Excess Knight shift theory of spin polarization in electron gas by magnetic impuritie

Ritchie, B.

English

NASA Technical Reports Server

General Disclaimer 
One or more of the Following Statements may affect this Document 
 
 This document has been reproduced from the best copy furnished by the 
organizational source. It is being released in the interest of making available as 
much information as possible. 
 
 This document may contain data, which exceeds the sheet parameters. It was 
furnished in this condition by the organizational source and is the best copy 
available. 
 
 This document may contain tone-on-tone or color graphs, charts and/or pictures, 
which have been reproduced in black and white. 
 
 This document is paginated as submitted by the original source. 
 
 Portions of this document are not fully legible due to the historical nature of some 
of the material. However, it is the best reproduction available from the original 
submission. 
 
 
 
 
 
 
 
Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19700018803 2020-03-23T19:33:42+00:00Z
X-641-70-117
PREPRINT
NASA TM X 3 $ 9 Co
SPIN POLARIZATION IN THE ELECTRON
GAS BY A MAGNETIC IMPURITY:
THEORY OF THE EXCESS KNIGHT SHIFT
BURKE RITCHIE
h
APRIL 1970
GODDARD SPACE FLIGHT CENTER
GREENBELT, MARYLAND
C4
:::]^ACMS 	 (THRU)
O	 (PAC $)
2T1
s (NAS CR OR TMX OR AD NUMBER) 	 (CATEGORY)
r
-:oin Polarization in the Electron Gas by a Magnetic Impurity:
Theory of the Excess Knight Shift
Burke Ritchie
Laboratory for Theoretical Studies, National Aeronautics and Space
Administration, Goddard Space Flight Center, Greenbelt, Maryland 20771
*NRC-NAS-NASA Resident Research Associate
:.oin Polarization in the Electron Gas by a Magnetic Impurity:
Theory of the Excess Knight Shift
Burke Ritchie
Laboratory for Theoretical Studies, National Aeronautics and Space
Administration, Goddard Space Flight Center, Greenbelt, Maryland 20771
*NRC-NAS-NASA Resident Research Associate
Abstract
In the absence of an applied magnetic field a process which
will permit the existence of a Fermi contact hyperfine interaction
at the Cu ba nucleus which does not average to zero when summed
over the randomly oriented spins of N conduction electrons is the
scattering of s-wave electrons from the long-ranged exchange poten-
tial of an open-shell impurity atom located at a distance R from
the Cu nucleus and then the Fermi contact interaction at the Cu
nucleus. This is a two-center analog of the virtual process which
is responsible for most of the Fermi contact hyperfine structure
of open-shell atoms (polarization of the closed s core). 	 The first
order energy due to the Fermi contact interaction of the conduction
electrons at the Cu nucleus which have first exchange scattered
from the 3d electrons of Mn is calculated as a function of R.
It is found to change sign between R=4.8299 (nearest neighbor
distance), R- 4.8299 and R=8, and R=8 and R-10 au, suggesting
dependence on impurity concentration experiments and experiments
in which compression or expansion can change the lattice distances.
The exchange potential is presented as a function of electron
velocity and R.	 For completeness the RKKY type spin density
waves are presented as a function of the distance from the impurity
for several values of R. Also for R-0 an estimate is made of the
exchange polarization potential due to the adiabatic polarization
of the impurity by the scattering electrons and found to be signi-
ficant.
2
4
4for the spin up eiectron and the absence of these potentials
for the spin down electron.
	 This is analogous to the virtual
process which gives the major contribution to the Fermi contact
hyperfine structure in open shell atoms due to the exchange of
an s electron with an open shell electron, its scattering into
an s excited state, then the contact interaction and the scattering
back into the original state. 5,6 This process is represented
diagrammatically by,
S	 S
Therefore many conduction electrons with randomly oriented spins
will experience a net nonzero Fermi interaction with a Cu nucleus
in the vicinity of an open shell impurity atom, in the absence of.
an
 applied magnetic field.
	 The strength of this interaction will
depend on the magnitude of the exchange potential and thus on the
distance between the Cu and impurity atoms. The interaction energy
can be of either sign and thus does not have to be in excess of
the no-impurity shift. The connection with the HWJG and other
theories now becomes more apparent. The above describes the micro-
system of a single Cu = impurity pair.
	 Since the other impurities
of the system can be distributed among 2S+1 degenerate states,
the Fermi interactions over the entire system will still average
to zero unless there is an applied field or short ranged magnetic
order among the impurity atoms (known to exist for Mn 7 ).
 This
paper will be concerned with a given pair for which the impurity
has a particular spin projection M S (M S S is the case chosen) and
thus with the zero-field shift localized in the region of this pair.
ter Fagg
t is cIcar w.'.' at
	 cDaaiiS when the field is turned on	 At zero
te.-nperature all o17 the Mn atoms occupy the lowest of the 2S+1
Zaaman-split states, and tie local shifts Cue to the Fermi
Interaction are of the ssme sign and add. 	 As the temperature
increases the levels become equally populated, and the local
shifts average to zero over the entire system.	 This is in
agreement with the observations of Sugawara.
1
k
4
5Theory
The first order energy due to the Fermi contact interaction
at the Cu nucleus is given.by ,
ivy *ko(r)12ueuN8'T/36(r)s•IJiPka(r)>
	
