A new approach to the study of an Anderson impurity in a linear chain

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A new approach to the study of an Anderson impurity in a linear chain

A.S. da Rosa Simões, J.R. Iglesias

To cite this version:

A.S. da Rosa Simões, J.R. Iglesias. A new approach to the study of an Anderson impurity in a linear chain. Journal de Physique, 1986, 47 (6), pp.967-972. �10.1051/jphys:01986004706096700�.

�jpa-00210294�

A new approach ^{to the} study ^{of an Anderson} impurity ^{in a} linear chain

A. S. da Rosa Simões and J. R. Iglesias

Instituto de Fisica, Universidade Federal do Rio Grande do Sul, ^{90000 Porto} Alegre, ^{RS, Brasil}

.

(Reçu ^{le 17} septembre ^1985, accepté sous forme définitive ^le 6février 1986)

Résumé.

Nous étudions la densité locale d’états et le moment magnétique ^d’une impureté d’Anderson dans une

chaîne linéaire au moyen d’une méthode de récurrence appliquée ^aux fonctions de Green du premier ^{et second}

ordre. Cette approche ^nous permet ^{d’obtenir une solution exacte de la «} correction de diffusion » de l’alliage ^ana- logue ^{dans le cas d’une} impureté, ^{Nous étudions à la fois} l’hybridation ^{locale et de} plus proche voisin, ^et comparons

nos résultats avec ceux obtenus par d’autres approximations.

Abstract.

The ^local density ^{of states and the} magnetic ^{moment of an Anderson} impurity ^{in a} linear chain are

studied by ^{means of a} recurrence technique, applied ^to the first- and second-order Green functions. This approach

enables to obtain an exact solution of the ^« scattering correction ^» of the alloy analogy ^{for the one} impurity ^case.

Both local and nearest neighbour hybridization ^are studied, and the results are compared ^with those obtained using ^other approximations.

Classification Physics ^Abstracts

71.55

1. Introduction.

The Friedel-Anderson model [1, 2], that describes the

magnetic behaviour of a transition metal impurity

in an sp-lattice, ^{has three basic} ingredients : ^an impu- rity ^{with a} non-degenerate d-level and a Coulomb

repulsion ^U between the _up- and down-spin states, ^a band of non-interacting conduction s-electrons and a

local hybridization between the d- and s-electrons.

Different approximations ^{have been} applied ^{to this} model, ^{in order to} study the formation of a local

magnetic ^moment [3] ^and Kondo effect [4].

Recently, ^{a real space «} decimation » method, ^which permits ^{to calculate the} one-particle Green’s functions of a disordered linear chain, ^{has been} developed [5] ^and applied ^{to the} periodic Anderson Hamiltonian within the alloy analogy approximation [6].

Here, ^we extend this technique ^to the calculation of the second order one-particle ^Green functions which appear when Coulomb interactions are taken into account. We study ^{the local} magnetic ^{moment of a} d-impurity ^{in a} linear chain described by ^a tight- binding Hamiltonian. In section 2, ^{the case of local} hybridization ^is analysed, and in section 3 hybridiza-

tion between the impurity-site ^{and its} nearest-neigh-

bours is considered. The conclusions are presented ⁱⁿ

section 4 and, ^{in an} Appendix, ^the problem ^of charge

transfer is discussed.

2. Local hybridization.

The Hamiltonian

describes a linear chain with an impurity ^{at the} origin.

The Hamiltonian of the impurity ^is

where d,,+,(d.,) ^creates (annihilates) ^a d-electron with

spin ^{6 at the} impurity (o)-site, E. ^{is the} d-energy ^level

and U the d-d Coulomb repulsion.

