Effect of the Morin transition upon electron transport in magnesium doped single crystal hematite
by Calvin Lee Ransom
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY in Electrical Engineering
Montana State University
© Copyright by Calvin Lee Ransom (1968)
Abstract:
The effect of the weak ferromagnetic-antiferromagnetic transition on the electron transport properties
has been observed in synthetic, . single-crystal, magnesium-doped hematite (nominally 0.5 percent
replacement of iron). Separate measurements of the transverse voltage were taken with current flowing
along the [111] direction and within the (111) plane. The predominant results are: 1) The activation
energy of the transverse voltage is quite different for current flow along the [111] direction and in the
(111) plane. The energies are -1.3 and 1.3 ev, respectively.
2) The transverse voltage produced for current flow along the [111] direction is altered as the crystal
undergoes the weak ferro- magnetic-antiferromagnetic transition at the Morin temperature (TM),
whereas no change is observed at TM for transport in the (111) plane.
3) No normal or anomalous Hall effect is observed for current flow in either the [111] direction or the
(111) plane; however, a second order dependence upon the applied magnetic field is observed for the
transverse voltage.
4) The data also indicates that the activation energy of the transverse voltage is dependent upon the
current density (more properly the applied electric field) within the sample under test.
The observed results are interpreted in terms of a model which considers the effect of the magnetic
state upon the electron transfer process within the crystal. The electron transfer energy is assumed to be
a function of ?, where ? is the angle between the quantization directions (determined by the large
internal magnetic fields) of the iron ions between which the electron is assumed to hop.
5) It was also observed that the transverse voltage of the samples studied was dependent on the
magnetic structure. The effect is believed to be the result of the sample being an array of resistors in
which each resistor is a magnetic domain and within the domain the electrical properties are a function
of its magnetic state, EFFECT OF THE MOfiIN TRANSITION UPON ELECTRON TRANSPORT
IN MAGNESIUM
DOPED
SINGLE CRYSTAL HEMATITE
by .
CALVIN LEE RANSOM
A thesis submitted to the Graduate Faculty in partial
fulfillment of the requirements" for the degree '
Of
DOCTOR OF PHILOSOPHY
<
in
Electrical Engineering
Approved:
■R e a d , Major Department
JA
^ha i r m a n ,
Examining Committee
MONTANA STATE UNIVERSITY.-Bozemail, Montana
December,' 1968
iii
ACKNOWLEDGMENT
I
my work.
would like to express my appreciation to those who aided in
Particular thanks go to Dr. J. P. H a nton, my advisor.
I
should also like to thank the National Aeronautics and Space Admini­
stration for granting me a fellowship during the
1966- 6? and 196?-68
academic years.
;
^
.
:
■
iv
- .
;■
:
■
TABLE OF CONTENTS
Chapter.
I
II
III
IV
,'
Page
Preface , ............. .......
.
.
.
.
.
Introduction
.
.
.
.
.
.
i
A.
Magnetic Behavior of Hematite....................
B.
Charge Transport by Hopping
C.
Double Exchange ■ .
D.
A More Detailed Look at the Hopping
M e c h a n i s m ...........................
The Experiment .
Besults
•
vii
7
................. 13
.................... ' .
16
.28
37
..............................
...................................... .
.
.
C o n c l u s i o n ........................................ ...
.
4l
.
34
Appendix A ................................................. $6
Literature Cited ..........................................
59
V
LIST OF FIGURES
Figure No.
. Page
1.
Illustration of a Tvvo-Sublattice S y s t e m ................. 3
2.
Crystal Structure of Hematite
...........................
8
3«
Magnetic Structure of Hematite
. ' ................
9
4.
Possible Morin Transitions
5.
Cyclic Chain on which Conduction Occurs by
Hopping . . . . . . . . . . . . .
6.
.................
.
.
.
. . . .
.
.
.
12
.
14
Coordinates Defining 0 ....................................... 17 -
. 7«
Coordinates Defining the Rotational Trans­
formation of the S p i n ........... 21
. 8,
Symmetric Wave Functions in a Two Dimensional
Crystal . . . . . . . .
...........................
24
Sublattice Magnetization Directions' of Hematite
26
9«
10.
Illustration of Allowed "Hops" for T •< T^
. . .
................31
1 1 . ' Illustration of Allowed "Hops" for T > T^ .................. 32
12.
"Spinn ihg" 3d Wave F u n c t i o n s ....................
13'.
4
c.
"Pictorial Hopping" from a 3d- to a 3d^ State
14.
Sample -Crystallographic Orientation
15.
Basic Experimental Configuration.
16.
17»
18.
19.
.
34
.
. 35
■ ...................... 38
• ■ ....................
A V as a Function of Temperature for Current Flow
.. in the [ 1 1 1 3 Direction
• .
. ■. .
. . '• •
A V as' a Function of Temperature for Current, F l o w '
in the (ill) Plane
. . . .
. . .
•
.
.42
• ^3
A V as 'a- Function of the B a s a l :Plane Magnetic
Field Direction
. . . . . . .
.......................
Effect of the Magnetic Field Strength' upon
AV
...
39
4$
. ' . 46
\
vi .
LIST OF FIGURES (continued). '
Figure No.
20.
■Hematite M - H. Loops
..
Page
. •,
.
.48'
21.
Shift of the Morin Transition as a Function
of the Magnetic Field . . . .
................. ’ . 50
22.
Shift of the Transverse Voltage as a Function
of the Applied Magnetic Field
...........................
51
23«.
Transverse Voltage as a Function of the Current .
Through the S a m p l e .................... ................... '52
24..
The Experimental Configuration ...........................
57 .
vii
PEEFACE
An interesting and as yet little investigated problem is the
consideration of c h a r g e ,transport in materials that possess a magnetic
state.
This thesis considers only a very limited portion of this prob­
lem, but it is hoped that it will, because of the interesting experi­
mental results obtained, provide motivation for more studies in this
area.
The work begins with a brief discussion of why the magnetic state
of a material may affect its conductivity.
Terms and concepts which may be unfamiliar to the reader are
introduced,
affording a glimpse of the overall problem.
arate sections,
Then, in sep­
these concepts are more completely described and simple
models are used to illustrate them.