1
where k represents the set of quantum numbers kRm (onlyR-m-0
contribute to this integral by the operation of the delta
function8(r)) and o is an index over the spin states of N
conduction electrons, each of which has a wave function
(dropping the spin indices),
1P k (r) - (2/RO)}kja,(i)"(2.t,+l)" 'tu k l (r)P^(cos6) /(4n) 	 2
where the radial waves are normalized by t'ie condition that they
vanish at the boundary of a sphere of radius R 0 ; hence,
i - R0/7rldkim	 3
f
6is the phase shift, defined as usual by the condition,
lim ukj (r) - sin(kr - 7r /2>i + 6R)/kr
r+c*
and a R are defined such that aRa R is the-Fermi distribution
function which at zero temperature (assumed throughout) if 1
for electrons whose energies are below e F and 0 for electrons
whose energies are above, where e  is the Fermi energy, taken
to be 7ev for the free electron model of Cu b . s and 1 are the
electron and nuclear spin angular momenta respectively, p  and
u N are A /2me c and 2.22617 eti /2mp c respectively, and
6(r) - i6(r)/r 2 (21+1)P 2 (cose)/4Tr	 5
The particle wave function is given in the first Born iteration
by,
4,k(r) - e ik-r - fdr'G(r,r')V(r')eik•r
4
G(r,r )	 1/41reik r-r	 6
r-r
t'
The potential is defined,
V(r)	 2(-Z1 /rMn
 - Z2/rCu +
}
Zfdr' {*.(r')(ltd(ms,ms '))Pi^^,^(r')
IL - r'I
t * ( r ' ) a ( m s . m s ' ) P i j V+ j (r')(e	 - k 2 /2)))	 7
where ms and ms are the spin projections of the scattering and
atomic electrons respectively, Z 1
.
 N 1 and Z 2 , N 2 are the charges
and numbers of electrons of Mn and Cu respectively, e  are the
orbital energies of the bound states* *
 (r) on Cu and Mn, and Pij
permutes the coordinates of the bound and scattering electrons,
giving exchange potentials for (+) singlet and (-) triplet
scattering respectively, The key point of the potential is the
fact that when the summation over spin states is performed in 1
there is cancellation of the products of the Fermi interaction
and the coulomb (direct) interaction and cancellation of the
products of the Fermi interaction and the exchange interaction
for all interactions except those products which involve exchange
with the Mn d electrons.
Rigorously we should consider the scattering as modified
by the residual 1/r Cu (coulomb tail) potential left when the 4s
electron is removed and the band structure (resulting from use
of Bloch waves) which the less completely screened potential can
support at close distances. The first consideration is not relevant
in the first Born method of solution used here (iteration with un-
distorted plane waves), and the negiiect of band effects is at least
qualitatively correct for the long-ranged exchange process considered
8here.	 Further there is an additional potential9 V pol i Vpol,exch'
which results from the long-ranged polarization of the target by
the scattering electron and is known to affect the phase shift by
about 10 ,in the low energy elastic scattering from hydrogen atoms.
We have estimated the contribution of Vpol,exch for the fictitious
case (R-0) when the Cu and Mn atoms are united and found the
exchange potential changed by about 10%. A careful study of this
potential at nonzero R is therefore indicated (see the appendix)
We specialize eq. 6 to r-0 (the only contribution to the
delta function interaction in 1), leaving only the leading term
In the expansion of G,
Jim ^ k (r) -0 - 1/4njd r leikr'/r' 1i ( r ')eik•r')x(2/R0)}kaOeia0 /(4n)}
ry 0
8
where V is obtained by keeping cnly the s-d exchange terms in
7, since ail others cancel when 1 is evaluated. We then define,
J(k,R)/k - - 1 /4nJdr'cos(kr')/r'V(r')j 0 (kr')
	
