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Disorder-induced local magnetic moments in weakly correlated metallic systems.

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Ann. Physik 4 (1995)43-52
Annalen
der Physik
0 Johann Ambrosius Barth 1995
Disorder-induced local magnetic moments
in weakly correlated metallic systems
Annette Langenfeld and Peter Wolfle
Institut fiir Theorie der Kondensierten Materie der Universittit Karlsruhe, Physikhochhaus,
76128 Karlsruhe, Germany
Received 31 August 1994, accepted 27 September 1994
Abstract. We consider the formation and the Kondo effect of local magnetic moments in the Anderson-Hubbard model with off-diagonal disorder. The existence of moments at sites weakly coupled to
the environment is deduced in effective medium approximation. The distribution of moments is
calculated both deep in the metallic phase and near the metal-insulator transition. We discuss the Kondo quenching of the moments and derive a distribution of local Kondo temperatures.
PACS numbers: 71.55.Jv, 72.10.Fk, 75.20.Hr
Keywords. Disordered electronic systems; Magnetic moments; Kondo effect.
Recent experimental studies of metallic samples of doped semiconductors have indicated that localized magnetic moments are formed in the metallic phase as these systems
approach the metal-insulator transition [ 1- 71. In fact the contribution of these
moments dominates the magnetic and thermodynamic properties at low temperatures.
The moments are thought to be intrinsic, i.e formed by the same dopant atoms that
form the impurity conduction band, rather than being due to additional magnetic ions.
There are two issues involved here: (i) the formation of moments, (ii) the eventual
quenching of the moments due to coupling to the environment or among each other
at low temperature. According to the conventional picture of moment formation [8],
a spindegenerate atomic level of an impurity placed sufficiently below the Fermi energy
is split by the Coulomb interaction into two levels. There is a level at the original position involving single occupancy and an upper level above the Fermi energy involving
double occupancy. Provided the coupling of the conduction electrons to the local impurity is sufficiently weak, the impurity will be occupied by a single electron to a good
approximation, and will hence form a spin local magnetic moment. In the case of
doped semiconductors the levels forming the conduction band and the magnetic impurity belong to the same system of dopant atoms, which therefore may be described
by a one-band model. The atomic levels are the same at each dopant site, but the hopping elements are not, as the dopant atoms occupy random lattice sites in the host crystal
lattice (we neglect possible short-range order correlation). In the metallic regime, i.e.
for sufficiently large doping concentration, an impurity conduction band is formed,
the Fermi energy lying in the center of the band, at the position of the atomic levels,
for the uncompensated materials. Even though at first sight there appears to be no
close similarity with above-described picture of the Anderson magnetic impurity, since
+
44
Ann. Physik 4 (1995)
the all important location of a level below the Fermi surface seems to be lacking, the
situation is in fact quite similar, as we shall show below.
An earlier work on the formation of magnetic moments in the same model system
has inferred the existence of moments from the instability of the paramagnetic state
[9]. Here we shall calculate the moments directly in effective medium approximation.
In this way we can determine the size of the moments in addition to their number for
a given distribution of hopping parameters tii. Local moments have also been shown
to form in an Anderson-Hubbard model in d = 00 dimensions and for a special model
of randomness in the hopping parameters [lo].
The second issue is the energy scale below which the moment is quenched due to interaction with the environment. We find that the exchange coupling of the moments
to the conduction electron spins is necessarily anisotropic and of antiferromagnetic
type. The associated local Kondo temperature, including potential scattering contributions, will be determined for a typical situation. We shall show that the randomness
of the couplings leads to a distribution of energy scales, in particular the local Kondotemperatures, with a large weight at very low energy. As a consequence, a sizeable temperature-dependent fraction of the moments remains unquenched even down to the
lowest temperature, giving rise to large contributions to the specific heat and to the spin
susceptibility, among other quantities.