These simple models are shown to
apply, to hematite and hence the hypothesis to be tested is formulated.
Chapter Two describes the experimental configurations.
Three presents and discusses the results of these experiments.
The conclusions are summarized in-Chapter Four.
Chapter
I
viii
ABSTRACT
The effect of the weak f erromagnetic-anti'ferromagnetic transi­
tion on the electron transport properties has been observed' in syn­
thetic, . single-crystal, magnesium-doped hematite (nominally 0.5 percent
“"replacement of iron)V " Separate measurements' of"the transverse voltage
were taken with current flowing along the
[ill]
direction and within
the (ill) plane.
The predominant-results are:
1) The activation energy of the transverse voltage is quite
different for current flow along the
[ill]
direction and-in the (ill)
plane.
The energies are -1.3 and 1.3 ev, respectively.
2)
The transverse voltage produced for current flow along the
[ill]
direction is altered as the crystal undergoes the weak ferromagnetic-antiferromagnetic transition at the Morin temperature (T^),
whereas no change is observed at T^ for transport in the, (ill) plane.
3) No normal or anomalous Kail effect is observed for current
flow in either the . [ill]
direction or the (ill) p l a n e ; ho w e v e r , a '
second order dependence upon the applied magnetic field is observed ■
for the transverse voltage.
4) The data also indicates that the activation energy of the
transverse voltage is dependent upon the current density (more properly
the applied electric, field) within the sample under test.
The observed results are interpreted in terms of a model which
considers the effect of the magnetic state upon the electron transfer
process, within the crystal.
The electron transfer energy is assumed to
be a- function of 9, where 6 .is the angle between the quantization
directions (determined by the large internal magnetic fields) of the
iron ions between which the electron is assumed to hop.
5)
It was also observed that the transverse voltage of the
samples studied was dependent on the magnetic structure.
The effect is
believed to be the result of the sample, being an array of resistors in •
which each resistor is a magnetic domain and within the domain the
' electrical properties are a function of its magnetic state.
I
INTRODUCTION
In considering charge transport in a material that may possess a
magnetic state, be it ferromagnetic, 'fe-rrimagnetic, or antiferromagnetic,
the question arises,
"What will be the effect of the internal magnetic
fields, associated with the magnetic state, upon the mobile charges?"
If one follows the approach suggested by Weiss in his successful
explanation of spontaneous magnetization one would consider the internal
magnetic field given by
= Ha +
XM
(I)
where H a is the externally applied magnetic field, M is the magneti­
zation of the material under consideration,
ori g i n , and
T a constant of unknown
is the internal magnetic field.
This suggests that the L o r e n t z 'force would be of the form
F = qv
x.
(Ha +
TM),
-
neglecting for the moment any possible electric fields,
of course,
(2)
.
q and v are,
the charge and velocity, respectively, of the mobile charge.
In ferromagnetic materials
'
:
,
y
T M - is of the order 10 - 10 oe.
It would thus seem that such fields cannot be neglected in considering
‘ ;(x)
possible Hall effects.
This is indeed found to be the case.
If the class of materials considered is extended to include anti­
ferromagnetics the magnetic fields now involve an additional term to
^
-2describe the effect at a two-sublattice antiferromagnet.
The concept of a sublattice is convenient and appropriate when
■discussing antiferromagnetism and ferrimagnetism.
It is -used to des­
cribe a crystal structure in which layers of atoms all possess a common
magnetic orientation=
T h u s 6 for example, a two-sublattice antiferro­
magnetic material may be imagined as a structure composed-of alter­
nating layers, each- layer consisting of atoms whose magnetic moments are
parallel within a layer but in each layer the magnetic moment is anti­
parallel to the adjacent layers.
This.is illustrated in Figure I.
In a two-sublattice model
sI=nfV sA-i
V
where
w
and Hg are the appropriate magnetizations of the two sub-
lattices.^'*
At this point, the concept of a molecular field begins to cause
serious difficulties when describing the motion of a charged particle
in the crystal lattice.
The fields must vary with position (something
not required in the simplest model of a ferromagnet) because there are
two sublattices to be considered,
.In the simplest approach,
that in
which the magnetic field within a given sublattice is constant,
the
Lorentz force acting upon, a charged particle w o u l d .depend upon within
which sublattice the. motion occurs.
As soon as there is to be transport of'electrons through the
alternating sublattices it becomes apparent that the simple model
-3-
Figure 1»
Illustration of a Two-Sublattice System Consisting
of Layers of Atoms.
Within each layer all of the
spins of the atoms are aligned.
I
n
j.M
'
postulated is much too s i m p l e .
. While it is still clear that the molecular fields may signifi­
cantly affect electron transport in antiferromagnetic materials,
the
models to be used in such a description now become unclear.
A group of materials that may allow these problems to become ten­
able,- although through- a more involved approach, are certain of the
3d
ox i d e s . .
The
3d oxides consist of materials formed of oxides of the elements
having incompletely filled 3d electronic shells.
are semiconductors, of-Fe 0
believe
Some- of these materials
and N i O , for example.
There is reason to
( 3) the Jd wave functions in these materials do not overlap or ■
only very weakly overlap, and thus a. type of conduction occurs.,in which
the mobile electrons spend most of their time localized on an Fe ion and are best described as being localized.
Therefore,
the. system is more
appropriately described as possessing discrete energy levels, rather
than energy bands, at its Fe or Ni host ion.
a transfer of the electron from ion to ion.
Then conduction occurs by
This- type of conduction is
frequently referred to as "hopping".
Such a transport mechanism is characterized by an- activation
energy associated with' the movement of these .self-trapped carriers.
Their mobility will be extremely low because they spend most- of-their
time trapped on the host ion.
'I.
-5H o w ever, if indeed the mobile electrons do spend a significant
portion of time at a localized site, it is quite likely that an elec­
tronic polarization of the surrounding neighborhood will result and the
mobile electron will, while localized, be required to satisfy H u n d 1s
rules«
The mobile charge together with its resultant lattice dis-
(4 )
tortion is known as a polaron.
If the itinerant electron is subject to H u n d is rules, an inter­
esting possibility arises if the material is antiferromagnetic.
The
stage is then set to satisfy the conditions required of double exchange.
