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Temperature and spatial dependence of the magnetization near a surface of a ferromagnet
Solid State Communications,
Vol. 13, pp. 347—351, 1973.
Pergamon Press.
Printed in Great Britain
TEMPERATURE AND SPATIAL DEPENDENCE OF THE MAGNETIZATION NEAR A SURFACE OF A FERROMAGNET* A. Liebsch, K. Levint and K.H. Bennemannt Physics Department, University of Rochester, Rochester, New York 14627, U.S.A. (Received 13 March 1973 by A.A. Maradudin)
The temperature and spatial dependence of the magnetization in a semiinfinite, single tight-binding band model ferromagnet is calculated. It is assumed that the spatial dependence of the magnetization arises primarily because of the downward shift of the ‘atomic’ energy level in the surface layer. This shift was found to occur in Ni in a previous calculation. The effects of the surface are approximated by considering a plane of defects in an otherwise translationally invariant metal. The surface layer is furthermore assumed to be paramagnetic as was previously suggested to be the case for Ni. It is demonstrated that the range of the magnetic disturbance arising from the surface increases with temperature and that the deviation in the magnetization of a given layer from the bulk magnetization is a damped oscillatory function of the distance of the layer from the surface. While the present theory assumes a highly over simplified model for the band structure and it is, therefore, not reasonable to apply these calculations directly to real metals, the results found here are qualitatively consistent with experiments on Ni.
1. INTRODUCTION 1
calculated. A simple one band, tight binding model for a ferromagnetic d band metal is used; thus s electrons are neglected.
.
IN A PREVIOUS paper (hereafter referred to as I), it was shown that near transition metal surfaces the ‘atomic’ d level, ~ lies significantly below that of the bulk. This was seen to have strong implications on the magnetic properties of the surface layer. In particular, it was shown that the surface layer in Ni is likely to be nonmagnetic.
It is shown, using for simplicity a simple cubic tight binding band model for the d band shape, that the magnetization in the first few surface layers dcviates from the temperature dependent bulk magnetization and that this deviation becomes greater as the temperature approaches the Curie temperature 7~. The net magnetization in the first six layers beyond the surface (‘dead’) layer is less than that in the bulk for all except very small (T ~ T~)temperatures. These results are in agreement with recent experiments.2 It is also pointed out, in analogy with the single impurity problem,3 that the change in magnetization of a given layer, relative to the bulk magnetization, is an osdillatory function of the distance of the layer from the
In the present paper further implications of the downward shift of the surface d level are explored: the spatial and temperature dependence of the magnetization in the first seven layers near the surface is *
Supported by U.S. Office of Naval Research. Present Address: Physics Department, University of Califorma, Irvine, California 92664.
surface. Alfred P. Sloan fellow, Present Address: Inst. Theoret. Physik., Freie Univ. Berlin, Berlin 33, Arnimall. 3, Germany. 347
348
TEMPERATUREAND SPATIAL DEPENDENCE OF MAGNETIZATION
2. THEORETICAL EXPRESSION FOR THE MAGNETIZATION AS A FUNCTION OF POSITION The tight binding one band model Hamiltoman diagonal in k~and k~,which describes electrons of spin a in a semi-infinite metal contained in the negative half ofx space, is given by (k11 1H01k11> = ~ (k11)e.,t (k11)
Vol. 13, No.3
where 1= i + / and G~,~bUlk(E, k11)
=
$
(2ir)’
xp [ik~(R1—Ri)] dk~
—
—
(4) It follows that the Green’s function for the Hamiltonian in Equation (1) may be written 2)G0oIE,k G~(E,k11) = G~°(E, k11) + ~-L’ 11) X im~
+
+ T~1(k11)Jn10(k11), (1) where i and j refer to the layer index and 1~’2)
(5)
(~ljl—~~&)X G~1(E,k11)
~(_1/2)[4
~ G~° (E, k11) + Gf~°(E, k11)(4
i~&j
—
11’2)GOO (E k
corresponds to a summation over all layers i * / in the negative half space. Here k11 = (Jc~,k~)designates a pair of coordinates in k space. Tu(k11) is the partial Fourier transform with respect toy and z of the hopping matrix element. The operators ê~(k11)and ~ (k~1)are the analogous Fourier transforms of the creation and annihilation operators in the ith layer and n~0(k11)= ô~(k11)ê10(k11).Finally, = ~0j + Un1_0 is the spin dependent energy level position in the ith layer which, as argued in I depends also on i. Because oflength, the short-range of theto Thomas—Fermi screening it is reasonable assume on the