Our work contrasts with a recent calculation of the local Kondo temperature [l 11 in
a disordered system which focussed on the random nature of the local density of states
at the local moment site, rather than the randomness of the exchange coupling
constant.
Mean-field theory of Anderson-Hubbard Model. The simplest model Hamiltonian
containing the essential elements sketched in the introduction is the Anderson-Hubbard
model
H=
1 (Ei-p)niU+ c tiici',cju+U c nitnil
iu
ijo
I
where ciu(ci',) is the annihilation (creation) operator for an electron with spin projection CJ in the atomic state of the dopant atom at site i and nio = ci',ciuis the occupation number operator. We will consider nearest-neighbor hopping on a d-dimensional
hypercubic lattice, with random hopping amplitudes tii- The statistical distribution of
tii will later be modelled by relating tii to the spatial distance between sites i a n d j , R,,
as tii = 2 E z exp ( - r i i / a * ) (1 + r j / a * ) , where E: and a* are the effective hydrogen
binding energy and Bohr radius, respectively, of a dopant atom in the semiconductor
(e.g. of phosphorus atoms in crystalline Si). The interaction energy Umay be estimated
to be less than the bandwidth of the impurity conduction band for doped semiconductors (e.g. for Si: P one estimates U/12t = 0.08, where t is the average hopping integral).
It seems therefore justified to apply Hartree-Fock theory, which amounts to replacing the interaction term Un,nL by U((nt)nl + (nl>nt). Following Anderson's original
treatment [8], one looks for a mean field solution breaking spin rotation symmetry,
(n,>-(nl> = M. This would amount to an unrestricted Hartree-Fock calculation,
which can only be done numerically for finite size systems. As an alternative to this
numerical treatment, it is worthwhile to discuss a simpler model problem, which can
be solved more or less analytically, while still capturing the essential physics. We observe that a dopant atom, which is weakly coupled to its neighbors, may be modeled
A. Langenfeld, P. Wtrlfle, Disorder-induced local magnetic moments in metallic systems
45
as an impurity in a perfect crystal in the approximation where the disordered environment is described by an effective medium. Models of this type will be studied in the
following. They should be valid in the limit where the concentration of magnetic
moments is small, which is frequently the case in experiment.
Impurity Model. We consider an impurity atom at the origin (i = 0) to be coupled
weakly to a regular hypercubic lattice in d dimensions yith nearest neighbor hopping,
as described by the Hamiltonian
c
n= - ?
Z
U
c t;l(c,+,c~"+c,+,c"")+Un~~n~~-c
c;cju-
"0"
(ii),"
n=l
i.j#O
0
2 u
(2)
U
where the last term is due to the choice of chemical potential p = - corresponding
2
to a half-filled band. The sum over n is over the Z nearest neighbor sites of the impurity. We have checked that the restriction to a hypercubic lattice is not essential, by studying the effect of including successively more neighbors with finite hopping probability. One finds that the influence of the fifth and higher neighbors is quite negligible.
We first consider the mean-field approximation of the interaction term. The resulting one-body Hamiltonian is given by
"
where
V f p = V" + Vi + v-f.
(4)
is a one-body scattering potential at the impurity site. Here V' = +U(n - O M - 1 )
= cos (k,a). For the half-filled
and Vi = - q- n fn e-jk.'~, with &k = - 2 t
band case the average density n at the impurity site is n = 1 due to particle-hole symmetry.
The local magnetization M i s obtained as a solution of the following self-consistency
equation
c
1
1
G&(u)G&(u)[T~~.(co)T'~J(u)]
71
1
= - - Im
71
[ J dw~ k ; ( A
7
-A
1)
+ 2gog, (B' - P I
+g:(cl-
c')]
The functions A , B , C are coefficients in the expansion of the T-matrix in terms of
the separable potential Vk: T f p ( w )=A"+B"(Vk+ V p ) + C " V i V p , u = t , l and are
given by
46
Ann. Physik 4 (1995)
-
isotropic model
3.0
Vt
2.0
1.o
0.0
U/t
Fig. 1 The region of moment formation in the T-
U plane for the isotropic and anisotropic models.