The conduction band is represented by ^a tight- binding Hamiltonian :

where Ci(CiC1) ^creates (annihilates) ^an s-electron with

spin ^{6 at the} i-site, EB is the s-atomic level, t ^{is the} hopping integral ^{and the sum} ( ij ) ^{runs over nearest} neighbours only. ^{The sites are} arranged ^{to form a}

linear chain. One must note that the sum over i includes the o-site. This means that we are assuming

the impurity having ^{also an} s-level, ^{at the same} energy of the rest of the chain. The limitations introduced by

this assumption will be discussed in the Appendix.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004706096700

968

In the Anderson model [4], ^the hybridization

between d- and s-electrons is described by ^{a term of the}

form

where Vo ^{is the} strength ^{of the} hybridization ^considered independent ^{of k and} C:a(Cka) ^creates (annihilates) ^a

conduction electron of spin ^{Q in a k-state.}

Fourier transforming, Hhyb ^{can be written as :}

so we are dealing ^{with a local} hybridization ^term.

In figure ^{1, a schematic} representation of the chain with the impurity ^{is shown.}

To obtain the local density ^{of states and the} magne- tic moment at the o-site, ^{we must} calculate the first- and second-order one-particle ^retarded Green func-

tions, ^{as defmed} by ^Zubarev [16]

and

being ^{z = w +} ib, ^the complex frequencies.

Using ^the equation ^{of motion} method, ^{we have :}

and

where the atomic Green functions ^{are :}

and the second order Green function is

Fig. ^1.

^Schematic representation of the chain with one

impurity ^{at the} origin. ^{In the case of local} hybridization, V1

^{0. With nearest} neighbour hybridization, V0

^0.

In order to describe in some detail the calculation of

Gd.d’(z), ^{we write} explicitly ^the expression ^for G:oda(z),

and ^we have two new Green functions, GIoQ(z) ^and G.!’(z) ^which satisfy :

Substituting equations (16) ⁱⁿ equation (15) ^{we obtain}

with

and

Equation (17) ^{has the same structure as} equa- tion (15), ^{that is, it} represents ^{a new} linear chain where the odd sites have been « decimated ^» [5]. ^The

new hopping ^energy t(z) represents ^the transfer of ^one electron from the o-site to the 2(2)-site, ^and {k(z)

is the Green function of an « effective » atom of the

new chain. Repeating ^this procedure ^{N times; when}

the number of decimations, N, increases, t1N)(Z)

decreases. For N -+ oo, t(N)(Z) -> 0 and g’ ()(z) ^- g’*(z) [17]. ^The renormalized Green function go (z)

contains the relevant information of all the atoms of the chain on the o-site.

After decimation, ^{the Green function} G:a(z) ^{is :}

and, ^when substituted in equation (8) ^{we obtain}

This equation ^{is an exact result and, as} equations (18)

are the same for a linear chain of identical atoms [5],

the value of g’*(z) which appears in the denominator of equation (20) ^{is the exact result of the local Green}

function of a pure linear chain (1).

To calculate r:odC1(z) ^from equation (10), ^{we need the}

(1) ^{It can also be} evaluated, ^for example, ^{from the know-}

ledge ^{of the} density ^{of states} by ^{means of}

expression ^for r::(1 (z) :

The last term of equation (21) ^is responsible ^{for the}

« resonance broadening » of the d-level. Here, ^we neglect ^{this term. Then, our} approximation ^is equiva-

lent to the ^« scattering correction » of the alloy analogy ^method [7]. ^{As a} consequence, ^terms of order

higher ^{than Y2 are not} considered. This is reasonable when studying the formation of local moments but, obviously, ^no Kondo effect will be found.

Neglecting ^{the last term of} equation (21), r:a(z)

is in a form suitable to be decimated, ^{and we obtain :}

Within the same method and approximations, ^we

have :

To perform ^{the numerical} calculations, ^{we have taken} EB ⁼ 0, ^t

^{1.0 and} the Fermi level at the centre of the conduction band, i.e., FF ⁼ 0. The d-d Coulomb

repulsion ^U has been taken equal ^to the bandwidth : U ⁼ 4 1 t 1. Finally, ^an asymmetry parameter q ^is defined as

to describe the relative position of the d-level with respect ^{to the centre of the band}

3. Numerical results.

Here we study ^the occupation ^{number at the} impurity site, the formation or not of a local magnetic ^moment

and the appearance of Friedel-Anderson virtual bound states in the d-local density ^{of states. The para-}

meters of interest ^are the asymmetry ’1, the hybridiza-

tion Yo and the Coulomb repulsion ^U.

