E^G = 2 £ h*k2/2m 
K<K1 
- < ^ * / / * > ^
 (8) 
Comparing with HF expression for the exchange energy of a 'free electron gas* one 
obtains for the potential 
VAP) = -(e2M-{3n2p(r)}V3.
 (9) 
Correlation energies can be included in the energy functional, having the form in LDA 
Vxc [p(r)} = Vx [P(r)} + Vc {P(r)} ( 1 0 ) 
where Vc represents the contribution to the total energy beyond HF limit. Spin is 
included as 
P = P T +P t 
where T and J, represents 'up' and 'down' spins. 
The Kohn-Sham equation for the electronic structure of matter is given by 
(-/72Y2/2/T7 + vN(r) + VH(r) + Vxc [P(r)}) #,<r) « E,4>,(r). (11) 
The KS equation is solved in LDA by using pseudopotential. The pseudopotential model 
treats matter as a sea of valence electrons moving in a background of ion cores 
composed of nuclei and inert inner electrons. 
The KS equation must be 'discretized' i.e. be reduced from a virtually infinite-
unknown problem to a finite-unknown problem. For crystals, plane wave basis has been 
quite effective. 
The plane wave used is of the following form. 
**('> - E G "<fcG> exP<y /1* + G l ' r> 02) 
where k = wave vector and it is a quantum number represented as k = to,, to2, to3, 
G = reciprocal lattice vector and 
ot (k.G) = co-efficient of basis. 
When expressed in a plane wave basis, the Hamiltonian is actually dense matrix. 
Specifically, the Laplacian term of the Hamiltonian is represented by diagonal matrix 
and the potential term gives rise to a dense matrix. 
For non-periodic systems the plane wave basis is combined with a 4super cell' 
method. The super cell repeats the localized configuration to impose periodicity to the 
systems. This preserves the "artificial" validity of k and Blocks theorem which eq. (12) 
obeys. 
1704 J D Dubey, J P Kushwaha and Bhimlal Prasad 
For the surface stress calculations we considered a symmetrical unit cell which 
included an atomic slab with five layers of Ge(Si) and mono-layer of As on each 
surface. The vacuum region between slabs was taken to be about five mono-layers in 
thickness. All of the atoms were allowed to relax except the two Ge(Si) layers in the 
middle of the slab. This choice of slab thickness was found to be adequate as the 
main contribution to the surface stress come from the first three atomic layers. Single 
particle wave functions were expanded with a plane wave basis upto a kinetic energy 
cut off of 12 Ry. The integration in the Brillouin Zone was performed using 8 special 
k points sampled within the Monkhorst pack scheme [15]. 
We considered three adsorption sites for the adsorbate As atom : (a) the non-
diffused configuration, (b) 50% inter diffusion of As into the 2nd substrate layer and (c) 
100% interdiffusion of As into the 2nd substrate layer i.e. As-atomic-layer replacing the 
top Ge(Si) atomic layer. The calculated bond lengths are : As-As « 0.252 nm, As-Ge 
w 0.250 nm, Ge-Ge « 0.244 nm, As-Si « 0.247 nm, Si-Si « 0.250 nm. 
For computing the surface stress for each of the three As adsorbed surface 
geometries the following three dimensional stress tensor expression was employed. 
ft* - (C/2) • < C = (C /2 )K / , + xSafi) (13) 
where C is the height of the supercell, \ is the finite basis correction, and <ra0 the 
three dimensional supercell stress tensor. The results of the present calculation are 
given in the Table 1. These result show that the lowest energy configuration of (a) is 
characterized by stress which is compressive along the surface dimer row direction and 
tensile in the orthogonal direction. A comparison of our two results show that the As-
adsorbed-Ge surface is more isotropic than the Si-adsorbed-Ge surface. The total-
energy minimization calculations show asymmetric dimers for Si(001) and Ge(001). The 
adsorption of As removes the original Si(Ge)-dimer reconstruction and forms symmetric 
dimers [16,17]. This is attributed to the decrease in stress anisotropy due to the 
introduction of defects on the surface. This inference is in agreement with that of He 
et al [18] who studied a Sb-adsorbed-Si(001) surface. 
Table 1. Calculated surface stress in (eV/(1 x 1) unitceli), for direction both perpendicular (g±) and parallel (g^ 
to dimer. The differences (g{l - g±) represent the stress anisotropy. 
As-capped (a) 
2nd layer (b) 
Si-capped (c) 
(tf-L) 
-0.59 
-0.85 
-0.92 
As/Si 
toil) 
0.67 
0.34 
-0.15 
ton - gi) 
1.26 
1.19 
0.77 
As-capped (a) 
2nd layer (b) 
Ge-capped (c) 
(g±) 
-0.38 
-0.47 
-0.61 
As/Ge 
to,,) 
0.75 
0.32 
-0.09 
toil-0-D 
1.13 
0.79 
0.52 
Acknowledgment 
A part of this work was financially supported by the University Grants Commission of 
Surface stress calculation forone-monolayeradsorption of As on the Si(001) and the Ge(00l) surfaces 1705 
India in the form a Minor Research Project granted to JPK. We also thank the 
Universite' Catholique de Louvain, Corning Inc., for the ABINIT code. 
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Surface Stress Calculation for One-Monolayer Adsorption of as on the Si(001) and the Ge(001) Surfaces

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