7
I.
Also shown is the result for the anisotropic model with fscaled by a factor -
fi
Here we have defined the following averages over momentum of the Green's function
C & ( l ' k ) " G f ( ~n) =
, 0 , 1 ; g2=
l'kIzcf(o).
We have solved Eq. ( 5 ) for M numerically for several sets of random hopping variables fn.In order to facilitate a qualitative discussion of the solution we discuss first
the two special cases: (i) the isotropic impurity, with = Tfor all n, and (ii) the extreme anisotropic case, with = ran,,,i.e. only one of the hopping amplitudes connecting the site &, = 0 to the environment is nonzero. In Fig. 1 the phase diagram for
these two cases is shown, indicating the region of parameters r a n d U in units of the
hopping amplitude t in the bulk, in which magnetic moments appear in the above
mean-field treatment. As expected, the critical values of fgrow for increasing U.Even
for small U,magnetic moments may appear for sufficiently small The extreme
isotropic case shows a much stronger tendency for moment formation, as one might
expect. Very roughly the two results scale with an effective hopping & = z'c where
z' is the effective coordination number (z'= 2 for the isotropic, z'=1 for the
anisotropic case). Note that in the extreme anisotropic case even a value of flarger than
t may yield a magnetic moment for moderately large U.
The general condition for the appearance of a moment is a sufficiently large interaction-induced shift of the impurity level, A q = i U ( l - M ) , relative to the Hartree
shift in the bulk heo= +U (assuming half-filling, i.e. ( n , > + ( n l >= 1). This shift is
caused by a reduced occupation of the impurity site nl = +(l -M) <+, due to suppressed hopping in the symmetry broken mean-field state. Of course, the spin rotation
symmetry can not be broken spontaneously for a finite system, but the splitting of the
local level is meaningful and should be interpreted as the formation of two Hubbard
levels. In Fig. 2 the value of the magnetic moment is shown for various values of U / t
as a function of the normalized impurity hopping parameter F/t, for the isotropic and
anisotropic models, respectively.
G f ( o ) = ( m - ~ k + i O ) - ' : g,=
r;,
A. Langenfeld, P. Wolfle, Disorder-induced local magnetic moments in metallic systems
1
M
0.5
1
I \
I
47
I
\
\
0
1
\
\
\
L
I
0
a) isotropic model, U/t=O.l
0 b) anisotropic model. U/t=O.l
0 b) anisotropic model. Un=l.O
0.0
0
0.25
0.75
0.5
1
tr
Fig. 2 Magnetic moment Mversus impurity hopping amplitude ;for various interaction strengths U.
Statktics of Local Moments. We now turn to the statistics of the random ensemble of
magnetic moments. For the anisotropic model and in the region not too close to the
metal-insulator transition, the distribution of hopping integrals may be directly inferred
from the probability P l ( r ) to find the nearest neighbor of a given site at distance r.
Assuming a completely random spatial distribution of dopants this is given by
PI(r) = 47rnr2exp- ( 4 n n r 3 / 3 ) ,where n is the density of dopants. In a real system the
dopant atoms are situated on the lattice sites of the host lattice (Si, for example), and
therefore can not come closer than a minimal distance rmin(in Sirmin= 2.3 A +a*).
The distribution P,(r) has to be normalized accordingly, which has been done in the
actual calculation reported below. The probability distribution of the hopping integrals
to the nearest neighbor site is obtained from this as
P',(?)
)(:-
= Pl(r(?))
where
r(?) = a*ln(t*/f)+a*, and t * is the hopping integral at distance a*,t* = ( 4 / e ) E 8 .
We have determined the critical value of ?as a function of U,and n, c ( U , n ) , below
which a magnetic moment forms (see Fig. 1). Here the weak dependence of & on n is
through the average value of the hopping t. The density nM of magnetic moments is
therefore obtained by integrating. the probability ISl (t'> over all t< <( U), or
equivalently
*
where r, = r ( < ) .