In figure 2, the mean value of the number of d-elec-

trons at the impurity ^{site and for one} spin ^direction

is plotted ^{as a function} of ’1 ^and V.. ^{The case} V. ^{= 0}

Fig. ^2.

^{Mean value of the} occupation number ( n,’ , > ^at

the impurity ^{site as a function} of il ^and V0, ^{for U}

^4.0.

corresponds ^to the atomic limit and n,’ >

^0.5

for - 2 Eo ^{0. One can see} that the total number of d-electrons is one over a wide _range of values of tj

and Vo. ^For V. :0 ⁰ the variation of ( nd, > ^{is rounded}

as it is to be expected ^{because the} hybridization

increases the width of the d-levels. When q ^{= 0 the}

total number of d-electrons is ( nd r > ⁺ nd 1 > ^{= 1.}

For q 0, ^{as the} Fermi level is fixed, ^a self-consistent calculation is required ^{in order to be sure that no}

extra charge ^{is added to the} system (or subtracted, when q > 0). ^This point ^{will be} discussed in the

Appendix.

The magnetic ^moment is defined as :

and it is zero for all values of _tl, V0 ^{and U, as it is to be} expected ^{from the} rotational invariance of the Hamil- tonian [8]. ^It is convenient to define the second moment, which ^measures the fluctuations of the

magnetic ^moment [9, 8]

The mean values ( nd6 > ^{are obtained from the} imagi-

nary part ^of G dl (z), ^defined by equation (22.1). ^The

mean value of the product nd, nd, > ^{is derived from} rdd,a(Z) (Eq. (22.2)).

The variation of ( m2 ) ^{as a} function of U for

Yo ⁼ 1.0, and q ^{= 0} is shown in figure ^{3. The mean}

square ^moment increases continuously ^{from 0.5} (for ^{U =} 0) ^{to 1.0} (for U Jit 8). ^The correlation ( nf nf ) ^{goes to zero when U increases indicating}

the existence of a local magnetic ^{moment in the sense} proposed by Kemeny (1966). Also, ^{we have} plotted m2 > ^{as a} function of Yo (see Fig. 4), ^{for different}

values of ’1 ^{and for U}

4.0. In the atomic case (V0 ⁼ 0)

the value of , m2 > is maximum when Eo is below the

970

Fig. ^3.

Mean square value of the magnetization m2 >

as a function of U, ^for V 0

^1.0 and q

^0.

Fig. ^4.

^{The value} of m2 > ^{as a} function of V 0 for ^diffe-

rent values of _17, with U

4.0.

Fermi level and E. ^{+ U above it} (I ^{11 I} 2.0) ^{and one}

has just ^one d-electron. In the case of full or no occu-

pation (I 11 I > 2) m2 > ^{= 0. If the} hybridization ^is

not zero the magnetism decreases because the loca- lized states have a finite _energy width which enlarges

when the hybridization increases. The symmetric ^case

is the one in which the magnetism ^is strongest ^{but all}

curves converge ^{to a common} value of ( m2 ) ^{= 0.5} (non-magnetic case) when Yo ^{is much} bigger ^{than the}

bandwidth. Similar results have been obtained by

reference [8], using ^{a different} approximation.

We can conclude that the Coulomb repulsion ^U,

that separates ^the d-levels, ^favours magnetism, ^{and the} hybridization V0, ^that enlarges ^{the virtual bound}

states, opposes ^to it.

It is interesting ^to plot ^the density of d-states at the impurity ^{site, to observe the} shape of the bound

Fig. ^5.

Density ^{of d-states for the} symmetric case 11

0

as a function of V0, ^{with U}

^4.0.

and/or virtual bound states. In figure 5, ^{we show} the

density of d-states for the symmetric case t1 ⁼ 0, ^{as a} function of V.. ^{Bound states} appear below and above the band region (which goes from - 2 to 2) ^{and the} weight ^{of those bound states increases} when Yo increases; also, ^the separation of the bound states from the band increases for greater ^{values of} V.. ^The

presence of bound states outside the band region ^{is an} expected result in the study ^{of one} impurity ^{in a linear}

chain [10] ^{and is an} additional confirmation of the exactness of the calculation. On the other side, ^virtual bound states appear ^near the band edges. ^This gives,

in total, ^four peaks ^{for the} d-density ^{of states.}

The virtual bound states are easier to identify ⁱⁿ

the density ^{of states} for t1 ^{= 1.0} (Fig. 6) ^when E. ^is

Fig. ^6.