4
basis of arguments given in I that, except in the surface layer i = ~0l = ~ bulk is temperature independent. The surface layer is assumed to be paramagnetic as was found to be the case in I for Ni and e~ can be written ~,
X
G~°(E,k11)+
m~
i
11) U6flm_ciG~j(E,kii),
im
(6) where 6 n0 is the change in the number of electrons of spin a in the ith layer relative to the bulk. Equation (6) is expected to be bulk1 a valid approcimation to equation and LThn,0 are small corn(5) whenever le~— pared to the bandwidth. The quantity 6n10 can be computed from the equation 2f f dk~d k~f(E 6n~0= — ir ~ Im_L, dE(2 irY Fc’~ (E + iO,k 11) — f~abu1k(E+ iO, k11)], (7) —
‘-‘ii
where f(E .i) = {l + exp [(E p)/kB T] }~‘and is the temperature dependent chemical potential. —
.i
—
~
=
~
+ 6E + U(n?~l~ n~’~l~).(2) —
Here, 6E which is assumed, simplicity, to be tern4 is thefor separation between the peratureand independent surface paramagnetic bulk energy levels. This quantity is sufficiently large and negative, but its magnitude is otherwise arbitrary, that the for surface layer density of states satisfies thesocondition local paramagnetism at temperature T = 0 (which condition is defined in I). For systems with two hybridized bands, 6E can be calculated from first principles, as was demonstrated in I. The symbols n~ikIpand
Using equation (7) the change in the magnetic moment of the ith layer cSmj(T) ~ 6n1~— 6n1~can be written in terms of a coupled set of equations which involve susceptibility functions F°.° +~ and the the change in magnetization in F~, the =other jJ layers 6m 1, where 0°= — ~ in~ dEf(E — .z) (2ir)-2 Fii AAjj
7
I dky -ITI dk 2 G°~’~ Ii GJF~t”~
-IT
and
2 f dk~,f dk~
~
refer to the values of n~1~k in the paramagnetic and ferromagnetic state, respectively. 0buik1,
~‘ij =
11) for the semiinfinite6system with allcubic e~,=tight 6~1k can beband obtained for a simple binding and is exactly given by
G~°(E, k11)
=
—
~
f°°dEf(E —
p)(2ir)
{G0bt~ [G0lk+G0~+G0~~k]}. ii
The Green’s function Gu°°(E, k
(8)
(i+j)1
(i+j)i
(9)
ii
Here we khave omitted for simplicity the argument (E + 10, 11) from all Green’s functions. 3. NUMERICAL RESULTS AND CONCLUSIONS
GI°J’~(E, k11) + G!~fl1&(E,k11),
(3)
The coupled equations for the 6m1 are solved here
Vol. 13, No.3
TEMPERATURE AND SPATIAL DEPENDENCE OF MAGNETIZATION
using a simple cubic tight binding band model for e(k)
W(cosk~+ cosk~+ cosk2), (10) where W is one-sixth of the bandwidth and the lattice constant is chosen to be unity. The maximum number electrons per atom containedin the band is taken to be 2, rather than 10 as in an actual d band. It is assumed that the of electrons per magnetic atom in the bulk isper n~~hlk + ~number = 1.6 and that the moment atom at T = 0 is 0.17 which is roughly half the maximum =
—
possible moment. l’his coincides with the situation in 6 with Ni in which there are ~ 1.2 holes in the d band a zero temperature moment of ~ 0.6. The Coulomb repulsion energy U/W is, thus,7 4.5 and the Curie temperature can be shown to be kBTC 0.17 W. The Fermi energy lies in a region in which the density of states is rapidly decreasing with increasing is 6 For this bandenergy model,asthe found to be the case for Ni. susceptibility functions F~° and H~°~ may be expressed in terms of integrals of Bessel functions and integrated numerically.8 Because H~contains as leading term a higher order Bessel function than F~° and is, therefore, somewhat smaller than F~°,8 it is not unreasonable to neglect H~°, in comparison with F~°. This approximation significantly reduces the number of integrals which need to be performed and should lead to qualitatively correct results. Physically, this approximation corresponds to the assumption that the dominant effect of the surface on all the other layers is to introduce a shift in the position of the energy level at the surface layer in an otherwise translationally invariant metal. Because of the tight binding character of the bands the magnetic properties of those layers i beyond the surface layer are rather insensitive to whether the sample is infinite or semiinfinite. The mathematical formalism of the problem, thus, parallels that which is used to calculate the magnetization near a single paramagnetic impurity in a ferromagnetic host.3 The present viewpoint differs somewhat from that of Mills er al.5 since it is believed here that the major effect on the magnetization in the vicinity of the surface arises from changes in both terms of the Hartree— Fock energy levels e~= ~ + Un~..0.The role of those hopping terms T11, which are present in the Harniltonian ofassumed the bulkto but in that of the semi-infinite is benot of secondary importance in themetal, present calculation. Because it was demonstrated in I that these
