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Knight shifts in solid noble metals
Volume
46A, number
4
PHYSICS
KNIGHT S. ABATI,
31 December
LETTERS
1973
SHIFTS IN SOLID NOBLE METALS* E. BORCHI,
P.G.
COPPI
and
S. DE GENNARO
Istituto di Fisica dell’lJniversit>, Firenze 50125, Italy Received
11 October
1973
The direct contribution to the Knight shift in so1.d noble metals is evaluated. The entire calculation is accomplished in the framework of the pseudopotentla \ formalism and the effective shape of the Fermi surface is taken into account by using the simple “eight-cone” model by Ziman. The theoretical results are discussed and compared with the available experimental data.
Theoretical calculations of the Knight shift in solid metals have been accomplished mainly for the alkaline and alkaline-earth metals (e.g. Das et al. [l] ). As the noble metals are concerned, the evaluation of the Knight shift presents some difficulties because of the distorted Fermi surface. However a simple model of the energy surfaces, the “eight-cone” model by Ziman [2], can be used to carry out calculations of the electronic properties. In this note we report on the results of a calculation of the direct contribution K to the Knoght shift for the noble metals Cu, Ag and Au, using the Ziman model and a pseudopotential formalism. Experimental measurements of the Knight shift in the noble metals have been accomplished by El Hanany and Zamir [3] for copper and by Narath [4] for silver and gold. The direct contribution K to the Knight shift is due to the hyperfine field of the s-like conduction electrons interacting with the nuclear spin via the Fermi contact interaction, K = $nxpQo(/ Gk(0)l 2), where xp is the Pauli susceptibility per volume unit of the conduction electrons, R, is the atomic volume and (1$k(0)i2) is the conduction-electron spin density at the nuclear site Ri averaged over all the Fermi surface states. In the OPW formalism $k (0) is given by
core s-wave functions at the origin, b,( lki) = (Ot(r)lexp(& *r)) is the orthogonalization parameter and the quantities C(k-G) are the coefficients of the OPW expansion. Now we assume b,(lkl) = b,(ik-GI) = b,(kF). Such an approximation, which allows remarkable simplicication of the calculation, was recently used in the evalcation of the spin density in solid cadmium [5] and was found to give only a slight enhancement of the spin density. The Knight shift is then given by K=g’Xpo~,‘o(llpbRi)12) where the enhancement Ni,
[I - C
bs(lk-Gl)Os(0)]
= [l --R,‘X:,
Enhancement
bF((L[j] 1’2 is the normaliza-
in part by the National
Research
factor
Table 1 and averaged
01,
Council.
func-
Ag Au
spin density
for the noble metals.
01,
nocl$~(o)lz~a)
52,(lrL/&0)12)b) ~-
433.43 653.19 1341.34
106.48 187.70 349.15
141.95 217.58 423.86
(I)
tion constant, t refers to the whole of the core states and s to the core s-states, O,(O) is the value of the * Work supported
and the pseudowave
tion Cp(Ri) = zGC(k-GjR;“‘exp [i(k-G)*Ri] . In the Ziman model the first Brillouin zone is supposed to be made up of eight circular cones pointing into the center along the diagonals of a cube. The axis of the cone is the reciprocal-lattice vector G corresponding to the nearest hexagonal (111) zone face. Taking the local z axis in the G direction, we define
cu
whereNk
factor O,& =
[l - ~sbs(kF)0s(0)]2
~o(I$~(0)12)
X exp[i(k-G)*Ri]
(2)
a) Obtained with u deduced measurements [ 31. b) Obtained with u deduced
from De Haas-van from optical
Alphen
measurements
16371.
255
Volume
46A, number
Direct contribution metals.
