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On anisotropic one-dimensional Heisenberg chain with impurities
Volume
29A, number
11
PHYSICS
ON ANISOTROPIC
LETTERS
ONE-DIMENSIONAL
Institut
of Physics
25 August 1969
HEISENBERG
A. SUKIENNICBI I Technical University,
CHAIN
Warsaw
WITH
IMPURITIES
I Poland
and
A. ZAGdRSKI Institute
of Experimental
Physics: Received
University
H=H,iV,
(1)
where Ho=-5
J
c f,g
(SxSx+SYSY+aSfZSgZ)-C~gH~ fg fg f
V=JC(SzSX+S;S631+aS:Sa”, 6 6 - yJ
, (2)
+
T(S;xS;+S;yS;+aS,$zS;)
- gHIJ.(S&z-S,)+ - KJ/2
T(S92
.
(3)
In the above formulae a = Jz/J, y = J’/J, 02 = S’/S, the prime refers to quantities connected with the impurity, K is a measure of the
one-ionic anisotropy, cause by the existence of a perturbating; K differs from zero in the nearest neighbourhood of the impurity only. Following the method applied in ref. 4 we obtain two types of impurity states: s-like states (even) and p-like ones (odd). It is possible to obtain a closed expression for the energy of an impurity state for the p-like mode only. The most interesting impurity energy levels are those lying beyond the spin wave spectrum in the ideal chain. Denoting the top and the bottom of the spin wave 664
Poland
9 July 1969
The impurity spin wave spectrum for a one-dimensional anisotropic (p-like) modes as a function of two anisotropy parameters.
In the last few years many papers have been devoted to an examination of impurities in the Heisenberg model of ferro-magnetism, both for three-dimensional models [l-3] and for onedimensional ones [4,5]. In the present note it is shown that taking into account anisotropy terms in the Hamiltonian of the one-dimensional chain of spins one may expect some new effects. We have taken the following form of the Hamiltonian of a chain with an impurity on the site 0:
of Warsaw.
chain of spins is obtained for odd
band by Et andEb.
respectively,
we get
E-~=JS[&~3-1)+K-1]~,‘[a(yu3-1)+K], a(Yo3-1)
(4) >l-K
for energies E of the impurity state lying above the top of the band and, E-Eb=JS[a(yu3-1)+K+1]2/[a(yu3-1)+K],
(5)
o(-y&l)<-1-K for E less thanEb. From these formulae one can see that for ya2 .: 1 (in particular for a vacancy) there may appear an impurity energy level above the top of the band for 0 6 a G max [O,(K-l)/(l-ya3)] or beneath the bottom of the band for a > > max[O, (K+l )/(l -yu3)]. In the region: max[O, (K-l )/( 1973) d CI Q max[O, (K+l)/(l -yu3)] the impurity level lies inside the band. If ~3 > 1 then there appears only the impurity level above the top of the band providing that a > max[O, (l-K)/( yu3-l)]. In the last case the impurity level appears also for an isotropic exchange integral (a = 1) for appropriate values of K. In the limiting case yu3 = 1 the energy of the impurity state lies above the top of the band and is independent of a, namely E = Et = (K- 1)2/K. It is worth while to point out that the finite onedimensional chain may represent a simplified model of a ferromagnetic thin film if one treats every chain-site as representing a whole ferromagnetic plane. Such a finite chain is part of an infinite one with two vacancies considered as two boundaries of the finite system. From this point of view localized modes in the case y = 0, u = 0
PHYSICS
Volume 29A, number 11
25 August 1969
LETTERS
correspond to surface spin waves (with in-plane wave v&tore equal to z&o) appearing in thin films if an appropriate anisotropy is taken into account [6,7]. On the other hand, one magnetic impurity in a linear chain represents a film with one monolayer different from others. Our results indicate the possibility of the existence of localized modes in this case as well.
References 1. T. Wolfram and J. Callaway. Phys. Rev. 130 (1963) 2207. 2. Li Yin-Yuan and Zhu Yan-Qing, Acta Physica Sinica 19 (1963) 753.
3. Y. Isyumov, Adv. Phys. 14 (1965) 569. 4. Li Yin-Yuan, Fang Li-Zhi and Gu Shi-Jie, 5. 6. 7.
We wish to express our sincere thanks to Prof. S. Szczeniowski for his valuable remarks concerning this paper.
Acta Physica Sinica 19 (1963) 599. R. M. White and C. M. Hogan, Phys. Rev. 167 (1968) 460. H. Puszkarski, Phys. Stat. Sol. 22 (1967) 355. L. Wojtczak, Rev. Roum. Phys. 12 (1967) 577.