349
changes are not insignificant in transition metals, the present viewpoint is believed to be a more appropriate one for the problem considered here. The energy integration in equation (8) which can be done analytically is performed first. The magnitude of the coefficients F~° in general decreases as li—/l increases and the sign of the F~° oscillates as a function of li—il. This behavior is similar to that found for 9 the dielectric function in an interacting electron gas. For ease in numerical calculation &m 1 will be assumed to be zero for i> 7. This approximation is so valid providing the temperature is sufficiently small that the magnetic disturbance does not extend beyond the first seven layers. l’his is found, using numerical methods, to be the case for kBT~0.14 W. The magnetic moment per atom m 1(T) as a function of the layer index i for 6E [defined in equation (2)] equal to — 0.2 W and for several temperatures is plotted in Fig. I. The results appear to be qualitatively insensitive to bE for reasonably small values of bE 11W. While the calculations only have meaning for discrete values of i, in order to represent the results in a convenient way, a continuous curve is plotted for m• (T) vs i. The surface layer i = 1 is paraniagnetic, as was discussed earlier. It may be seen from the figure that for a given temperature T the sign of the deviation from the bulk magnetic moment mbUn~(T)oscillates with i. At all temperatures the deviation from mb~(T)is largest in the first few layers. As the temperature increases toward T~the oscillations in each layer i about the bulk moment value became more extreme. For TI T~3~0.88, the second layer is paramagnetic. In Fig. 2 is plotted the average moment per atom m,( T) as a function of temperature for several layers i. The dotted line indicates the bulk magnetic moment. Except at very low temperatures the deviation from the bulk moment in the layers beyond the surface layer is largest in the second layer and the net magnetization in these layers is less than that of the bulk. It is clear from the figure that the average moment in the second layer is zero at some fmite temperature T below T~.While our approximations are only valid for T< T 0, by extrapolating the curves to higher temperatures it appears that the average moment in the third and fourth layers is non-zero above T0. 2 indicate that in Ni at room Recent experiments temperature the loss of magnetization in the layers near the surface is equivalent to two ‘dead’ layers
350
TEMPERATURE AND SPATIAL DEPENDENCE OF MAGNETIZATION
I
—
I
I
I
Vol. 13, No.3
I
0.20 ,T/ W
0.15 m;
~-0.10
0. 5 0~20~:::
-
cfl4
m
1
0.10
0.05
-
-
0. 0
-~
0.05
-
~\BuIk
-
N2
0.C
I
-~
Bulk
-
7
6 5 LAYER INDEX
4
3
2 0.0 I
0.0
I
0.05
~.
0.10
0.15
w
I. Magnetic moment per atom in the ith layer vs layer index i for various temperatures. For convenience, continuous curves are drawn through the discrete points,
TTc
FIG.
although at T = 0 only one layer is magnetically inactive. While it is not reasonable to make quanti. tative comparisons, the present model calculation is roughly consistent with these experiments. It should be pointed out, however, that the model calculation indicates a much less rapid decrease in the magnetization with temperature than has been observed. In reference 2 it was demonstrated that ~ excellent fit to the experimental curve of the net magnetization versus temperature for two Niwould samples seven and seventeen layers respectively be of obtained if the surface is viewed as a- layer of excess d charge in a ferromagnetic gas of d electrons and if s electrons are completely neglected. This excess d charge was assumed to arise from an s —~d charge transfer. The apparent inadequacies of this explanation
FIG. 2. Magnetic moment per atom vs temperature for various layers i. The dotted line indicates the bulk magnetic moment. of the dead layer experiments have been pointed out in I. By taking long wavelength limits and assuming the coherence length behaves like (T~ T) the authors of reference 2 show that the deviation of the magnetization from the bulk value decays exponentially in space. This spatial dependence of the magnetization should be contrasted with the damped oscillatory dependence found in the present model. However, because the coherence length believed to 5 the longiswavelength vary spatially near the surface, approximations made in reference 2 do not seem appropriate. —
~,
Acknowledgements We wish to thank R. Bass for his assistance in some numerical aspects of these calculations. —
REFERENCES 1.