Double exchange was first suggested by Zener
(5)
by Anderson and H a s e g a w a ^ ^ and_de G e n n e s ^ ^ .
and later expanded upon
Double exchange involves
the transfer of an electron from one ion site, to another ion site when
the two ions have different quantization directions.
The energy re­
quired to transfer the electron is then angularly dependent "upon the
relative orientation of the ion sites.
This is exactly the case for
to occur by hopping,
of - F e ^ C y :
Conduction is thought
the transport of charge occurring among the Fe 3d
electrons; it is antiferromagnetic at certain temperatures,
thereby
magnetically orienting the spins of the Fe ions, but it has the added
feature of undergoing a phase transition in which the spins of the Fe
ions reorient themselves but no dimensional changes- occur within the
crystal.
During this transition the material goes from an antiferro­
magnetic to a weakly ferromagnetic state over a temperature of approxi­
mately one degree centigrade.
This transition,
to be discussed in
•
“ 6-
.
'
(8)
detail later, is commonly referred •to as the Morin •
will be designated as occurring at
'
transition and
This typically occurs in the
range of -10 to -40°G
Some conductivity data is available' for hematite,
oC -Fe^O^, but
most of it has included only studies of polycrystalline materials.
Morin's data.^ ^
on sintered hematite doped with titanium
"suggests" a change, in the Hall voltage as the temperature of the sample
passes through
.
He stated in that paper that single-crystal data
would be forthcoming but to the best of this author1s knowledge it has ■
never been published.- ■ In later work, Morin
(3)
claims no Hall effect
should be present.
More recent work does predict a Hall effect for
polar on m
o
o
t
i
n
15).
W e ’shall now discuss in more detail the magnetic and electrical
conduction properties of hematite and how the conduction process may be
influenced by the.magnetic state of the material.
’
,
./
t
~7A.
Magnetic- Behavior of Hematite
The crystal structure of a unit cell of hematite is illustrated
in Figure 2.
To discuss the magnetic properties of hematite only the orien­
tation of the magnetic moment.of the Fe ions need be known*
Figure 3
shows the orientation of these ions' magnetic moment with respect to the
crystallographic axis, above and below the Morin temperature.
Thus, it becomes apparent that below .T^ the sum of. the magnetic '
moments of the Fe ions within the unit cell completely cancel and the
Not so above T^ 5 where there is a net
material is, antiferromagnetic.
moment, in the (ill) plane, in- the amount of
weak ferromagnetic
per unit cell.
M
4
Mj
Sin od
(4)
is the magnetic moment of one of the .Fe^+ -ions.
This ■
canting phenomena is-known as weak ferromagnetism, ocbeing the canting angle
Dzialoshinski
has shown that a material may exhibit weak ■
■ferromagnetism under certain symmetry configuration.
Hematite satisfies.
these requirements.
.
The energy term Dzialoshinski obtained is of the form
%weak = 5 -
^
(5)"
where D is a constant vector parallel to the trigonal'axis and M and M
-U
. '- a. b
are the magnetic moments of the two sublattice's, respectively-, . It should
.
-8-
Figure 2.
Crystal Structure of o ' -Fe^O^. ^The open^and
shaded circles represent
^ C^ and Fe^' I
respectively.
-9-
Projection of
Magnetisation onto
(111) Plane
T > T1
Figure
3.
Magnetic Structure of ex' -Fe^O^.
magnetization of iron ions.
Arrows respresent
-IO"-
be noted Dzialosbinski stated that this phenomenological theory is
possible by symmetry arguments, but-he did not specify its origin.
Consideration of spin-orbit coupling and the exchange interaction using
the perturbation theory of quantum mechanics will indeed produce such
(17)
a term''
Artman, Murphy, and Foner
considered the temperature dependence
of the magnetic dipolar field energy and the fine structure anisotropy
energy and were able to show that these energies are very close;.in mag­
nitude but compete for- the directional orientation of the magnetic moment
o f ' t h e 'Fe ions within the crystal.
Using antiferromagnetic resonance to
obtain the constants needed in their theory,
they were able to predict
quite closely the, ratio., of the. Morin temperature ■to the- Neel temperature,
The Neel temperature is the temperature at which the material becomes
paramagnetici
.
There have been a number of studies of -the magnetic properties
of h e m atite.
20
’
21)
Perhaps the most systematic: studies were those of Lin
,■
1
. -He studied the magnetization, susceptibility arid transition
temperature.
The last paper
(21 )
"
dealt with synthetic single; crystals
while the others concerned themselves with natural crystals.
Because of
the difficulties encountered in growing synthetic crystals', most studies '
are on natural crystals.
Among 'those studies there is naturally much
inconsistancy of the data and it is obviously difficult to make studies
of'the effect of doping in such crystals.
used-quite extensively.
Sintered, crystals have been
These are polycrystalline and thus have grain
boundaries which have drastic effects on demagnetizing fields in magnetic
materials and should be avoided in studying fundamental properties of
magnetic materials<unless,
of course, one is interested in just such
effects.
Two types of deviations from the idealized Morin transition do
appear in such studies:
irIine width".
(l) the Morin transition may possess a large
The transition occurring over as much as a
hO°C range^
;
(2) a shift of the Morin temperature to lower values as the materials
are doped
(22)
<
These two effects may occur s i m u l t a n e o u s l y T h i s be­
havior is illustrated in Figure 4.
The origin of this modified behavior is not understood.
One
(23)
author ■
has claimed to observe that the widened transition is actually
composed of many small shifted transitions, and he suggests that the
total response ,is that of many separate regions of the crystal undergoing
the normal narrow transitions.
This may be imagined much like the-flipping of many domains similar
to the Barkhausen jumps observed in ferromagnetic materials.
-12-
Increasing
Magnetization
Shifted
, Transition
Broadened
Transition
Ideal
Transition
Figure 4.
Possible Morin Transitions,
B.
Charge Transport
Consider a ring containing iron ions in which one ion has a
.different charge than the others.
Such a structure has the
This is illustrated in Figure
5.
same energy as one in which the F e 2+ ion is at
■
the jtb site rather than the itb site.
2,
This is true even if the Fe
causes the adjacent ions to become polarized or shift positions on the
ring.
.
