AC hopping conductivity in silicon doped yttrium-iron garnet

Mat. Res. Bull. Vol. 6, pp. 959-966, 1971. Pergamon Press, Inc. Printed in the United States.

AC HOPPING CONDUCTIVITY IN SILICON DOPED YTTRIUM-IRON GARNET

R. E. Fontana, Jr. and D. J. Epstein Department of Electrical Engineering and Center for Materials Science and Engineering Massachusetts Institute of Technology Cambridge, Mass. 02139

(Received July 26, 1971) ABSTRACT The conductivity of silicon doped YIG has been measured at selected frequencies from DC to microwaves and as a function of temperature over the range 20 - 200°C. The conductivity consists of a DC term plus a frequency dependent term. The latter is interpreted as arising from the localized hopping of an electron among the four Fe 3+ cation sites surrounding a Si 4+ impurity. A numerical estimate of the conductivity due to this process agrees reasonably well with experimental data. The hopping is characterized by a temperature activated relaxation time constant; the activation energy is 0.28 ev and the time constant at 30°C is approximately 10 -9 sec.

Introduction Yttrium-iron garnet (YIG) is a cubic crystal (space group la3d - 0hI0) characterized by the following structural-chemlcal formula

c

d

a

The trlvalent iron cations are distributed over two different sets of lattice sites: a d sublattice in which each cation is tetrahedrally surrounded by oxygen anions and an a sublattlce where the coordination is m

octahedral.

The yttrium ions are in ~ sites, characterized by a dodeca-

hedral oxygen environment.

The addition of silicon leads to the modified

formula [Y31" [Fe3_ 6 c

d

a

959

960

YTTRIUM-IRON GARNET

Vol. 6, No. 10

which indicates that, to preserve charge balance, the Si 4+ ions, which ! structurally enter the ~ sites, force the reduction of some of the octahedrally coordinated Fe 3+ ions to Fe 2+.

Each Fe 2+ ion is more correctly

viewed as an Fe 3+ cation plus an additional valence electron.

Under appro-

priate circumstances this extra electron can be detached and made mobile over the sublattice of Fe 3+ cations, thereby providing a mechanism for electrical conduction.

Experimentally it is found that undoped YIG is an excellent

insulator (i) with a DC resistivity, at room temperature, in excess of 1012 ohm-cm. The addition of Si in concentrations ~ = 0.05 drops the resistivity by about 8 orders of magnitude (2). For what follows it is important to describe the energetlcs of the additional electron created by Si doping.

At low temperature the coulomb

attraction of the Si 4+ cation will localize the electron on those Fe 3+ cations closest to the d site occupied by the silicon ion.

Each d site has, in fact,

four nearest neighbor ~ sites, so that at low temperature we have a constellation [Si4+'4Fe3+'electron] which acts as a donor center.

As the temperature

is increased the donor electron becomes ionized and can participate in electrical conduction.

From measurements of DC Hall effect (2) we have found that

conduction is indeed n-type, as anticipated.

Moreover, the Hall mobility

depends only weakly on temperature, a feature strongly suggestive of band mobility.

A reasonable model for DC conduction in SI-YIG is, therefore,

one in which electron conduction occurs in a band, formed primarily by Fe 3+ orbltals, where the charge carriers are derived from thermal ionization of donor complexes of the form [Si4+'4Fe3+'electron]. The present study was initiated because we postulated that in addition to the foregoing conduction mechanism there was the possibility of electron transfer confined within the donor center.

If this were so we might,

under the influence of an AC electric field, expect to see a conductivity contribution arising from local hopping of the donor electron over the four Fe 3+ cations forming the donor complex.

The results presented below bear out this

hypothesis. Measurements The electrical conductivity of a single crystal sample of Si-YIG (6 = 0.09) was measured at selected frequencies in the range DC to 11.47 GHz and for temperatures between 20 - 200°C (Fig. I).

All measurements were

made on the same sample, a bar 0.040" x 0.040" x 0.200". DC measurements were made using a four-termlnal method.

At i0 MHz

Vol. 6, No. 10

YTTRIUM-IRON GARNET

961

the sample was supported between parallel plates and the conductivity was obtained by measuring the dielectric loss tangent on a modified General Radio 716C capacitance bridge.

The 300 MHz data were obtained by placing the sample

at the end of a coaxial transmission llne and determining its properties from standing wave measurements.