LEVIN K., LIEBSCH A. and BENNEMANN K.H., Phys. Rev. B7, 3066 (1973). A brief summary of this work was published in Bull. Am. Phys. Soc. 16, 584 (1971). The present paper was originally reported as part of this longer article in a University of Rochester preprint (July 1972).
Vol. 13, No.3
TEMPERATURE AND SPATIAL DEPENDENCE OF MAGNETIZATION
351
2.
LIEBERMANN L., CLINTON 1., EDWARDS D.M. and MATHON J.,Phys. Rev. Lett. 25, 232 (1970); LIEBERMANN L., FREDKIN D.R. and SHORE H.B.,Phys. Rev. Lett. 22, 539 (1969). There is some controversy about the validity of these experimental results. See SHINJO T., MATSUZAWA T., TAKADA T., NASU S. and MURAKAMI Y.,Phys. Lett. A 36 A,489 (1971).
3. 4.
MORIYA T., Proc. mt. School ofPhysics ‘Enrico Fermi~Varenna, 1967, Academic Press, New York (1967). In the present formalism it is easy to allow the surface layer energy level shift to be determined self-consistently by including coupling to the bulk. However, because we are interested in the effects of a magnetically ‘dead’ surface layer on the other layers near the surface, bE is taken to be a fixed parameter. Using a very similar model, the effects on the surface layer magnetization of coupling to the bulk have been independently computed by FULDE P., LUTHER A and WATSON R.E., [preprint].
5.
6.
DOBRZYNSKI L. and MILLS D.L.,J. Phys. Chem. Solids 3~),1043 (1969). This behavior for the Green’s function is characteristic of tight binding bands. MILLS D.L., BEAL-MONOD M.T. and WEINER R.A., Phys. Rev. B5, 4637 (1972). HODGES L., EHRENREICH H. and LANG N.D.,Phys. Rev. 152, 505 (1966).
7.
PENN D.R.,Phys. Rev. 142,350(1966).
8.
It can be shown that ~ (~1)~”fdEf(E~p°)Tdt sin EtJ~(t)J21+2~+1 (t) 0) T~itsinEt {J 2ir~ E (—1)~7dEf(E— p 21+2~(t)[1 + (—I)’] +J2(~4f)~2~~1 (t)} n —oc 0 0 = p — ~obujk After performing theE integration at T = 0, it follows that the suswhere 1 = I functions ceptibility —/ > 0 and canp be written as summs of the one particle Green’s functions C(k, m,n) tabulated by WOLFRAM T. and CALLAWAY J. [Phys.Rev. 130, 2207 (1963)]. The magnitude of the C(k,m,n) for fixed p decreases as the indices increase. PINES D. and NOZIERES P., The Theory of Quantum Liquids, chapter 5, Benjamin, N.Y. (1966). H~
9.
Die Abhangigkeit der Magnetisierung von der Temperatur und vom Abstand zur Metalloberflãche ist berechnet für den Fall eines Ferromagneten, der in der tight-binding Naherung beschrieben wird. Es wird angenomnien, dass die Veranderung der Magnetisierung durch elne Senkung des ‘atomaren’ Energieniveaus in der Oberflachenschicht verursacht wird. In Ni ist auf Grund dieser Senkung die erste Atomschicht paramagnetisch, was in einer fruheren Veroffentlichung demonstriert wurde. Der Bereich der magnetischen Störung ninixnt mit det Temperatur zu, und das Verhalten det Magnetisierung ist durch eine räumlich oszillierende Funktion wiedergegeben. Die verwendete Methode entspricht derjenigen, mit der die Magnetisierung in der Umgebung einer paramagnetischen Verunreinigung in einem Ferromagneten berechnet wird. Wahrend das Verfahren em sehr vereinfachtes Bandmodell benutzt, stimmen die gefundenen Resultate qualitativ mit experimentellen Ergebnissen für Ni überein.