9
where we have kept the leading term in the plane wave expansion
of e ik ^'r .	 This is equivalent to spherically averaging the potential
so that the 1-0 partial wave, which is the only waVe'which"contri-
butes to 1, is uncoupled to other partial waves.
If the spin up states ara given by 8, then the spin down states
are given by,
lim * k ( r ) - 1x(2/R0)ika0a160/(470}
r7% 0 —
}
10
9which is a consequence of the choice M S W S for the 6  ground
state of Mn. Thus to first order in J eq. 1 becomes,
k
D E S /1' a, 11 N a 4/ir6dk k J (k,R)
when the exchange potentials take the upper sign (singlet
scattering).	 The total shift is given by,
AE - 1/46E 5
 + 3/4AE t
	12
where AE  results when t!, e potentials take the upper sign
(triplet scatteri.ng ).
10
.RKKY Spin Density
The spin density function is defined here as p+-pt,
where,
PT+ s 1*kf+12
a	 IVk(r)	 e ik r - e ikr/4ar ldrIe- ik r (V(r ' ) +2 (8n/3ue11N26(r' )msm1)
V k (r^ )	 13
for large distances from the Cu nucleus. For close distances we
can write,
^ 
(r) _ e ik r - e-ik r/4wfdr eikrk /r' (V(r ' ) +2 (8n /3Uei1N26(r')msmi)
*k (r )	 14
Now substitute 14 into the right hand side of 13 and keep only
cross products of the Fermi and V interactions. When the modulus
is taken and the difference of spin down and spin up densities
summed over k, the result to first order in J is,
11
Q(r) _ .4/'r65k sin(2kr)/r 2J(k,R)x4/31, U m s m i x1 /4n
r = (r2Mn + R 2 - 2r Mn RcosBMA
	
15
we average this expression over the angles of Mn in the usual
way and define the pefturbation t6-:the electron density resulting
from the two-center scattering process as a function of the
distance from the impurity atom as,
p(r) = 1/41rrdodesinea(r) 	 16
This function is plotted in Figs. 1, 2 and 3 for several
values of R.
It is important to note that even though eq. 13 results
from the large r expansion of the Green's function (6), it
is the only surviving term in the limit r^=0 when the Fermi
interaction in 13 is evaluated, and likewise for the small
r expansion of the Green's function to give 14. Thus the
result (15):%s not asymtotic for large r.
12
Numerical Procedures
The exchange integrals defined by J(k,R) are the two-electron,
two-center type, and the integrations are performed numerically
in prolate spheroidal coordinates defined by,
&	 (rMn+rCu)/R
n	 ( rMn r Cu ) /R	 17
where R is along the polar axis of the reference frame. The
inverse distance (r-r'I are given in spheroidal coordinates
by the Neumann expansion l0,
2 /REE( 2j+1) ((j - l m ) )! / ( j+ l m ) )! ) 2 P[ ml (, )Q[ m j' (^ )P[ mI (n )P!ml (n )j m
	J 	 < j
	
> j
	
1 j
	 2
eim(O1-02)x(-1)m	 18
where P and Q are the associated Legendre functions of the first
and second kind respectively. The results were checked by taking
R-.01 in the spheroidal coordinates and checking against the Rao
integrals in spherical polar coordinates. Note that in the R-0
limit only the j-2 term of the Neumann series is nonvanishing in
the integral defined by 9 owing to the angular integrations over
and s and d pair. For RHO, however, the expansion must be carried
13
to convergence , and the contribution for each term through
j-8 is tabulated in Table 2. The overlap terms in eq. 7
were estimated to contribute about 1% at R-1 to the energy
shift and were neglected (note that only the m-0 d electron
has an overlap with the s waves).
Inspection of Table 2 shows that for R greater than about
4 the convergence is very slow. The numbers for the higher R's
are not fully converged, but are adequate to establish that the
oscillatory variation of the shift with R is not an artifact of
a nonconverged series. Also it is felt that this convergence is
adequate given the set of physical approximations used in this
calculation.
All integrals were evaluated using the analytic Hartree-
Fock d orbitals of Mn calculated by Clementi.t1
14
Results and Discussion
Values of J(k,R) and AE s are tabulated in Tables 1 and 2.
Two interesting features of these results are-the very long
range of the exchange potential and the oscillatory dependence
of the energy shift on R. The long range is familiar in the
evaluation of atomic hyperfine structure by perturbation
theory 5 ' 6 in which better than 90% of 'the contribution to the
core polarization exchange diagram (see the Introduction) comes
from the continuum excited states of the atom. This is due, of
course, to the unboundedness of the continuum functions; thus the
boundedness of the integrals is governed by.how tightly the d
electrons are bound. The oscillat-:^fy behavior of AE  can be
inferred from the oscillatory dependence of J(k,R) on k for
larger R by inspection of Table 1, and is to be expected when
the interaction is small at large R, leaving a kind of distorted
overlap of d orbitals with continuum orbitals. -This means that
there will be a complicated dependence of the shift on impurity
concentration and suggests dependence on concentration experiments
and/or experiments in which compression or expansion of the metal
can change the lattice distances.
	