The dependence of and, therefore, r, on the doping concentration n is weak and
will be neglected. The functional form of nMas a function of n is then explicitly given
<
48
Ann. Physik 4 (1995)
t:
. As we will show,
by (7). nM has a maximum at nmax= - r e
nmaxis likely to
3)-1
be outside the metallic regime. In other words, nM = n exp ( - n/n,,,) is a decreasing
function of n in the relevant regime.
For the system Si: P, assuming the effective mass value m * = 0.33 m, and a background dielectric constant E = 12, the effective Bohr radius is u* = 36uB = 19 and
E * = 2.3 x
Ry = 31 meV. The average hopping integral in the concentration
range n = 5 x 10" cm-3 is obtained as t = 28 meV. From Fig. 1 we read off t',= 0.91,
at U = I , which leads to a value re = 32.4 A and nmax= 1.5 x 10" ~ m - The
~ . fraction
of doping atoms carrying a magnetic moment is just exp ( - d n , , ) . For the experi~ cm-3 we find n M / n equal to
mentally studied [5] values of n , 3.6, 4.4 and 5 . 6 10l8
9'70, 4.3% and 2.4'70, respectively. These values compare well with the experimental
data of 7.5'70, 3% and 2% at these concentrations.
The density of moments nM depends obviously on the distribution of the few next
nearest neighbors as well. We have calculated n,,, for the realistic case of 2 d nearest
neighbor hoppings fn, distributed independently according to the distributions P,(r)
for the n-th nearest neighbors.
One finds that the values of n M / n are somewhat lower in this case, 5 % , 2.8%, and
1.3% at the specific doping concentrations. As will be discussed below, a small fraction
of the moments will be Kondo quenched in the temperature range of the experiments,
reducing the apparent concentration of moments.
In the region close to the metal-insulator transition, the assumption of a regular lattice in which an impurity is embedded becomes questionable, since the disorder in the
environment of the impurity will influence its.coupling to the impurity. One may define
an effective hopping integral to the impurity refr = f by relating it to the local density
1
of states at the Fermi energy pi(E,) = --Im(Gii(E,)J as p = (t/2Zp) and p =
A
Ic
(aZr/2p) for the isotropic and anisotropic models, respectively, where a =
[1+(n/2)2]-' = 1/4. The probability distribution of p has been calculated in Ref. 13
*
(
(t
as P@) =-exp - k l n 2
e")).
Here po is the average DOS, and u is a
G
P
dimensionless measure of disorder, u = In (ado), where ooand o are the Drude conductivity and the renormalized conductivity, respectively. The density of local
moments in this regime would be given by
where pc = (Zt/2Z2t7) with
model, respectively.
z = a - 1'2
or
= Z for the anisotropic and isotropic
Local Spin Hamiltoniun. The preceding discussion is based on a meanfield treatment
of the interaction U which is expected to work in the weak coupling regime. However,
even if Uc t in the bulk, at the impurity one may be in the strong coupling regime U > f,
where Hartree-Fock theory is not reliable. Nonetheless, as in the case of the original
Anderson magnetic impurity model we may expect the mean field theory to be suffi-
A. Langenfeld. P. WOlfle, Disorder-induced local magnetic moments in metallic systems
49
cient in predicting the existence of magnetic moments. However, the properties of the
magnetic moment systems are not adequately described by mean field theory.