CU Ag AU
4
PHYSICS
Table 2 K to the Knight
shift for the noble
K’ird
K % b)
Exp. Knight
0.203 0.31s 0.598
0.253 0.356 0.699
0.2375 i 0.0004 0.522 i 0.0003 1.64 i 0.02
shift (%)
a) and b) as in table 1.
the dimensionless variables z = 2k, and u = Y~/(ri~G~/4m), where V, is the Fourier component of the lattice pseudopotential. The Fermi surface of the noble metals cuts the side surface of the cone at an ordinate zl. This same energy surface meets the zone boundary at z2 = 1. By knowing the pseudopotential form factor u, the values of zJ and eF (the Fermi energy) are obtained along the same lines as reported in ref. [2]. Using the “eight-cone” model we obtain after some calculations (1ik(O)i’)
= 0~~(l~(lii)12)
= (3)
Q2-l O& 0
1 [
ln _ _ ~_~~ ~___U z
r!,. 1
( Zl_
1+[(1
-z1)
-2~1/2 +u
]
>I
In table 1 we report on the enhancement factor 02, and the averaged spin density (1Gk(0)12) multiplied by a0 for the noble metals Cu, Ag and Au. We have calculated sZ,(/ Gk(0)12) using for u both the values deduced from the Haas-van Alphen experiments, as reported by Ziman [2], and the values deduced from the optical measurements of the band gap at L [6,7]. Values of O&, are taken from Micah et al. [8], who use Herman-Skillman core wave-functions. These wave-functions are not relativistic and this fact probably causes and underestimate of the O& value for Au. A general feature of the averaged spin density, i.e. marked increase of R,(i il/k(0)12) with the atomic number, is clearly reproduced in table 1. In evaluating the Knight shift one needs, in addition to the averaged spin density, the Pauli susceptibility xp. We have used for the Pauli susceptibility of the noble metals the results previously obtained by two authors of the present note [9]. The effect of the ions on the non interacting electronic susceptibility was taken into account using the Ziman model and a pseudopotential formalism, and the corrections due to the electron-electron interaction were estimated 256
LETTERS
31 December
1973
from Silverstein’s work [IO] *. The theoretical results given in ref. [9] were found to agree well with the experimental susceptibilities [I 11 Our theoretical results about the Knight shift in the noble metals are listed in table 2. To allow a comparison with experiment, the experimental data [3,4] are also reported on the last row of table 2. The results reported on table 2 show that the theory generally gives a reasonable account of the magnitude of the Knight shift in the noble metals. For copper the direct contribution to the Knight shift is in very good agreement with the experimental value, then suggesting that the exchange core polarization, (ECP), contribution shou!d be substantially small. In the silver case, in order to obtain agreement between theory and experiment, a large part of the Knight shift should arise from ECP effects. For gold, even calling for a large ECP contribution, our results are too small with respect to the experimental data. On the other hand we think more reasonable to ascribe this discrepancy to the use of non relativistic wave functions for the atomic core states. several authors [ 1 l- 13 ] pointed out that Silverstein’s theory underestimates the exchange and correlation effect. Using the formula (3.1) of ref. [lo] for the Pauli susceptibility xp and taking into account the exchange and correlation effect as evaluated by Tosi et a~.,[ 121 one obtains values of xp which are nearly 204 larger than those reported in ref. [ 91
* Recently
References [ 11 1’.Jena, T.P. Das and S.D. Mahanti, Phys, Rev. B 1 (1970) 432. [2] J.M. Ziman, Advan. Phys. 10 (1961) 1. [ 31 U. El Hanany and D. Zamir, Phys. Rev. us (1972) [4] A. Narath, Phys. Rev. 163 (1967) 232. [5] P. Jena, T.P. Das, G.D. Gaspari and S.D. Mahanti.
30.
Phys. Rev. Bl (1970) 1160. [6] 1.1’. Cornwell, Phil. Mag. 6 (1961) 727. [7] G.P. Pelts and M. Shiga, J. Phys. C2 (1969) 1835. [8] ET. Micah, G.M. Stocks andW.H. Young, J. J’hys. C2 (1969) 1653. [91 E. Borchi and S. De Gennaro. Phys. Rev. B5 (1972) 4761. [lOI S.D. Silverstain, Phys. Rev. 130 (1963) 1703. 1111 R. Dupree andC.J. Ford, Phys. Rev. B8 (1973) 1780. I121 G. Pizzimenti, M.P. Tosi and A. Villari, Lett. Nuovo Cim. 2 (1971) 81. 1131 R. Dupree and D.J.W. Geldart, Solid State Commun. 9 (1971) 145.