Note that the interaction is
almost at a node for the nearest neighbor distance R - 4.8299 and
is larger and of opposite sign at the next nearest neighbor
distance R=6.8305.
In the interpretation of the results for larger R it is
important to appreciate the smallness of the interaction and
thus the tentativeness of the numbers of this calculation, from
the point of view of the absolute numerical accuracy but especially
from the point of view that better approximations to the physics
need to be considered. Some of these include the exact numerical
^5.
15
solution of the relevant equations to obtain scattering wave
functions more accurate than the first Born iteration used
here, the consideration of band effects, the adiabatic polar-
ization of the target (see the appendix), electron correlation,
and the deviation of the d orbitals from those of the free
atom. Also in Cu-Mn it is very important to consider the
short ranged magnetic order (clustering) of the solute atoms.
Acknowledgements. The author wishes to thank Drs. F. H. M.
Faisal and R. Drachman for helpful discussion, Dr. Merle
Riley for the u=e of his integrator, D. ifsys, and Mr. A.
Silver for able-computing assistance,
References
1. T.	 Sugawara,	 J.	 Phys.	 Soc.	 of	 Japan,	 14,	 643
	 (1959)•
2. A.	 J.	 Heeger,	 L.	 B.	 Welsh,	 M.	 A.	 Jensen	 and	 G.	 Gladstone,
Phys.
	
Rev.,	 172,	 302
	
(1968).
3. For	 a	 discussion	 see	 C.	 Kittel,	 Quantum Theory	 of	 Solids
(John Wiley	 and	 Sons,
	
inc.,	 New	 York,
	
1961),	 P.	 360.
4. M.	 A.	 Ruderman	 and	 C,	 Kittel,	 Phys.	 Rev.,	 96,	 99	 (1954);
K.	 Yosida,	 ibid.	 106,	 893
	 (')57);	 and	 Reference	 3.
5. Hugh
	