*****
IMPURITY
SCATTERING
IN PURE
BISMUTH
J. M. NOOTHOVEN VAN GOGR Philips Research
Laboratoriest
N. V. Philips Gloeilampenfabrieken, Received
Eindkoven,
The Netherlands
9 July 1969
As a re< of their theory on screening of fixed charges in bismuth Brownell and Hygh calculated the related effects on charge-carrier mobilities. Fair numerical agreement is shown with values found by interpolation of experimental results on doped bismuth.
Recently Brownell and Hygh [l] computed the relaxation frequency tensors for the electron and hole band in pure bismuth, assuming ionized impurity scattering. For that purpose they calculated the screening charge densities and the potential matrix elements of the impurities [2] and took into account the peculiarities of the bismuth band structure [3]. Their theory predicts for each ellipsoid the values of the elements of a certain tensor s. Apart from a factor e, this tensor s equals the inverse of the mobility tensor of the ellipsoid concerned. The s tensors were found to be proportional to Z2A, in which 2 and A describe respectively the charge and density of the ionized impurities. The results were compared with the experiments on damping of Alfv&-waves in pure bismuth by McLachlan [4]. Fair agreement was found as far as the various ratios of the tensor elements were concerned. Since, however, neither the valencies nor the densities of the impurities in the experimental sample were known, no absolute check was possible. As a matter of fact, the pure material is rather unsuitable for comparison with a theory
on impurity scattering. Firstly no reliable way is available to determine the product .#A, while, secondly, it is precarious to disentangle the rble of impurity scattering from those of various other mechanisms. Sometimes, however, more insight into pure bismuth can be gained by experiments on doped samples. For example, tin and tellurium are known to form monovalent acceptor and donor ions respectively in the bismuth lattice [5]. If one of these elements occurs as dope, its density can be derived via the excess charge-carrier density from the Hall coefficient saturation value. In doped samples both A and 2 are thus known, the latter equalling unity, and the computational results for the elements of the s tensors can be compared with the mobility values measured. (With 9 N pure bismuth as starting material and doped densities higher than lo17 cme3, the residual impurities are neglected). In samples doped with either tin or telliurium, mobilities and charge-carrier densities of holes and electrons at 4.2oK were determined from galvanomagnetic measurements at intermediate fields [6]. Due to limitations of the experimental 685
29A, number
11
PHYSICS
ON ANISOTROPIC
LETTERS
ONE-DIMENSIONAL
Institut
of Physics
25 August 1969
HEISENBERG
A. SUKIENNICBI I Technical University,
CHAIN
Warsaw
WITH
IMPURITIES
I Poland
and
A. ZAGdRSKI Institute
of Experimental
Physics: Received
University
H=H,iV,
(1)
where Ho=-5
J
c f,g
(SxSx+SYSY+aSfZSgZ)-C~gH~ fg fg f
V=JC(SzSX+S;S631+aS:Sa”, 6 6 - yJ
, (2)
+
T(S;xS;+S;yS;+aS,$zS;)
- gHIJ.(S&z-S,)+ - KJ/2
T(S92
.
(3)
In the above formulae a = Jz/J, y = J’/J, 02 = S’/S, the prime refers to quantities connected with the impurity, K is a measure of the
one-ionic anisotropy, cause by the existence of a perturbating; K differs from zero in the nearest neighbourhood of the impurity only. Following the method applied in ref. 4 we obtain two types of impurity states: s-like states (even) and p-like ones (odd). It is possible to obtain a closed expression for the energy of an impurity state for the p-like mode only. The most interesting impurity energy levels are those lying beyond the spin wave spectrum in the ideal chain. Denoting the top and the bottom of the spin wave 664
Poland
9 July 1969
The impurity spin wave spectrum for a one-dimensional anisotropic (p-like) modes as a function of two anisotropy parameters.
In the last few years many papers have been devoted to an examination of impurities in the Heisenberg model of ferro-magnetism, both for three-dimensional models [l-3] and for onedimensional ones [4,5]. In the present note it is shown that taking into account anisotropy terms in the Hamiltonian of the one-dimensional chain of spins one may expect some new effects. We have taken the following form of the Hamiltonian of a chain with an impurity on the site 0:
of Warsaw.
chain of spins is obtained for odd
band by Et andEb.