There may exist a barrier which must -be overcome to -move the ■
electron from the Fe2+ site to an' adjacent Fe^+ , thereby shifting t h e '1
Fe
site, but the system is the same regardless of the original position
of the Fe
2+
ion, assuming steady state conditions.
Under proper conditions it would be possible to p r o d u c e 'a holeelectron pair (Fe2+ - Fe^"* ) on a ring originally containing only- Fe^+
ions.
Similarly, a hole or an electron could be introduced.by the sub­
stitution of an ion which has one fewer or one more electron, respectively,
- 5,
than Fe
.
,
If this hole or electron were somehow' moved -many sites from
the dopant position,
the ion being assumed- stationary■on’ the- chain,
then
conduction could occur, as outlined above, among the Fe ions,.only.
This is precisely how hematite is made 'semi-conducting- —
either
by doping or by the creation of intrinsic hole-electron pairs at h i g h .
temperatures^"*"”"*"2 ^.
of approximately 10
10
Pure hematite at room- temperature, has a resistivity
■
ohm-centimeters. ■ The conduction.process is thought '
to occur by a hopping of Jd electrons from Fe ion to Fe ion.-
-14-
Figure 5«
Cyclic Chain on which Conduction Occurs
by Hopping.
The decision of what dopant to use in obtaining a hole or an ■
■ electron depends upon three factors-:
(I). the easiest
to solve is the
choice of sign of the carrier., -This involves choosing an atom whose
valence is such that when placed in the lattice it produces a l o c a l l y
deficient or excess electron.
(2) The interstitial atom must be small
enough to fit into the lattice without excessive distortion of the
lattice dimensions, an extreme case causing so much distortion of.the
lattice that the crystal does not "grow".
(3) The last factor to be
considered is difficult to discuss analytically. . The carrier associated
with the dopant must not be so tightly bound to that site that it requires
excessive energy to "activate" it.
Two elements that have been successfully used to produce conduc­
tion by holes and electrons are magnesium and titanium, respectively.
Typical dopings are less than one percent atomic substitution for iron.
Successful growth of single crystals large enough to be used expert- •
mentally require dopings of less than one-half percent Mg and Ti, in this
a u t h o r ’s experience.
ohm-centimeter.
For such dopings,
typical resistivities are 10-100
i
-
C.
Double Exchange-^ ^
^
16
-
'
• .
-
The description of double exchange to be presented here will
follow that outlined by Anderson arid H a s e g a w a ^ .
illustrated in Figure
Consider the electron
6 being transferred from ion i to ion j .while
subjected to the following constraints:
(Cl)
The directions’ of "S^ and I?" are fixed (for example, by the
. local internal magnetic field in the crystal).
(C2)
The electron to' be transferred is either on the itb
or the jti> ion and while there is subject to H u n d 1s
rules.
' To describe the energy associated with- the transfer of the electron
from site i to site j the matrix theory of quantum mechanics is used.
basis functions are those of the isolated ions.
nated by
The
Tb y are to be desig­
^d.^ and I d , the d referring to the d electrons of iron, for
i
.
j
those of the i —
and j -
sites, respectively.
When the itinerant electron is localized on the i ^
ion it will
be
assumed that its spin orientation relative to the ion's spin is described,
by the hamiltonian
'
H = -2 J S v s
"i
'
(6)
where J is the Heisenberg intraexchange integral, S. is the spin of the .
"ttl
i ~ ion, and "s" is the spin of the electron.
Hund's rules reduce.this to.
two possible states
■E = + JijSiI. = + J S '
(7)
Itinerant
Electron
Spin of electron
Spin of ion
Figure
6.
Coordinates Defining 0
-18corresponding to whether the electron's spin is parallel, or antiparallel
to S
While at site j the electron's energy will similarly be given by
E = + J IS .[ = + J S
(8 )
.
What is the sign of-- J? . Hund's rules state that if the electronic
shell is less than half full,
the next electron added to the atom will
"go in" with its spin parallel to the others and antiparallel if the
electronic shell is more than half full.
Hence', the lowest energy when
the shell is l e s s ■than half full would require J to be positive and i f '
more than half full, J must be negative.
We shall see later that the
sign of J is irrelevant in considering which states will have the lowest
energy when the electron is tranferred from one site to the' other.
When considering the- electron to be on one site or the other
there is no dependence upon the relative orientation of the ion which
does not contain the extra electron.
Ho w e v e r , if a transfer is made from
site A to site B, then t h e .energy associated with this transfer
Ep = < Bf Hpl A>
(9)
must somehow account for the fact that the state given by
may not both refer to the- -same.coordinate system.
IB^> and
[A >
Compensation for this
may be done in two ways:
(1) Transform the energies for site B into 'a coordinate
system parallel to A's, or
,
'
(2) Transform- the transfer matrix elements so they refer
-19correctly to the different directions- associatedwith the sites A and B.
The latter procedure will be used.
' .
For" this, it is necessary to intro­
duce the rotation operator given b y ^ ^
iJ
Eu (0 ) = e
for a rotation
.
•
'
0
(10)
U
by an amount
0
about an axis u.
of angular momentum along u, taking -6 =
is the component
1.
The two possible spin states of the electron when referred to
the direction of
ponding to
8^ by
will be designated
a z and /3
ion i are represented by
|d^ a/^>
and
| d /3^
.
|d^o)>
by cl and /3 .
Those corres­
Thus the eigenstates for the electron on
and
|d
, and those on
j by .
.
Inserting the proper discrete values of the allowed angular momentum
equation (7) becomes
ECdi 0. )'= -J Si = -J S
E(di
p ) = -J(Si-H) = J(S-H)
(11) '
.
(12)
Similarly, when the electron is at the jtb site
E(d . c£z) = J'S. = -J S
(13.)
E (d .■ /3Z) = J(S .. ) = J(S-H)
J '
J+l
(i4)
i
-20The transfer matrix elements, if
<a. a
<d. «
< h / 3
and S^. are parallel, would he
I HT I d . O
= t
(15)
I HT I iJ
.= t
(16)
= O
(17)
— O
(18)
I3T
HT I dj
The last two equations simply state that the transfer-, only occurs
when the initial and final spin states are the same on the respective
ion sites (again, a requirement needed-'to satisfy Hand's rules). ■
Vftiat does the transformation of the transfer matrix elements in­
volve?