P.	 Kelly,	 Phys
	
Rev.,	 173,	 142	 (1968).
6. E.	 S.	 Chang,	 Robert	 T.	 Pu,	 and	 7-,	 P	 Das,	 Phys.	 Rev.,	 174,
s 1 (1968) .
';;. 7. Reference	 2,	 p.	 309.
r" 8. Reference	 2,	 P.	 315.
9. A. Temkin, Phys. Rev., 116, 358 (1959)•
10 J. C. Slater, Quantum Theory of Molecules and Solids.
(McGraw Hill Book Company, New York, 1963), P. 266.
1 1 .	 E. C 1 emen t i , J. Chem Phys . , 41 ,. 295 0964).
r^x
nV
16
17
N
N
co
n
7
V
Y W
^D
co
M
LA
C
m
7
v
dN
Vd
d
Ln
I-
0U...
N
OC
Y
v
7
N ? - N O LA
N LA ? M 01 ?
00 co LA 00 01n %D O I^ M O NLA O 00 O O ON N O O O O O
M 00 P^ --zr -T 00 0101 01% co O t^ ^
^ — LA M O
M 01 00 P% O M
M 01 C14 O O ON N O O O O O
O
01N
M
LA
LA
LA
C11
n
^
O «- O %D M 00 t^-LA M N fl O M U%00 M ? %D ^ ?? ^ LA %0 l!1 011 MM O ^ LA .- O ON N ^ O O O O
O1 co 1^ ? C14 ^D
-T %D co %D LA N01 %0 O %0 co co coA M M M ? M N
M N co Ln M .^ O
^ ^ O O O O O
Y O	 ^	 N	 M ?
	 %D	 co
H GC
t
I
18
r
fit•
N
-T ?
O co coLn ^' O1 O ^D ^D ^E -T N LA O O O O
^ •"• O M O O O O
N M %D N O O O O O
O ^' Q .n
co O O O O
O O O OO O O O
O O O O
Ln co
' n O O1p O O O0 0 0 00 0 0 0
o O o 0
co %D M 1^
N M -' LA 1
O O LA co N 1^O O O O •- %D N M
^D O O O O O O •-O O O O O O O dO O O O O O O O
• O O O O
LA M 100 O1 -T 00 -T
••- f^ M N r• 111LA O O N M O M O
O O O O O O O OO O O d O O• O O
O O O O O O O O
M N N N O
LA N T N S ? Ol LA
? 1.0 ON %D co -' O •--O O O O O O O OO O O O O O O OO O O O O p O O
N O N (14 M
00 M %O co LA I^ LA coM O LA Il. N M M — O
- LA O O O O O
O O O O O O O
O O O O O O O O
o m co ON %a LA O1
co M — d O1 M O
N LA M ? LA n N O
%0 N O O p O O
%D O O O O O O
O O O O O O O O
r LA LA N O
^D ? ? ? N ? O J1
N ^D ? I^ V% N O
M O O O O O O
.? M .- O O O O O
O O O O O O O O
LA N 01 ? co co Go
LA LA r- %0 41 LA %D +"O ON O %D N co r- O O
m M co .T ^- O O O
O ^-- O O O O O OO O O O O O O O
^-	 N	 M	 -W	 O► 	 co	 O
N	 ^•
co
^ N
L.
4-
O
C
O
.vi
C
mCL
x
v
W
r
N
C
x
db
C
d
t45
1n
•1
a
v
b
d
N
ud
d
L
O
4-
Z
Z
3
IA
W
d
N
d
10
aG
19
Figure Captions
Fig. 1.	 p(r)/.a where a=I/yn-u eu N x3/5 as a function of r, the
distance from Mn, averaged over the angles of Mn, for several
values of R.	 Calculated using J vhlues.:converged'through'jw2.
Fig..2.	 p(r)%(-where a -I ...	 N x3 /5 as a function of r, the distanceell
from Mn, averaged over the angles of Mn, for several values of
R. Calculated using J values converged through j-2.
Fig. 3. p(r)/a where a- %P e u N x3 /5 as a function of r, the distance
from Mn, averaged over the angles of.Mn, for several values of
R. Calculated using J values converged through j•2.
1
R
eni
0
N
O
p	 N	 Ci	 h	 ^O	 h	 m	 P
O
46
01 9 9 1 1 9 s § 0 § 9 9 A8, 9 "
S a
06
20
Appendix. Adiabatic exchange polarization potential.
The contribution of the adiabatic exchange polarization
potential is estimated as foi ,lows. When the Mn target is
polarized by a static electron at r 2 , the solution to the
first order egyati;on from Rayleigh-Schrodinge .r: perturbation
theory- ;(.for a perturbation. on the m-O'-orbital. in particular,
where nondegenerate perturbation theory can be used because
the d orbitals are not split by the dipole potential of
polarized orbital theory 9 )' is•;
- -(2) 7/2/(720)'x(5/ 4 70u 1 (r)/rP 1 (F • r2 r2P 2 (cos9) c (r < r 2 )	 1
wheree is a step function 1 f-or r<r 2 .a-nd 0. for . r>r 2 where the
radial part of the 3 d orbital (unnormalized) is taken 'to
be r 2exp(-r) (based on a binding energy of abo.ut .6 a.u. as
calculated by Clementi). We do not take Clementi's HF function
for the unperturbed orbital because then we could not find
an analytic solution to the fl rst. order eq.uati.on. The first
.order radial solution.is,
u 1 (r) - e-r(r2+r.3/2+r4/5+r5/15+11X6%.(.l4x15)+1lr7/(4x14xl5)
+22r8/(9x4xi4x15-.)+154r.9/(35x9x4x.14x15.}+308.r10%(11x35x9x4x14xl5)
+ 0(10 -4 )r 1i )
	
2
i
21
The exchange polarization potential is generated by substituing
1 for one *j in eq. 7 of the text. J(k,R-O) is about 10% of
J(k,R-O) calculated with unpol.arized orbitals.
	
It is clear
that this effect should be looked at more quantitatively.
Y
A.
i4 ^u'


Spin polarization in the electron gas by a magnetic impurity - Theory of the excess Knight shift

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server