At microwaves a cavity perturbation method was

used in which the complex conductivity was derived from measurements of changes in cavity Q and resonant frequency produced by introducing the sample into the cavity. Frequency Dependent Conductivity The data in Fig. i show that at a given temperature the microwave conductivity is substantially larger than the DC value.

The frequency de-

pendence, shown more explicitly in Fig. 2, can be fitted to a curve of functional form

(~)2 Ot

= O0 + O f

= 00 + 0

(2)

1 +(~)2

in which the total conductivity o t is broken down into two terms: conductivity o ° and a frequency dependent component of.

a DC

As the frequency

is increased the latter term monotonically increases from zero, at m = 0, to an asymptotic value o

as m ~ ;

the behavior in between is that of a

relaxation process characterized by a time constant T.

This kind of

frequency dependence is typical of the dielectric conductivity accompanying dipole relaxation (3).

In the present case, it appears that we are

dealing with a spatially quantized dipolar motion associated with the transfer of an electron over the set of four Fe 3+ cations that cluster tetrahedrally about a Si 4+ ion.

The electron (charge -e) at the Fe3+slte

determines the dipole "tail"; the net charge +e located at the Si site fixes the dipole "head".

As the electron hops from one Fe 3+ site to

another, within the donor complex, the dipole "tail" moves over the four quantized positions defined by the Fe 3+ tetrahedron.*

A statistical

average over the different orientations of the Fe 3+ tetrahedra occurring within the unit cell yields for the mean dipole moment of the donor electron, in an applied DC electric field E, i (ea) 2 E < P > = 32 kT

(3)

* If YIG did not have a spontaneous magnetization the four Fe 3+ sites would be energetically equivalent, with respect to donor electron occupancy. The presence of a magnetic axis renders the sites inequivalent (4,5),but the degree of inequivalence is not large enough to affect the present discussion.

962

YTTRIUM-IRON GARNET I

I

I

Vol. 6, No. I0

I

I

I

I

2.7

2.9

5.1

5.5

10-2

i

E cO

10"5 a D.C. •

0

L~

I0 MHz

+ :500 MHz o

8 . 9 8 GHz

•

11.47 GHz

10-4

2 1

2.5

25

iO00 / T (°K) FIG. Conductivity

i

as a function of temperature

6---~

I

I

at a number of frequencies.

I

I

I A

I w

E

U I

J

4

II

E I

0

©

0/

2

S S

X

b

m

0

m

m

% 0

n

u

i

m

m

~

I

I

I

I

I

10 6

I0 7

I0 e

10 9

I0 I°

Frequency ( H z ) FIG, 2 Conductivity

as a function of frequency

at 30°C.

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YTTRIUM-IRON GARNET

963

O

Here

a

is the lattice constant

(12.39 A), k the Boltzmann constant and T the

absolute temperature, In an AC field < p > becomes complex because the transfer of charge between Fe 3+ sites occurs via an irreversible thermodynamic process characterized by a relaxation time T.

Accordingly, we obtain a complex polariza-

bility * =

E

=

(ea) 2 1 32kT (i + J0~)

(4)

'

and, for the system of N e donor electrons, a complex dielectric constant e* = e'-je" = e O + N e~*.

The dielectric conductivity

(related to the imag-

inary part of £* by of = we") consequently becomes Ne (ca) 2 of - 32kTr

(~r) 2

(0;[)2

l+(0x)"/ = o

l+(~r) z

From our measurements shown in Fig. 2

we can obtain a value for T

at 30°C and can, therefore, proceed to calculate a value for o know Ne, the number of non-lonized donors.

(5)

provided we

The total number of donors

(equal to the Si concentration) is given by N d = NoP6/M = 4.226 x 1021 cm -3, where N

is Avogadro's number, p the density of YIG (5.17 gm/cm 3) and M the o molecular weight (738 gm/mole). Thus, for 6 = 0.09, the number of donor centers is N d

=

3.80 x 1020 cm -3.

+

The number of ionized donors N d is equal

to the number of conduction electrons N

= N d - N . From our previous c e measurements of DC Hall effect we know that the Hall mobility # ~H = 0.I cm 2 volt -I sec -I.

If we assume that the drift mobility ~D does

not differ drastically from ~H' we can obtain a reasonable estimate of N c from the DC conductivity O O = Nce~ D.