The relevant effective Hamiltonian of a localized magnetic moment coupled to a
system of conduction electrons is of the spin-exchange type. It may be derived from
(2) by a Schrieffer-Wolff transformation, eliminating the doubly occupied and empty
impurity states in second order perturbation theory in the hopping In the general
model with hopping integrals fn(n = 1, . . . z ) connecting the n-th nearest neighbor site
with the impurity at site 0, the effective s - d Hamiltonian is given by
uu'
c
Here go= ,,,,~c&ic0,,~is the spin operator at the impurity site, J p p is an exchange
coupling, Vik, is a scalar scattering potential and r'is the vector of Pauli matrices. J i p
and Vzp are defined by:
1
Jii,= - hphp, ,
U
Z
hi =
c eii.R t-n
(10)
n=l
The first term in Vpp. accounts for the blocking of the impurity site and is the dominant one, whereas the second term,
U
- -,
2
originating from the chemical potential
-
shift leads to a moderate effect ( U / Z t < 1 !) and the last term J is also moderately
small.
One may define an effective exchange coupling including the effect of potential scattering by [15]
J$?,= Jpp +
c4 Tpk,Gp,
The scattering matrix
Tkk'
(O
(12)
= 0)Jplp,
is obtained as a solution of
In the approximation of an isotropic conduction band, Tpp and consequently JZY,
can be expressed in terms of the scattering phase shifts 71:
JZY, = CI J T f f p / ( l . &;' )
J eI f f =
JIe - " l / cosrl,
(14)
71
One may show the s-wave phase shift 'lo
to be equal to -, since the blocking of the
2
impurity site corresponds to an infinitely strong repulsive potential at the impurity site,
50
Ann. Physik 4 (1995)
which dominates the additional terms
-g2 and --U1 hl;hp. Thus, the s-wave compo-
nent of the exchange coupling, Jo, renormalizes to J;"=O. In the isotropic model
considered above, the remaining coupling will be due to extended s-wave components
(I = 4 or higher), which will be necessarily very small.
For the anisotropic model one may show that the p-wave component dominates, and
Jf& = Jt" cos2q l e i ( f - E 7 - R lwhere
,
Rl is the closest neighbor site. The real part of the
complex valued J& has been taken as the dominant part here. We will restrict the
discussion to this model in the following. One may estimate J:" from J k p =
tz exp i(L- R')-R,,for the case where hopping occurs only to the nearest neighbor
U
at R1,by performing a "partial wave decomposition" in terms of cubic harmonics.
One finds a decomposition in terms of the basis function of the representations r,,r;
with coefficients given by J f p - -
1
84
+36rk- 3rj]. Clearly
[ - 14171-7r2-14r3+31'4+3r5-r;-r'2-21-;
dominates, with basis function given by ~ ( x=)x, which
1
has p-wave symmetry. Thus the dominant component is J1 = - Jo, where we define
2
tl
Jo = -. The I = 1 phase shift is solely due to the potential - Jo exp i(Z- l'). R,,
U
R
and hence we find ql = tan-'y, where y = --NoJl, with No the DOS at site R1and
3
(-)
t l
2 u
J;" = l
(1 + y 2 ) - I
Kondo Effect. In the approximation, where the potential scattering is absorbed into
an effective exchange coupling, the Hamiltonian H = Ho+ V, where Ho is the free
conduction electron part and V is the exchange contribution, i.e.
may be subjected to a "poor man's scaling" analysis [15]. The scaling of the coupling
constant under reduction of the conduction electron band width 2Ec is obtained by
successively integrating out the high energy shell. As a result one obtains the differential equation
dS" =
1
- No (J:" )2
dEc
o-EC+A
(No is the density of states at the Fermi level). For small energies o,and small energy
shift A , compared to the scaling energy E,, integration yields the scaling invariant
energy
which we identify with the Kondo temperature.
A. Langenfeld, P. Wolfle, Disorder-induced local magnetic moments in metallic systems
51
We are now in the position to calculate the distribution of local Kondo temperatures
in a diluted system of random magnetic moments induced by the statistical fluctuations
of the hopping amplitudes f In terms of the distribution function P ( f ) considered
before, the distribution of Kondo temperatures may be expressed as
where TK is defined in terms of the effective exchange coupling J;ff in (18), and J;f'
is given as a function of p/U in (15).