respectively,
we get
E-~=JS[&~3-1)+K-1]~,‘[a(yu3-1)+K], a(Yo3-1)
(4) >l-K
for energies E of the impurity state lying above the top of the band and, E-Eb=JS[a(yu3-1)+K+1]2/[a(yu3-1)+K],
(5)
o(-y&l)<-1-K for E less thanEb. From these formulae one can see that for ya2 .: 1 (in particular for a vacancy) there may appear an impurity energy level above the top of the band for 0 6 a G max [O,(K-l)/(l-ya3)] or beneath the bottom of the band for a > > max[O, (K+l )/(l -yu3)]. In the region: max[O, (K-l )/( 1973) d CI Q max[O, (K+l)/(l -yu3)] the impurity level lies inside the band. If ~3 > 1 then there appears only the impurity level above the top of the band providing that a > max[O, (l-K)/( yu3-l)]. In the last case the impurity level appears also for an isotropic exchange integral (a = 1) for appropriate values of K. In the limiting case yu3 = 1 the energy of the impurity state lies above the top of the band and is independent of a, namely E = Et = (K- 1)2/K. It is worth while to point out that the finite onedimensional chain may represent a simplified model of a ferromagnetic thin film if one treats every chain-site as representing a whole ferromagnetic plane. Such a finite chain is part of an infinite one with two vacancies considered as two boundaries of the finite system. From this point of view localized modes in the case y = 0, u = 0
PHYSICS
Volume 29A, number 11
25 August 1969
LETTERS
correspond to surface spin waves (with in-plane wave v&tore equal to z&o) appearing in thin films if an appropriate anisotropy is taken into account [6,7]. On the other hand, one magnetic impurity in a linear chain represents a film with one monolayer different from others. Our results indicate the possibility of the existence of localized modes in this case as well.
References 1. T. Wolfram and J. Callaway. Phys. Rev. 130 (1963) 2207. 2. Li Yin-Yuan and Zhu Yan-Qing, Acta Physica Sinica 19 (1963) 753.
3. Y. Isyumov, Adv. Phys. 14 (1965) 569. 4. Li Yin-Yuan, Fang Li-Zhi and Gu Shi-Jie, 5. 6. 7.
We wish to express our sincere thanks to Prof. S. Szczeniowski for his valuable remarks concerning this paper.
Acta Physica Sinica 19 (1963) 599. R. M. White and C. M. Hogan, Phys. Rev. 167 (1968) 460. H. Puszkarski, Phys. Stat. Sol. 22 (1967) 355. L. Wojtczak, Rev. Roum. Phys. 12 (1967) 577.
*****
IMPURITY
SCATTERING
IN PURE
BISMUTH
J. M. NOOTHOVEN VAN GOGR Philips Research
Laboratoriest
N. V. Philips Gloeilampenfabrieken, Received
Eindkoven,
The Netherlands
9 July 1969
As a re< of their theory on screening of fixed charges in bismuth Brownell and Hygh calculated the related effects on charge-carrier mobilities. Fair numerical agreement is shown with values found by interpolation of experimental results on doped bismuth.
Recently Brownell and Hygh [l] computed the relaxation frequency tensors for the electron and hole band in pure bismuth, assuming ionized impurity scattering. For that purpose they calculated the screening charge densities and the potential matrix elements of the impurities [2] and took into account the peculiarities of the bismuth band structure [3]. Their theory predicts for each ellipsoid the values of the elements of a certain tensor s. Apart from a factor e, this tensor s equals the inverse of the mobility tensor of the ellipsoid concerned. The s tensors were found to be proportional to Z2A, in which 2 and A describe respectively the charge and density of the ionized impurities. The results were compared with the experiments on damping of Alfv&-waves in pure bismuth by McLachlan [4]. Fair agreement was found as far as the various ratios of the tensor elements were concerned. Since, however, neither the valencies nor the densities of the impurities in the experimental sample were known, no absolute check was possible. As a matter of fact, the pure material is rather unsuitable for comparison with a theory
on impurity scattering. Firstly no reliable way is available to determine the product .#A, while, secondly, it is precarious to disentangle the rble of impurity scattering from those of various other mechanisms. Sometimes, however, more insight into pure bismuth can be gained by experiments on doped samples. For example, tin and tellurium are known to form monovalent acceptor and donor ions respectively in the bismuth lattice [5]. If one of these elements occurs as dope, its density can be derived via the excess charge-carrier density from the Hall coefficient saturation value. In doped samples both A and 2 are thus known, the latter equalling unity, and the computational results for the elements of the s tensors can be compared with the mobility values measured. (With 9 N pure bismuth as starting material and doped densities higher than lo17 cme3, the residual impurities are neglected). In samples doped with either tin or telliurium, mobilities and charge-carrier densities of holes and electrons at 4.2oK were determined from galvanomagnetic measurements at intermediate fields [6]. Due to limitations of the experimental 685