^{Same as} figure ⁵ four 11

^1.0.

inside the band. One of them appears ^at the _energy

Eo = - 1.0 and has ^a form that looks like ^a Lorent- zian [1]; ^{the width} increases with V; ^{as it is to be} expected. ^{A bound state is} always present ^at E. ^{+ U}

3.0 and another ^one starts to appear below the band-

edge when V. Z ^0.8.

4. Non-local hybridization.

From symmetry consideration, ^one expects ^{that the} on-site hybridization ^{between s- and} d-electrons vanishes. In this section we consider the case when the d-electrons hybridize ^{with the} s-orbitals of its nearest

neighbours, ^i.e. V 0 ⁼ 0, V, 0 ^0.

The hybridization ^{term of the} Hamiltonian (1) ^is,

now,

The equations of motion for G."(z) ^and Fdd’(z) ^{read :}

Applying ^{the same} procedure ^{above described} [5] ^we

obtain

and

After an infinite number of decimations we find

The same parameters ^{of section 2 are} utilized in the numerical calculations.

When ( m2 > ^is plotted ^{as a} function of U and V,,

no sensible difference is found with respect ^{to the case} of local hybridization; just the form of the density

of states is modified. In figure 7, ^the density ^{of states is} represented ^{for two values of} Vi, ^{0.5 and} 1.0, ^{and for}

11

1.0, ^{U =} 4.0. The main difference between these

curves and figure 6 is that the virtual bound state at energy E. ^is asymmetric ^and sharper than in the case of local hybridization. ^However, this difference in shape

does not affect the magnetic properties. Therefore, ^a model which assumes an « eflective » local hybridiza-

tion seems to be valid to describe the properties ^{of a} magnetic impurity.

Fig. ^7.

Density ^{of d-states as a} function of the nearest

neighbour hybridization ^term Vi, with 11

^{1.0 and U}

^4.0.

5. Conclusions.

The main feature of the present ^{work is to show that}

a « decimation » method [5] provides ^a simple ^and powerful technique ^{to treat the} problem ^{of one} impu- rity ^{in a linear} chain, particularly ^{to calculate the first}

and second-order one-particle Green functions. This allows us to calculate the densities of states and the

mean values of the occupation number nf ) ^and

the correlation function n1 n1 >. ^{We made} just ^one approximation (in Eq. (21)) ^{and the results} correspond

to an exact solution of the scattering correction of the

alloy analogy [7], applied ^{to the one} impurity ^case.

Concerning ^{the numerical results, we} have obtained

a mean value of the magnetization m > ⁼ 0, ⁱⁿ agreement ^{with the} rotational invariance of the system [9]. Also there is no abrupt transition to the formation of a magnetic ^{moment as} in reference [1].

Instead, ( m2 ) ^increases continuously (starting ^with

infinite derivative) ^{as a} function of U, ^{as in refe-}

rence [11,12].

It is found that the mean square value of the magne-

tization m2 > converges ^to the non-magnetic ^value

when the hybridization ^is large compared ^{to the band-}

width.

We have treated the case of non-local hybridization

too, ^that corresponds ^{to a more realistic situation.}

The results so obtained constitute an a posteriori

verification that the local hybridization, although

forbidden by symmetry considerations, essentially gives ^{the same} physical results. We remark that the real _space treatment presented in section 4, ^is very

simple. ^{This is not the case when one} calculates the Green’s functions in k-space ^as in references [1, 11, 8].

These authors did not consider the case of non-local

hybridization.

Finally, ^{one notices that the} technique here utilized enables to treat more detailed models, i.e., ^a degenerate impurity [13], ^{and to} calculate other properties, ^{as the}

contribution of the impurity states to the magnetic

972

susceptibility ^{and the} specific ^heat. Progress ^{on both}

directions are now in course.