Consider the coordinate system shown in Figure 7«
The' rotation
(25)
operator takes the form
i
E u (O) =
6/2 CT
e
I Cos 6/2 - i
where
( 19)
U
(20 )
C T u Sin 6/2
CTu = 2 Sy (i.e., the Pauli spin matrices have been introduced).
For t h e •coordinate system chosen the rotation is about the y axis
and hence
.
o - i
(21 )
OV
i
and.
0
i
“ 21-
S. lies in the
x - z plane
Figure 7.
Coordinates Defining the Rotational Transform­
ation of the Spin.
I
—22—
Ii
o V
i =
(22 )
\o
i/
The spin coordinates now transform as
a'
a = Cos G/2
/9 =
-Sin
+ Sin 0/2
0/2 a' + Cos 0/2
( 23)
/S' ' ■ '
.
(24)
This resembles-a rotation of a coordinate system x' - y' through
an angle
0 to produce a new.system, x - y, except for the factor of 0/ 2 .
This results because the "thing" being transferred has, being an elec­
tron, a spin of one-half;
Taking this into account results in an energy
matrix as shown below:
( 25).
Eigenfunctions
a ?
I
tip a
" '
df/B
I
CL.
_ I
I -J S^
dp /3
, ‘ 0
dj /
.
I
t Cos 0/2
,
0
I
—
P'
J
t Sin 0/2
'I
-
J(SiU )
■
1 -t Sin. 0/2.
I t Sin 0 / 2 1
I
■t Cos 0/2
—I
ti.a' 1 t Cos 0/2 I -t Sin 0/2, -J S.
I
a
'
t Cos 0/2'
I
0
,
0
i
j ( s i + i)
-23The thing to be noted, in equation (25) is that the off-diagonal elements,
i.e. those associated with the transfer of the electron from one site to
another, are now angularly dependent upon the relative angular oriention
of the sites between which the electron m o v e s .
Let us now illustrate h o w these off-diagonal elements may vary with
the direction of the electron motion in a crystal, assuming that this
motion occurs by a hopping from one site to the next so that the two-ion
model may be used.
8.
Consider the two-dimensional lattice shown in Figure
The arrows indicate the quantization direction of the ions.
Suppose,
for simplicity, the corresponding wave functions are the same along and
symmetrical a b o u t .the x and y axes as illustrated by the fourleaf appearance
of the wave functions.
If the electron were to go from one ion to the
next along the [10] direction then 9 = 0
0
-J S
t Cos 0/2 t Sin 0/2
J (S-KL)
0
-t Sin 0/2 t Cos 0/2
t Cos 0/2 -t Sin 0/2
t Sin 0/2
-J S
0
and (25) becomes
-J. S
t Cos'0/2
0
J (S+l)
0
-
0
J (S+l)
t
0
0
t
..0
(26 )
t
0
- JS
0
t
0
J (S+l)
I roii
Figure
8 . Symmetric Wave Functions in a Two Dimensional Crystal.
The arrows represent the magnetic moments of the atoms.
I
- 25■However, if the electron -transfer, is along the [01] direction then
6 = TT
and (25) becomes
J- -J S
O
0
J (S+l)
t Sin tt/2
0
t Cos 7r/2
0
• J (S+l) '
- ■
-t Sin tt/2 t Cos n/2
t Cos tt/2 -t Sin tt/2
-J S
77/2 I
t Cos Tr/2 t Sin
- J S 0
0
J (S+l)
0
t
-t
0
( 27)
L
0
-t
-J S
0
t
0
0
J (S+l)
■ J
It will be shown in the next section that the expression for
charge transport involves the use of off-diagonal elements, which do
appear to be different for this simple two-dimensional model.
When the unit cells of.'hematite are joined to form a crystal the
structure generated is quite similar to that of Figure 8.
The Fe ions -
•form a two-sublattice structure which is illustrated in Figure 9•
Within, each layer of the sublattice the.magnetic moments of the ions
are parallel to each other.
•
For charge transport within the { i l l ) plane,
•
Q = O both- above and
below the Morin t r a n s i t i o n , whereas .current flow along the [ill] direction
has. © = IT
for T <
but 9 = II- 2a for
< T ^ ^Neel''
.
>
a
.
'typically of
-26-
plane
t
iJ
< tm
=
and
I
eb
I
T,
Keel
are
-L (ill) plane
Figure 9*
T., < T <
and Mg lie in (ill) plane
Sublattice Magnetization Directions of Hematite.
the order of 5°.
T h u s , if double ..exchange is to be/a significant factor
in discussing the energy of mobile electrons in hematite, then as the
temperature is varied through T^ the value of 8 is unchanged for trans­
port within the. (ill) plane and h e n c e ■the. matrix elements dicribing-the
system are those shown in equation (26). 'H o w e v e r , for electron transport
along the [ill] direction with T < T ^ 5 6 = II; but for T > T^, 8 = TI - 2a.
T h u s , below T^ the energy matrix is that of equation (27) but above T^
it becomes
_
-J- S
0
t Sin a
t Cos
a
-t Cos a
t Sin
a
0
J (S+l) .
.
(28)
t S i n a -t Cos
a
t cos«
a
t Sin
0
-JS
T h u s , in going from T < T^
0
to
J (S+l)
T>T
the energy m a t r i x , for current
flow in. the [ill] direction, undergoes a change in the two by two ■
submatrices off the main diagonal.
If the expression for the charge
transport in hematite is dependent upon these off-diagonal elements it
appears there may be a change of the transport properties along the [ill]
direction at T^ (because the off-diagonal elements change at T^) whereas'
for transport along the (ill) plane no change would occur because there
is no c h a n g e ,in the corresponding matrix elements a T^.
- 28D.
A More Detailed Look at the Hopping Mechanism'
'
A classical model of hopping, considered on a microscopic- basis,
would describe a current as a succession of electron movements (trans­
fers) over a discrete distance (from ion site-to ion site).
Thus, in
writing an expression for the current as
J = . q n v
(29)
it would seem appropriate to replace the velocity, v, by a term,
v = CJ
where CxJ
a
B
(30)
*a-»B
is the probability per unit time ,for transfer from site
a to site B, which may be degenerate; i .e .,^ there may be equivalent
sites in the crystal, and d^_^g is the vector distance traversed in
the transfer.