Vol. 13, pp. 347—351, 1973.
Pergamon Press.
Printed in Great Britain
TEMPERATURE AND SPATIAL DEPENDENCE OF THE MAGNETIZATION NEAR A SURFACE OF A FERROMAGNET* A. Liebsch, K. Levint and K.H. Bennemannt Physics Department, University of Rochester, Rochester, New York 14627, U.S.A. (Received 13 March 1973 by A.A. Maradudin)
The temperature and spatial dependence of the magnetization in a semiinfinite, single tight-binding band model ferromagnet is calculated. It is assumed that the spatial dependence of the magnetization arises primarily because of the downward shift of the ‘atomic’ energy level in the surface layer. This shift was found to occur in Ni in a previous calculation. The effects of the surface are approximated by considering a plane of defects in an otherwise translationally invariant metal. The surface layer is furthermore assumed to be paramagnetic as was previously suggested to be the case for Ni. It is demonstrated that the range of the magnetic disturbance arising from the surface increases with temperature and that the deviation in the magnetization of a given layer from the bulk magnetization is a damped oscillatory function of the distance of the layer from the surface. While the present theory assumes a highly over simplified model for the band structure and it is, therefore, not reasonable to apply these calculations directly to real metals, the results found here are qualitatively consistent with experiments on Ni.
1. INTRODUCTION 1
calculated. A simple one band, tight binding model for a ferromagnetic d band metal is used; thus s electrons are neglected.
.
IN A PREVIOUS paper (hereafter referred to as I), it was shown that near transition metal surfaces the ‘atomic’ d level, ~ lies significantly below that of the bulk. This was seen to have strong implications on the magnetic properties of the surface layer. In particular, it was shown that the surface layer in Ni is likely to be nonmagnetic.
It is shown, using for simplicity a simple cubic tight binding band model for the d band shape, that the magnetization in the first few surface layers dcviates from the temperature dependent bulk magnetization and that this deviation becomes greater as the temperature approaches the Curie temperature 7~. The net magnetization in the first six layers beyond the surface (‘dead’) layer is less than that in the bulk for all except very small (T ~ T~)temperatures. These results are in agreement with recent experiments.2 It is also pointed out, in analogy with the single impurity problem,3 that the change in magnetization of a given layer, relative to the bulk magnetization, is an osdillatory function of the distance of the layer from the
In the present paper further implications of the downward shift of the surface d level are explored: the spatial and temperature dependence of the magnetization in the first seven layers near the surface is *
Supported by U.S. Office of Naval Research. Present Address: Physics Department, University of Califorma, Irvine, California 92664.
surface. Alfred P. Sloan fellow, Present Address: Inst. Theoret. Physik., Freie Univ. Berlin, Berlin 33, Arnimall. 3, Germany. 347
348
TEMPERATUREAND SPATIAL DEPENDENCE OF MAGNETIZATION
2. THEORETICAL EXPRESSION FOR THE MAGNETIZATION AS A FUNCTION OF POSITION The tight binding one band model Hamiltoman diagonal in k~and k~,which describes electrons of spin a in a semi-infinite metal contained in the negative half ofx space, is given by (k11 1H01k11> = ~ (k11)e.,t (k11)
Vol. 13, No.3
where 1= i + / and G~,~bUlk(E, k11)
=
$
(2ir)’
xp [ik~(R1—Ri)] dk~
—
—
(4) It follows that the Green’s function for the Hamiltonian in Equation (1) may be written 2)G0oIE,k G~(E,k11) = G~°(E, k11) + ~-L’ 11) X im~
+
+ T~1(k11)Jn10(k11), (1) where i and j refer to the layer index and 1~’2)
(5)
(~ljl—~~&)X G~1(E,k11)
~(_1/2)[4
~ G~° (E, k11) + Gf~°(E, k11)(4
i~&j
—
11’2)GOO (E k
corresponds to a summation over all layers i * / in the negative half space. Here k11 = (Jc~,k~)designates a pair of coordinates in k space. Tu(k11) is the partial Fourier transform with respect toy and z of the hopping matrix element. The operators ê~(k11)and ~ (k~1)are the analogous Fourier transforms of the creation and annihilation operators in the ith layer and n~0(k11)= ô~(k11)ê10(k11).Finally, = ~0j + Un1_0 is the spin dependent energy level position in the ith layer which, as argued in I depends also on i. Because oflength, the short-range of theto Thomas—Fermi screening it is reasonable assume on the
4