In explicit form of &TK) is given by
D
R
where x = In -, D = Zt, a = -(a * ) 3 n , = (N,,t22 U).Inserting the parameter valTK
6
ues discussed above, one finds a = (1.8 to 2 . 5 ) ~lo-', /?= 1. P(TK) is dominated by
the T i factor originating from the Jacobian 1 df/dTK1. The divergence as TK-0 is
D
weakened by the factors involving ln-,
such that P(TK) is integrable, and may be
TK
normalized to 1 over the interval O< TK< TEu. The maximal value of TK is given by
the maximal value of
f as
. For the parameters appropriate
TE" = D exp
to Si: P, TgaXG 6 K . Surprisingly it turns out that P ( T K ) a T i aover a large range of
TK values,
K a TK< TZu, where a = 0.9. The scaling behavior leads to corresponding power law behavior in temperature, e.g. for the specific heat, C y a T 1 - a .
A weak temperature dependence of a magnetic impurity contribution to C y with
effective exponent a = 0.2 has been observed in experiment [ 5 , 181.
Conclusion. We have been able to show that the formation of magnetic moments in
a single-band Hubbard model with off-diagonal disorder, i.e. with random hopping
amplitudes to, can be understood within a simple mean-field picture. Weakly coupled
sites are occupied less (or more, depending on the spin) on the average and hence acquire a Hartree energy shift which is less (or more) than the average, so that these local
levels move below (above) the Fermi energy. As a consequence these levels are occupied
by a single electron (the Coulomb repulsion U shifts the level for double occupancy
above the Fermi energy), and form a spin
local magnetic moment. We have
calculated the concentration of moments as a function of both the effective local hopping amplitude f a n d of U.The model considered here is applicable to doped semiconductors. Using realistic values of U and of the random distribution of f w e find a concentration of local moments, which compares well with the experimental data on Si: P.
We also addressed the question of the effective spin model for these moments, and
derived the effective exchange Hamiltonian. It turns out that potential scattering is very
effective here in renormalizing the s-wave component of the spin exchange coupling of
the local moment to the conduction electron spins to zero. The largest component left
is for most of the local moments the p-wave component J,. Since J , is antiferromagnetic, we expect a Kondo effect.
+
52
Ann. Physik 4 (1995)
We demonstrated that the exchange coupling generates a Kondo scale TKdefined in
terms of the effective coupling J;“. From the statistics of the hopping amplitudes 17
via the definition of J!ff in terms of ( the distribution of local Kondo temperatures
may be calculated. It has been suggested that certain distributions of TK may lead to
“non-Fermi liquid” behavior [ 161. The consequences of the distribution function for
TK derived here for observable quantities such as the specific heat, the magnetic
susceptibility and the thermopower have not been derived yet. Work in this direction
is in progress. A preliminary report on first results for the specific heat will be published elsewhere [ 181.
There are several open questions not addressed in the present work. One is the role
of spin-spin interactions. One should expect the RKKY interaction between the localized moments to lead to the formation of spin singlet clusters, as proposed for the insulating phase [17], or even to the freezing of the spins into a spin glass. An estimate
of the relevant energy scale for Si: P yields values below 1 K, possibly indicating that
the RKKY interaction is less important than the Kondo effect. A second concern is the
effect of disorder on the conduction electrons involved in the Kondo effect. It has been
suggested recently [lo] that in this case the local conduction electron density of states
should enter the dimensionless Kondo coupling N(0)J.This quantity is known to fluctuate strongly as the metal-insulator transition is approached, leading to a peculiar distribution of TK’s.
We have enjoyed useful discussions with H. von LOhneysen and M. Lakner. This work has been supported by Sonderforschungsbereich 195 der Deutschen Forschungsgemeinschaft.
Note added in proof: the Anderson-Hubbard model with site-diagonal disorder has been recently
studied in unrestricted Hartee-Fock approximation [19].
References
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[14] P. W. Anderson, Phys. Rev. B 1 (1970)4464
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