The net current would be the average of all such hops in the
crystal; i.e.,
<
2
I na ^
a-B
a->B
>
(31)
all
possible
transition
where n
is the number of particles per unit volume at sites equivalent
to a.'
Using first order time -dependent perturbation theory the transition
probability is given by
(26 )'
-
.
/ . '
CO
a~>B
^
2ir/a
where V^a is the matrix element connecting states
i.ec,'V^a .= < b | V |
at
a.y
I
I
| aT>
and
jbj>
(32)
;
, and yO ^ is the density of final states
b.
Thus it becomes apparent how the transfer matrix elements in­
fluence the current in a hopping process.
See comments after (25).
The above description of current flow is by no means unique.
It should be just as valid to use Equation (33) where the quantum
■mechanical equivalent of v is used.
(1)
This may be done.,in two ways:
v = (l/i-ti) [ r, H ]
(33)
where v is the time rate of change of the position vector,
r, and
hence is obtained by commuting v with the Hamiltonian.
(2)
v = (l/m)[ p + q(A / c ) ].
= (l/m)[ (h/i)V
where
V
(34)
+ q(A/c)]
is the gradient operator and A is the m a g n e t i c .vector
potential as ’'seen" by the moving charge.
If the exact Hamiltonian■can ■
be described and the resulting set of eigenfunctions found,
three.descriptions of the -current should be valid.
then all
The e x a c t •solutions
cannot, however, be obtained and must be approximated.
This implies
t h a t .varying degrees of approximation must be accepted and thus the
values of J obtained by -the three approaches outlined above w i l l , in
-JO-
general, yield different, results.
The first approach,
considering
the probabilities for. an electron"transfer, directly incorporates the
concept of hopping- on a microscopic basis and has thus been chosen for
the discussion of how current "flows" in such materials.
Some of the multiple paths for current flow along the
[ill]
direction or possible sites for hopping are illustrated.in Figures 10
and 11»
For the case T < T^, there are two allowed hopping paths -.- '
(remember we required
<d^
4
..j Hp | dg /3 >
= < dp#
j Hpl CL, d]>
= 0)
as shown by the-dotted lines representing the connecting matrix elements
Vp and V^, respectively.
For T > T^ there are four possible paths.
The only hops consider­
ed are to the nearest neighbor sites in the direction of current flow. ■
'
•
.
-
,
A question arises with respect to matrix elements such as V-, V_, and
„
Vg.
:
Are they equal?
■
■
In the absence of external fields it appears
that they are.
Consider for the moment the'angular and. time1 dependence of the
■basis functions used to describe the 34 electrons.
.This is of the form
(G, 0, t) =0(0) (^(0)
°
^
. (3$)
-31direction of three fold
symmetry in hematite
Figure 10.
Illustration of Allowed "Hops" for T <
The spins of the ions are along the
[ill)
direction, © being antiparallel to 0.
-32flll] dii ection - three fold
axis of hematite
Figure 11.
Illustration of the Allowed "Hops" When T >
T, .
M
The spins of the ions lie in the (ill) plane
ajid are canted as shown.
Transitions such as
Vp , etc., have been omitted to keep the sketch
uncluttered.
-33but
i I
cP (0) = e
'
Putting E =
..
(3o)
Hico Q- we obtain
i(coet +■ m^0)
^fr = (9 G:e e
(37)
which represents a wave-like rotation of the wave function about the
z-axis with angular frequency CO q A ’
-^ •
Sketches of such functions are
shown for the various Jd states in Figure 12.
The itinerant, electron
spends most of its time trapped and only occasionally jumps.
Thus, it
gets a rather "fuzzy picture" of its neighbor as shown in' Figure IJ
which illustrates the possible
i
V^, and
it
4
for a hop from a Jd
5
Jd
to a
■'
state.
Similar sketches can be drawn for the other transitions in
the crystal. ' From the symmetry exhibited one expects all hops to ,equi­
valent sites to involve identical matrix elements because of the time
average symmetry of the wave functions.
What is the difference between the two-sublattice hopping put
forth earlier and this seemingly more complicated model? ."Strictly
speaking,
the two-sublattice model-forbids hopping in the
[ill]
direction since the matrix elements of
V
=
(3 8 )
connecting antiparallel spin
for temperature's above the Morin temperature.
The more''detailed model
still adheres to this requirement of the transfer matrix elements■but
-34-
Figure 12.
"Spinning" 3d Wave Functions.
-35-
Figure 13.
’’Pictorial Hopping" from a 3d
4
to a 3d
5 State.
-36considers a transfer which has a lateral component,
In the absence of
external fields the average of such transfers must yield a net trans­
verse component equal to zero.
position on the
[ill]
Thus the charge is transferred from a
axis of one unit cell to that of an adjacent
unit cell, rather than along.the
T h u s , a net current flow along the
[ill]
[ill]
direction at each unit cell.
direction is allowed.
THE.EXPEEIlffiNT.
Single crystals of hematite doped with magnesium w e r e ■grown by
the methods outlined by Besser
(27)
' .. The doping was one-half percent
(nominal) atomic substitution -for iron.
Two samples were cut from the same single crystal being separated
"by only 12 milli-inches due to kerf loss of the saw blade and approxi­
mately two milli-inches were removed when polishing the surfaces of
the crystals.
Both samples are regular parallelpipeds with dimensions
of 0.050" x 0.050" x 0.150".
The orientation of crystal with respect
to the rectangular sides of the sample is illustrated in' Figure 14.
Silver contacts were applied by evaporation of silver in a
vacuum followed by firing at 3 0 0 0C for 20 minutes.
square ends were 0.050" in diameter.
were 0.010" x 0.050".
The contacts on the
Those on the rectangular sides
The 0.010" width was .chosen to m i n i m i z e •t h e .
shorting caused by this contact.
Tne contact orientation is illustrated
in Figure 15.
The i n d u c e d 'voltage was measured while passing a constant current
through the sample in t h e .longest direction and measuring the voltage
difference produced between the smaller rectangular contacts.
The samples were mounted so they could be rotated through
36O 0
while subjected to an external magnetic field and a known' temperature.■ ■'
Because there are .many possible orientations of current .,flow.,
.voltage measured, and1 magnetic field directions with respect to .,
%
-38-
Figure 14.