basis of arguments given in I that, except in the surface layer i = ~0l = ~ bulk is temperature independent. The surface layer is assumed to be paramagnetic as was found to be the case in I for Ni and e~ can be written ~,
X
G~°(E,k11)+
m~
i
11) U6flm_ciG~j(E,kii),
im
(6) where 6 n0 is the change in the number of electrons of spin a in the ith layer relative to the bulk. Equation (6) is expected to be bulk1 a valid approcimation to equation and LThn,0 are small corn(5) whenever le~— pared to the bandwidth. The quantity 6n10 can be computed from the equation 2f f dk~d k~f(E 6n~0= — ir ~ Im_L, dE(2 irY Fc’~ (E + iO,k 11) — f~abu1k(E+ iO, k11)], (7) —
‘-‘ii
where f(E .i) = {l + exp [(E p)/kB T] }~‘and is the temperature dependent chemical potential. —
.i
—
~
=
~
+ 6E + U(n?~l~ n~’~l~).(2) —
Here, 6E which is assumed, simplicity, to be tern4 is thefor separation between the peratureand independent surface paramagnetic bulk energy levels. This quantity is sufficiently large and negative, but its magnitude is otherwise arbitrary, that the for surface layer density of states satisfies thesocondition local paramagnetism at temperature T = 0 (which condition is defined in I). For systems with two hybridized bands, 6E can be calculated from first principles, as was demonstrated in I. The symbols n~ikIpand
Using equation (7) the change in the magnetic moment of the ith layer cSmj(T) ~ 6n1~— 6n1~can be written in terms of a coupled set of equations which involve susceptibility functions F°.° +~ and the the change in magnetization in F~, the =other jJ layers 6m 1, where 0°= — ~ in~ dEf(E — .z) (2ir)-2 Fii AAjj
7
I dky -ITI dk 2 G°~’~ Ii GJF~t”~
-IT
and
2 f dk~,f dk~
~
refer to the values of n~1~k in the paramagnetic and ferromagnetic state, respectively. 0buik1,
~‘ij =
11) for the semiinfinite6system with allcubic e~,=tight 6~1k can beband obtained for a simple binding and is exactly given by
G~°(E, k11)
=
—
~
f°°dEf(E —
p)(2ir)
{G0bt~ [G0lk+G0~+G0~~k]}. ii
The Green’s function Gu°°(E, k
(8)
(i+j)1
(i+j)i
(9)
ii
Here we khave omitted for simplicity the argument (E + 10, 11) from all Green’s functions. 3. NUMERICAL RESULTS AND CONCLUSIONS
GI°J’~(E, k11) + G!~fl1&(E,k11),
(3)
The coupled equations for the 6m1 are solved here
Vol. 13, No.3
TEMPERATURE AND SPATIAL DEPENDENCE OF MAGNETIZATION
using a simple cubic tight binding band model for e(k)
W(cosk~+ cosk~+ cosk2), (10) where W is one-sixth of the bandwidth and the lattice constant is chosen to be unity. The maximum number electrons per atom containedin the band is taken to be 2, rather than 10 as in an actual d band. It is assumed that the of electrons per magnetic atom in the bulk isper n~~hlk + ~number = 1.6 and that the moment atom at T = 0 is 0.17 which is roughly half the maximum =
—
possible moment. l’his coincides with the situation in 6 with Ni in which there are ~ 1.2 holes in the d band a zero temperature moment of ~ 0.6. The Coulomb repulsion energy U/W is, thus,7 4.5 and the Curie temperature can be shown to be kBTC 0.17 W. The Fermi energy lies in a region in which the density of states is rapidly decreasing with increasing is 6 For this bandenergy model,asthe found to be the case for Ni. susceptibility functions F~° and H~°~ may be expressed in terms of integrals of Bessel functions and integrated numerically.8 Because H~contains as leading term a higher order Bessel function than F~° and is, therefore, somewhat smaller than F~°,8 it is not unreasonable to neglect H~°, in comparison with F~°. This approximation significantly reduces the number of integrals which need to be performed and should lead to qualitatively correct results. Physically, this approximation corresponds to the assumption that the dominant effect of the surface on all the other layers is to introduce a shift in the position of the energy level at the surface layer in an otherwise translationally invariant metal. Because of the tight binding character of the bands the magnetic properties of those layers i beyond the surface layer are rather insensitive to whether the sample is infinite or semiinfinite. The mathematical formalism of the problem, thus, parallels that which is used to calculate the magnetization near a single paramagnetic impurity in a ferromagnetic host.3 The present viewpoint differs somewhat from that of Mills er al.5 since it is believed here that the major effect on the magnetization in the vicinity of the surface arises from changes in both terms of the Hartree— Fock energy levels e~= ~ + Un~..0.The role of those hopping terms T11, which are present in the Harniltonian ofassumed the bulkto but in that of the semi-infinite is benot of secondary importance in themetal, present calculation. Because it was demonstrated in I that these