Acknowledgments.

We would like to acknowledge discussions with C. Lacroix and P. M. Mors. This work was supported

in part by ^Brazilian agencies CNPq ^{and FINEP.}

Appendix.

When one plots ^the occupation ^{number at the} impu- rity site ( nda ), ^{as a function} of ’1 ^and Yo, ^the charge

transfer from or to the impurity ^{is not} usually ^discuss-

ed [11, 8].

Nevertheless, if in the symmetric ^case (il ⁼ 0) ^the charge conservation is assured, ^{this is not automati-} cally ^{the case} for different values of _’1, and a self-con- sistent adjustment ^{of the} parameters ^is required ^to

ensure that the charge ^{transfer will not violate the}

conservation of the total charge.

Equations (22) ^and (23) ^{enable us to calculate the}

densities of states and the occupation ^{number on the} impurity ^{site. However,} the presence of the impurity

modifies the densities of states in all lattice sites. Then,

to assure the conservation of the total charge ^{we must}

consider the charge ^added (or subtracted) ^{on the} impurity ^site plus ^the charge subtracted (or added) ^to

the rest of the chain.

Calling gij(z) the Green function for the s-electrons

on the _pure lattice

and G;j(z) the Green function for the s-electrons when there is one impurity ^{at the} o-site, ^{we define}

where the sum over i runs over all lattice sites, includ-

ing the o-site.

It is _easy to find, ⁱⁿ the case of local hybridization,

that [14]

and

The imaginary part ^of AG"(z), ^when integrated

from - oo to SF, gives ^{the net} charge transferred from the impurity ^{to the lattice, An’ :}

and, ^{if we assume} that the original ^number of electrons of the impurity ^{is one} d-electron and one s-electron,

the condition of conservation of charge ^{is :}

This condition is certainly valid in the symmetric ^case (because ^An’

0), but it is not at all evident that it could be satisfied for _any value of il ^and Yo. ^{To assure}

the validity ^of (A. 6) ^we need other adjustable para- meters, ^for instance, the _energy of the s-level at the

impurity ^site, which does not necessarily ^coincides

with EB. However, ^to adjust ^this energy ^one has to know how much charge ^was transferred to or from the

impurity, ^{or to assume a} given value for this charge

transfer [15].

This _may eventually ^{introduce some} modifications in the shape of the densities of states and in the values

of m2 ) when q ^{=F 0.}

References

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[2] FRIEDEL, J., Nuovo Cimento Suppl. ⁷ (1958) ^287.

[3] BLANDIN, B., Magnetism, Ed. G. Rado and H. Suhl

(New York, ^Academic Press) 1973, ^Vol. 5, p. 57.

[4] KONDO, J., Solid State Physics, ^{Ed. F.} Seitz, ^{D. Turn-} bull and H. Ehrenreich (New York, ^Academic Press) 1969, ^Vol. 23, p. 183.

[5] GONÇALVES ^DA SILVA, C. E. T. and KOILER, B., ^Solid

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[12] KIWI, M., PESTANA, ^{E. and} RAMÍREZ, R., Phys. ^Status

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[13] HEWSON, ^A. C., Phys. ^{Rev. 144} (1966) ^420.

[14] IGLESIAS-SICARDI, ^J. R., GOMES, ^A. A., JULLIEN, R., COQBLIN, ^{B. and} DUCASTELLE, F., ^{J. Low} Temp.

Phys. ²⁷ (1977) ^593.

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J. R., Solid State Commun. 52 (1984) ^87.

[16] ZUBAREV, ^D. N., ^Sov. Phys. Usp. ³ (1960) ^320.

[17] ^{In the case of an ordered linear} chain, ^lim $$(N) ~ 0 if z

N~~

is real. That’s because the states are extended

over all the chain. The convergence of $$(N) ~ 0

is ensured by taking ^a complex ^z. See, ^for example,

OLIVEIRA, ^P. M., CONTINENTINO, ^{M. and} ANDA, E.,

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