Sample Crystallographic Orientation.
i
-39-
Constant
Current
Hematite
Sample
Differential
Voltmeter
--
Magnetic Field
Figure 15.
Variable
Temperature
Basic Experimental Configuration.
-40crystallographic directions these orientations will be shown on each
figure where the results are presented and-will- thereafter be referred
to by the respective figure n u m b e r .
The values of magnetization were obtained using a Foner vibrat-
(28 )
ing-sample magnetometer^
.
Magnetic, fields were obtained from a
Varian 3600 magnet with field dial control of the field magnitude.
The data were taken directly using as x - y recorder having such
inputs as a voltage proportional to; the angle of the sample- with
respect to the applied magnetic field., the temperature of the sample,
the magnetization of the sample, and the transverse voltage A V .
The values of the magnetization were obtained using a Foner
vibrating-sample magnetometer
(28)
.
Magnetic fields were obtained from
a Varian 3600 magnet with field dial control of the field magnitude.
A more detailed discussion of the experimental configuration and how
the experiment was conducted is given in Appendix-A.
RESULTS
The experiment shown diagrammatically in Figure'15 determines
the transverse resistivity,
field dependence.
temperature, magnetic field and electric
That is, the parameters being studied are described
by the resistance tensor yQ
given by.
P x x
P
Pyx
P y y
P
ZX
Xy
P
P
P y z
P- zz
in which each yO ^ . is a function of temperature, applied magnetic
field, magnetic state of the sample, and applied electric field.
the experiment is characterized by its terminal behavior,
Pyx
1X
-
Pyy
Iy +
P ; yz
If
then
z
(38)
P yz 1z
PL&,
R]
r,.
assuming the differential voltmeter has an infinite input impedance.
It is, t h e n ,
yOL that this experiment characterizes,
Figures 16 and 17 give
cur'rent flow along the
[ill]
A V
.
as a function -of temperature .for
direction and in the (ill) plane,
respectively.
Each of the curves may be used to determine the' activation.energy
of
yQ_ when the current is along the [ill]
direction and In the-(ill)
T==T
T T =
T i T] , I I ! U T + :
- ..U - L L - H H - L f
— -L r p j - H l H - I - L - A R t i t i j
'
A
. :
=ITd
-LLU
I i'
rr~iLi= = I,
Li.
Sample # jAp- Vu _____
Recorder
Y
A m p Gain X / /3 Y
Data
X ^ ^ Y <r~
Bias Current
7 , c‘
Offset
At *2 /At
T. C. Zero
4 L y—
Mult.
Null
T. C.
Date / c / ~
b
f' =I=Iit
=U-LLM-I=T t t p i i i
1
i
-HL LitM
it =Il-i'.tbI
=H=j_i i t d± |.it I''i:t= tit:
I
j. .!--IL—
U-;_L|-L _j—!--L_Ll-Pl-LL;.-!--l
. t p T T . l Z -i- L. LL i ; ' I'';'
.1
=L=i±l±iL!=u=
i— i
I- :
I— L
* ■r
-— !=T-I=Ii
f—— ■— i— , — |
—I— f—;
I
-j
-j-!...I—j...L._|
=:=u:Ll
rr ^ u I = M
— .
I-
-" H tt
I
i t t t j : j i T T r= iI nt it =l ' ' . I = I i u t i t
r- I—r - i h
__ ______
r-1-pp-
i/O - 1
.Li-
" r r r r r - , r;
-T !
—I ! U-I J— j —i —.J ! I 1'--; j—( j
Tj=-H-T H -H -M -U —
- H t u :U H = I : t M ' ! T f i T t I t l t t i l t t j =ifI H 4A.Ni,".*
■'*
-Ir--I-HTfiTi
--—I
'1
.
|—I—I ~ i-1—[" I —!
—1
I—I I i—i—1I—'-"H
Tf
ITi-L..: -
.tftitt
IiiSiStr
I
'f U L
-' -lL nrfri;:.'
L,.,-"'1L..', vi'll
i
— -— - ■- '----- -- — -
l i t
h"~rri~'ii~I'=—j-r-jH*
L
i ! : '
1
I : I i I
1
—H Ti—
U ti
I u t = L T : Ta''''
- : ' T
.on)
= ( 5 millivolts/inch)
-Ll_
■— f-•-— r
AT
L-L
I_ •_ _ .
' '
. ■
Transverse Voltage
fpri-n-4-1
-:-f-
-I"
B d ± t i : h : :H i t t n i '
-
.... F
:%
1
— t-1
—I
—r
LtfcHUU
J
H
f
o
U
l
.U=
q :
M
~I
! : I I I : I I ;
...
-H
Ii^utpi=;.'! i
—H - H
i T nV
f HH IH i H - H H i•r f T F r f i - r f u f f
Hj=-TLTrrr
U T T iUT i TnT i ,T l - u
-TJ T T
L LTUiT=T 1= = H T i = I i m -J- , 1- I
,-j— T—
-L S
B-L----r-;— - M - - - = U T T U = H f T *
FT
L m i- L u . -,u - m - l u - H ui—L.j—l-t-f-r-L
LLJ-LH-!
t
i_I
1
HU
m u
- - . 6. 6.
.
= LiLfiS
...
--U 4 H f-I-IL
_____'
__
\ ________ i___r n
; - 1 -________ .. .;=!=f!=i:uHiti-T!-m:a
....
I*1"IH-Trv
:
:
. .... '
:
,
LI I I -L fH -I- I H l H iJ ffi" : I u E-:! LL
\-M-1..<—j---I—j
—I—L-!.L-I--J--L I—i—!—(.-1—I—I—I-!-.-!—! —1—•
— f—I
J-H- HT:=! =I==L1'' LH-J-I..
.
!-LLj -j-T ff [- "-hf+tl-FTr
r -'
i 1 H U - E f U u T i f l U T H t i t i L - L - i - --1 f-H-Hr-T-t—r
Th H f t
rf
=IT=T=.
TL L L H t
H-i
1 TTrtiH.
=TC
-H- H
t i t IH
11
lUmneraturpL-:
U t 1
-hi
I-IfTjm
'H':j-ffi-;f-ti-m
Tti 1 :
iitti4
U_L;_
....