349
changes are not insignificant in transition metals, the present viewpoint is believed to be a more appropriate one for the problem considered here. The energy integration in equation (8) which can be done analytically is performed first. The magnitude of the coefficients F~° in general decreases as li—/l increases and the sign of the F~° oscillates as a function of li—il. This behavior is similar to that found for 9 the dielectric function in an interacting electron gas. For ease in numerical calculation &m 1 will be assumed to be zero for i> 7. This approximation is so valid providing the temperature is sufficiently small that the magnetic disturbance does not extend beyond the first seven layers. l’his is found, using numerical methods, to be the case for kBT~0.14 W. The magnetic moment per atom m 1(T) as a function of the layer index i for 6E [defined in equation (2)] equal to — 0.2 W and for several temperatures is plotted in Fig. I. The results appear to be qualitatively insensitive to bE for reasonably small values of bE 11W. While the calculations only have meaning for discrete values of i, in order to represent the results in a convenient way, a continuous curve is plotted for m• (T) vs i. The surface layer i = 1 is paraniagnetic, as was discussed earlier. It may be seen from the figure that for a given temperature T the sign of the deviation from the bulk magnetic moment mbUn~(T)oscillates with i. At all temperatures the deviation from mb~(T)is largest in the first few layers. As the temperature increases toward T~the oscillations in each layer i about the bulk moment value became more extreme. For TI T~3~0.88, the second layer is paramagnetic. In Fig. 2 is plotted the average moment per atom m,( T) as a function of temperature for several layers i. The dotted line indicates the bulk magnetic moment. Except at very low temperatures the deviation from the bulk moment in the layers beyond the surface layer is largest in the second layer and the net magnetization in these layers is less than that of the bulk. It is clear from the figure that the average moment in the second layer is zero at some fmite temperature T below T~.While our approximations are only valid for T< T 0, by extrapolating the curves to higher temperatures it appears that the average moment in the third and fourth layers is non-zero above T0. 2 indicate that in Ni at room Recent experiments temperature the loss of magnetization in the layers near the surface is equivalent to two ‘dead’ layers
350
TEMPERATURE AND SPATIAL DEPENDENCE OF MAGNETIZATION
I
—
I
I
I
Vol. 13, No.3
I
0.20 ,T/ W
0.15 m;
~-0.10
0. 5 0~20~:::
-
cfl4
m
1
0.10
0.05
-
-
0. 0
-~
0.05
-
~\BuIk
-
N2
0.C
I
-~
Bulk
-
7
6 5 LAYER INDEX
4
3
2 0.0 I
0.0
I
0.05
~.
0.10
0.15
w
I. Magnetic moment per atom in the ith layer vs layer index i for various temperatures. For convenience, continuous curves are drawn through the discrete points,
TTc
FIG.
although at T = 0 only one layer is magnetically inactive. While it is not reasonable to make quanti. tative comparisons, the present model calculation is roughly consistent with these experiments. It should be pointed out, however, that the model calculation indicates a much less rapid decrease in the magnetization with temperature than has been observed. In reference 2 it was demonstrated that ~ excellent fit to the experimental curve of the net magnetization versus temperature for two Niwould samples seven and seventeen layers respectively be of obtained if the surface is viewed as a- layer of excess d charge in a ferromagnetic gas of d electrons and if s electrons are completely neglected. This excess d charge was assumed to arise from an s —~d charge transfer. The apparent inadequacies of this explanation
FIG. 2. Magnetic moment per atom vs temperature for various layers i. The dotted line indicates the bulk magnetic moment. of the dead layer experiments have been pointed out in I. By taking long wavelength limits and assuming the coherence length behaves like (T~ T) the authors of reference 2 show that the deviation of the magnetization from the bulk value decays exponentially in space. This spatial dependence of the magnetization should be contrasted with the damped oscillatory dependence found in the present model. However, because the coherence length believed to 5 the longiswavelength vary spatially near the surface, approximations made in reference 2 do not seem appropriate. —
~,
Acknowledgements We wish to thank R. Bass for his assistance in some numerical aspects of these calculations. —
REFERENCES 1.