46A, number
4
PHYSICS
KNIGHT S. ABATI,
31 December
LETTERS
1973
SHIFTS IN SOLID NOBLE METALS* E. BORCHI,
P.G.
COPPI
and
S. DE GENNARO
Istituto di Fisica dell’lJniversit>, Firenze 50125, Italy Received
11 October
1973
The direct contribution to the Knight shift in so1.d noble metals is evaluated. The entire calculation is accomplished in the framework of the pseudopotentla \ formalism and the effective shape of the Fermi surface is taken into account by using the simple “eight-cone” model by Ziman. The theoretical results are discussed and compared with the available experimental data.
Theoretical calculations of the Knight shift in solid metals have been accomplished mainly for the alkaline and alkaline-earth metals (e.g. Das et al. [l] ). As the noble metals are concerned, the evaluation of the Knight shift presents some difficulties because of the distorted Fermi surface. However a simple model of the energy surfaces, the “eight-cone” model by Ziman [2], can be used to carry out calculations of the electronic properties. In this note we report on the results of a calculation of the direct contribution K to the Knoght shift for the noble metals Cu, Ag and Au, using the Ziman model and a pseudopotential formalism. Experimental measurements of the Knight shift in the noble metals have been accomplished by El Hanany and Zamir [3] for copper and by Narath [4] for silver and gold. The direct contribution K to the Knight shift is due to the hyperfine field of the s-like conduction electrons interacting with the nuclear spin via the Fermi contact interaction, K = $nxpQo(/ Gk(0)l 2), where xp is the Pauli susceptibility per volume unit of the conduction electrons, R, is the atomic volume and (1$k(0)i2) is the conduction-electron spin density at the nuclear site Ri averaged over all the Fermi surface states. In the OPW formalism $k (0) is given by
core s-wave functions at the origin, b,( lki) = (Ot(r)lexp(& *r)) is the orthogonalization parameter and the quantities C(k-G) are the coefficients of the OPW expansion. Now we assume b,(lkl) = b,(ik-GI) = b,(kF). Such an approximation, which allows remarkable simplicication of the calculation, was recently used in the evalcation of the spin density in solid cadmium [5] and was found to give only a slight enhancement of the spin density. The Knight shift is then given by K=g’Xpo~,‘o(llpbRi)12) where the enhancement Ni,
[I - C
bs(lk-Gl)Os(0)]
= [l --R,‘X:,
Enhancement
bF((L[j] 1’2 is the normaliza-
in part by the National
Research
factor
Table 1 and averaged
01,
Council.
func-
Ag Au
spin density
for the noble metals.
01,
nocl$~(o)lz~a)
52,(lrL/&0)12)b) ~-
433.43 653.19 1341.34
106.48 187.70 349.15
141.95 217.58 423.86
(I)
tion constant, t refers to the whole of the core states and s to the core s-states, O,(O) is the value of the * Work supported
and the pseudowave
tion Cp(Ri) = zGC(k-GjR;“‘exp [i(k-G)*Ri] . In the Ziman model the first Brillouin zone is supposed to be made up of eight circular cones pointing into the center along the diagonals of a cube. The axis of the cone is the reciprocal-lattice vector G corresponding to the nearest hexagonal (111) zone face. Taking the local z axis in the G direction, we define
cu
whereNk
factor O,& =
[l - ~sbs(kF)0s(0)]2
~o(I$~(0)12)
X exp[i(k-G)*Ri]
(2)
a) Obtained with u deduced measurements [ 31. b) Obtained with u deduced
from De Haas-van from optical
Alphen
measurements
16371.
255
Volume
46A, number
Direct contribution metals.