This calculation, carried out at

T = 30°C, gives N

= 1.5 x 1016 cm -3. Comparison with N d reveals that the c degree of donor ionization is less than 0.01%. Thus N~ = 0 and N e = N d.

Taking T = 7.5 x 10 -10 sec -I (from Fig. 2) and N d = 3.~0 x 1020 cm -3, we obtain o (30°C) = 1.5 x 10 -3 ohm -I cm -I, a result fairly close to the experimental value 0.4 x 10 -3 ohm -I cm -I.

We might, in fact, expect our

theoretical estimate of ~oo to predict too high a conductivity because in our model the dipole length is taken to be the distance between the Si 4+ and Fe 3+ lattice positions.

The effective dipole length is undoubtedly somewhat

reduced owing to the coulomb attraction exerted on the donor electron by the effective charge +e at the donor core.

T Because YIG is ferrimagnetic it exhibits ordinary and extraordinary Hall components. The Hall mobility was determined from the ordinary Hall c o n s t a n t .

964

YTTRIUM-IRON GARNET

Activation

Vol. 6, No. I0

Energy

The data in Fig. 1 show that the DC conductivity

follows a temperature

activated law , o

=O

o

with W

= 0.3 ev. The asymptotic o written in a similar form:

-Wo/kT e

o

(6)

limit of the total conductivity may be

, -Wt/kT Ot~ = OO + O~ = Ot~ e From Fig. 1 we find W t = 0.26 ev. the activation

However,

(7)

it is of more interest to find

energy associated with the transfer of donor electrons.

To

do so, we write o~ in Eq. 5 as Ne (ea) 2

B

(8) 32kTT and explore the functional

Tm

dependence

of

o T(T) by observing

with temperature of (ot~ - o o) T. Treatment shown in Fig.

the variation

of the data in this fashion,

3, leads to the result w T

which demonstrates

/kT (9)

= TWe e

that the relaxation time is temperature

expected for a hopping process. IO°

l

The activation barrier W I

l

I

I

I

I

I

e

activated, = 0.28 ev.

l

o

E

u 7

E I0 "l

10"2 2.0

I

I

2.2

2.4

2.6 2.8 5.0 I 0 0 0 / T (° K )

I 5.2

5.4

FIG. 3 Determination of the activation energy for electron hopping; the slope yields W e = 0.28 ev.

as

Vol. 6, No. i0

YTTRIUM-IRON GARNET

965

Suma~ The results reported here together with our previous measurements

(2)

of DC conductivity support the following interpretation of the conductivity mechanism in silicon doped YIG.

DC conduction occurs in a low mobility

band with the observed activation energy arising not from carrier mobility, but from the thermal generation of donor electrons.

The donor centers are

located at silicon impurity positions and consist of the complex [Si4+'4Fe3+'electron].

In addition to the DC conductivity there is an AC

component resulting from the hopping transfer of the donor electron over the four Fe 3+ sites of the non-ionized donor complex.

Because the degree

of ionization is small, the number of electrons participating in hopping is essentially temperature independent; the activation energy for AC conductivity, therefore, is due almost totally to the thermally activated hopping of the donor electron over the energy barriers between the Fe 3+ ions within the donor center. A cknpwledgements The research described here was supported by the Advanced Research Projects Agency of the Department of Defense under contract DAHCI5-07-C-0222. In addition, one of the authors (R.E.F.) received support under a National Science Foundation fellowship. The experimental technique used in this study is that of "dielectric spectroscopy".

The name, coined by Professor Von Hippel, describes a

methodology pioneered by Von Hippel and brought to a state of sophisticated development in the Laboratory for Insulation Research at M.I.T., which he founded over thirty years ago.

We are pleased to acknowledge that some of

the facilities of this laboratory were used in the present work. References i.

J. Graczyk, S.M. Thesis, Dept. of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Mass., unpublished.

2.

D.J. Epstein, "Magnetic and Dielectric Loss in Magnetic Insulators", Technical Report AFML-TR-70-210, Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio (August 1970).

3.

A. Von Hippel, "Dielectrics and Waves", p. 177, John Wiley and Sons, N.Y. (1954).

4.

R.P. Hunt, J. Appl. Phys. 38, 2826 (1967).

5.

T.S. Hartwick and J. Smit, J. Appl. Phys. 40, 3995 (1969).