LEVIN K., LIEBSCH A. and BENNEMANN K.H., Phys. Rev. B7, 3066 (1973). A brief summary of this work was published in Bull. Am. Phys. Soc. 16, 584 (1971). The present paper was originally reported as part of this longer article in a University of Rochester preprint (July 1972).
Vol. 13, No.3
TEMPERATURE AND SPATIAL DEPENDENCE OF MAGNETIZATION
351
2.
LIEBERMANN L., CLINTON 1., EDWARDS D.M. and MATHON J.,Phys. Rev. Lett. 25, 232 (1970); LIEBERMANN L., FREDKIN D.R. and SHORE H.B.,Phys. Rev. Lett. 22, 539 (1969). There is some controversy about the validity of these experimental results. See SHINJO T., MATSUZAWA T., TAKADA T., NASU S. and MURAKAMI Y.,Phys. Lett. A 36 A,489 (1971).
3. 4.
MORIYA T., Proc. mt. School ofPhysics ‘Enrico Fermi~Varenna, 1967, Academic Press, New York (1967). In the present formalism it is easy to allow the surface layer energy level shift to be determined self-consistently by including coupling to the bulk. However, because we are interested in the effects of a magnetically ‘dead’ surface layer on the other layers near the surface, bE is taken to be a fixed parameter. Using a very similar model, the effects on the surface layer magnetization of coupling to the bulk have been independently computed by FULDE P., LUTHER A and WATSON R.E., [preprint].
5.
6.
DOBRZYNSKI L. and MILLS D.L.,J. Phys. Chem. Solids 3~),1043 (1969). This behavior for the Green’s function is characteristic of tight binding bands. MILLS D.L., BEAL-MONOD M.T. and WEINER R.A., Phys. Rev. B5, 4637 (1972). HODGES L., EHRENREICH H. and LANG N.D.,Phys. Rev. 152, 505 (1966).
7.
PENN D.R.,Phys. Rev. 142,350(1966).
8.
It can be shown that ~ (~1)~”fdEf(E~p°)Tdt sin EtJ~(t)J21+2~+1 (t) 0) T~itsinEt {J 2ir~ E (—1)~7dEf(E— p 21+2~(t)[1 + (—I)’] +J2(~4f)~2~~1 (t)} n —oc 0 0 = p — ~obujk After performing theE integration at T = 0, it follows that the suswhere 1 = I functions ceptibility —/ > 0 and canp be written as summs of the one particle Green’s functions C(k, m,n) tabulated by WOLFRAM T. and CALLAWAY J. [Phys.Rev. 130, 2207 (1963)]. The magnitude of the C(k,m,n) for fixed p decreases as the indices increase. PINES D. and NOZIERES P., The Theory of Quantum Liquids, chapter 5, Benjamin, N.Y. (1966). H~
9.
Die Abhangigkeit der Magnetisierung von der Temperatur und vom Abstand zur Metalloberflãche ist berechnet für den Fall eines Ferromagneten, der in der tight-binding Naherung beschrieben wird. Es wird angenomnien, dass die Veranderung der Magnetisierung durch elne Senkung des ‘atomaren’ Energieniveaus in der Oberflachenschicht verursacht wird. In Ni ist auf Grund dieser Senkung die erste Atomschicht paramagnetisch, was in einer fruheren Veroffentlichung demonstriert wurde. Der Bereich der magnetischen Störung ninixnt mit det Temperatur zu, und das Verhalten det Magnetisierung ist durch eine räumlich oszillierende Funktion wiedergegeben. Die verwendete Methode entspricht derjenigen, mit der die Magnetisierung in der Umgebung einer paramagnetischen Verunreinigung in einem Ferromagneten berechnet wird. Wahrend das Verfahren em sehr vereinfachtes Bandmodell benutzt, stimmen die gefundenen Resultate qualitativ mit experimentellen Ergebnissen für Ni überein.