CU Ag AU
4
PHYSICS
Table 2 K to the Knight
shift for the noble
K’ird
K % b)
Exp. Knight
0.203 0.31s 0.598
0.253 0.356 0.699
0.2375 i 0.0004 0.522 i 0.0003 1.64 i 0.02
shift (%)
a) and b) as in table 1.
the dimensionless variables z = 2k, and u = Y~/(ri~G~/4m), where V, is the Fourier component of the lattice pseudopotential. The Fermi surface of the noble metals cuts the side surface of the cone at an ordinate zl. This same energy surface meets the zone boundary at z2 = 1. By knowing the pseudopotential form factor u, the values of zJ and eF (the Fermi energy) are obtained along the same lines as reported in ref. [2]. Using the “eight-cone” model we obtain after some calculations (1ik(O)i’)
= 0~~(l~(lii)12)
= (3)
Q2-l O& 0
1 [
ln _ _ ~_~~ ~___U z
r!,. 1
( Zl_
1+[(1
-z1)
-2~1/2 +u
]
>I
In table 1 we report on the enhancement factor 02, and the averaged spin density (1Gk(0)12) multiplied by a0 for the noble metals Cu, Ag and Au. We have calculated sZ,(/ Gk(0)12) using for u both the values deduced from the Haas-van Alphen experiments, as reported by Ziman [2], and the values deduced from the optical measurements of the band gap at L [6,7]. Values of O&, are taken from Micah et al. [8], who use Herman-Skillman core wave-functions. These wave-functions are not relativistic and this fact probably causes and underestimate of the O& value for Au. A general feature of the averaged spin density, i.e. marked increase of R,(i il/k(0)12) with the atomic number, is clearly reproduced in table 1. In evaluating the Knight shift one needs, in addition to the averaged spin density, the Pauli susceptibility xp. We have used for the Pauli susceptibility of the noble metals the results previously obtained by two authors of the present note [9]. The effect of the ions on the non interacting electronic susceptibility was taken into account using the Ziman model and a pseudopotential formalism, and the corrections due to the electron-electron interaction were estimated 256
LETTERS
31 December
1973
from Silverstein’s work [IO] *. The theoretical results given in ref. [9] were found to agree well with the experimental susceptibilities [I 11 Our theoretical results about the Knight shift in the noble metals are listed in table 2. To allow a comparison with experiment, the experimental data [3,4] are also reported on the last row of table 2. The results reported on table 2 show that the theory generally gives a reasonable account of the magnitude of the Knight shift in the noble metals. For copper the direct contribution to the Knight shift is in very good agreement with the experimental value, then suggesting that the exchange core polarization, (ECP), contribution shou!d be substantially small. In the silver case, in order to obtain agreement between theory and experiment, a large part of the Knight shift should arise from ECP effects. For gold, even calling for a large ECP contribution, our results are too small with respect to the experimental data. On the other hand we think more reasonable to ascribe this discrepancy to the use of non relativistic wave functions for the atomic core states. several authors [ 1 l- 13 ] pointed out that Silverstein’s theory underestimates the exchange and correlation effect. Using the formula (3.1) of ref. [lo] for the Pauli susceptibility xp and taking into account the exchange and correlation effect as evaluated by Tosi et a~.,[ 121 one obtains values of xp which are nearly 204 larger than those reported in ref. [ 91
* Recently
References [ 11 1’.Jena, T.P. Das and S.D. Mahanti, Phys, Rev. B 1 (1970) 432. [2] J.M. Ziman, Advan. Phys. 10 (1961) 1. [ 31 U. El Hanany and D. Zamir, Phys. Rev. us (1972) [4] A. Narath, Phys. Rev. 163 (1967) 232. [5] P. Jena, T.P. Das, G.D. Gaspari and S.D. Mahanti.
30.
Phys. Rev. Bl (1970) 1160. [6] 1.1’. Cornwell, Phil. Mag. 6 (1961) 727. [7] G.P. Pelts and M. Shiga, J. Phys. C2 (1969) 1835. [8] ET. Micah, G.M. Stocks andW.H. Young, J. J’hys. C2 (1969) 1653. [91 E. Borchi and S. De Gennaro. Phys. Rev. B5 (1972) 4761. [lOI S.D. Silverstain, Phys. Rev. 130 (1963) 1703. 1111 R. Dupree andC.J. Ford, Phys. Rev. B8 (1973) 1780. I121 G. Pizzimenti, M.P. Tosi and A. Villari, Lett. Nuovo Cim. 2 (1971) 81. 1131 R. Dupree and D.J.W. Geldart, Solid State Commun. 9 (1971) 145.