" h h l ""r
fU ffl
=HHHEEfuM
-— — • — L — i-
LHUHtitiijtiti+= - U L # # i-=ttW
m titiu m
Figure 16.
Temperature Dependence of the Transverse Voltage for Current Flow in the
[ill]
Direction.
i
ro
1
-H -rH -'H " r t
fft'
rc i
!'Si
-
Recorder X A o m Y srn
A m p Gain X /O Y /o
Data
X era You'
Bias Current
B .vS ryl O
Offset
At 2 UL
4
T .C. Zero
Mult.
Null
T. C.
Date Io/y A
i I
1IlliiiFffls
iI
-f P - T
—•—|— j-
I',
^
^.
r.
'-±u!±bd3±irc::h:
....
-I--U-LL.:_!_L_!_‘.J. L-L-I-L-I-U.; -L-
-I-U Lu
14-!-U u l X i X i X u ;-.|4
OJ fS
.
L iU i- S .:;- ::
I-—.—
I•I
h i-
'u
O - ^sVT-*— I—
rr,'
-L;-!UL--LipXI-1-
1OTrrU
mm
Ui I i L u L i u
_i_l
{-!-r—I- ;
-j—j-1-I-p-j-J-j—1-.:— j—
L X f u i h
Silii
ff $
Sxti
1
|] X r j - u j
1
-LU
-l-i ! » I
l± n % i
X fr"
j—
-U-Lj
x > r-i-i" |- r | m
LXLXXLXL
TP
P I
Ti !
x
B
uXXLXLi—! 1L L - U x X X b X L L
— :—
,.:P .u
- u -----;—
"XL
-T- r—
I.
—
i t : ! !
IXIX u X lX
X K ur
p rsX r r h p -
Figure 17.
-f—p
. ______ _
' ttt
XiTpjxjBH
Ip - p - t
LLUi
i i i l-Tup
Li-T-;-' L u i-r[ - H - - K - H - T r X
~[TP~K
r rpH X~rT~K
j X r iXf
xH± ' i x p A l x c
L u i L f-I-L f-'
BI
-t#Fi
B h h r X iXf j
X x jiiB n r X
s
LEE
-L
I
—j—j— ■
i
— j—
L L L lX b
P -Lt
K
- 16.6
:iBix;
.,X u hx
BBB
h rx tx x
HXvYiE w ‘
...
;....— .— j— I— •.— •— i—
.
•pi
— r— .— I—
I— j -I--L-L-X
i — *— .— •.— -—
K f u - r s X x W
XL
—
L
X
r
HXrLLrH-
KLlXr'— rI
■ .............
vBui’t
, ;X h X
X i- H X p t '
TxtrH-LB ;
i
JbtiLXsIii:
- I :.9
"
-Ka- -!."Xl ___S i
L X i b d -LI "I
L.l
L L Li-Ur H - X r r J I"i” !![Bt t L t u l l b H t " B
*— * — - —: -— u — j— —
j— __L
Xl-Ut-I '
-L
.1
—4— I— I—I—■
—i
CH LU'
LL-
.
a s
-
-LQ ■ o :
. f-4-; > - - - \
I— T- !O -• *f~t ‘
— I
— -j
.
-LLLLLj
.|._.PL_L !ii-X-L-f-bu-U-u-
•H - i X h j X t r P I " LK -L rf—L
K"Bxx t,TX hL l L
i .L l x n j - p - X b t i B r P
O
X f-i-Xf-XI
XJBtiBi
Bh
X h P x X x x
,.L u B tL L
BiBp
W
.r - u :L-.:.-,TejnpcraiurJL1U K
—i—i—I
S i'-Thj--PXr
T U r f z r X efK f -W X K S
P
IP
T
U
X
-U-X K K F X ' I II
ir
1
^
-i-
Temperature Dependence of the Transverse Voltage for Current Flow in the
(ill) P l a n e .
plane.
These energies are
Ea ^111I =
Ea
Note that these
There is
•
.•
_ i .3 ev
=
• and
1.3 ev, respectively.
are determined for a specific current density.
a rapid change in
Figure 16 but not in Figure I?.
A
V at the Morin temperature in
This is exactly as expected if a double
exchange were a significant factor in describing the conduction process.
Figure 18 is' a plot of
A V for current flow along the [ill]
direction and an external magnetic field applied in the (ill) plane.
& is the angle between the magnetic field direction and the contacts
used to measure
A V.
The temperature was fixed at 2 5 °C; i.e., T > T^.
The results are not as expected in a normal Hall experiment. " The
resulting voltage has a 2 0 dependence rather than just Q; i . e . , the re­
sulting transverse voltage is not linear in applied magnetic field but
is of second order.
Figure 19
to
the magnetic
The origin of this behavior is not known.
shows the effect of rotating the sample with respect
field when the magnetic field is in the plane formed by
the direction of current flow along the
direction- defined, by
A V.
[ill]
direction and the
Again, T = 2 5 °C. > T^.
H e r e ■the- influence of the large anisotropy confining the.magneti­
zation ■of hematite to the basal plane is evident.
■
•
I
Figure l8.
A V as a Function of the Basal Plane Magnetic Field Direction
Recorder X sc- i Y g-j
Amp Gain X____
Data
X tT/-,
Bias Current
Offset o.p?o At
T-C. Zero
Mult
Null_____
T-CDate — /£•
; Tj
—-'I-
Figure 19«
Effect of the Magnetic Field Strength upon
AV
Vsriy^
-47' The anisotropy energy within.the basal plane is very weak,' being
. equivalent to tens of oersteds.
-The magnetization' is,, h owever, confined'
to this plane except for very large, fields, on the order of IO^ o e ^ \
Under such conditions an M-H loop in which H is; applied in a plane
containing the
Figure. 20(a).-
[ill]
direction would be similar to that shown, in
But the data of Figure 18 indicates we should be con­
sidering the second order field and hence -more appropriate is the
|M| -
H loop shown in Figure 2 0 ( b ) »
Remembering that the sample consists of many magnetic domains,
the results are obvious.
When H is along the
fill]
direction the
magnetization is still confined to the (ill) plane but © is randomly
oriented in the plane resulting in no net magnetization.
As soon as-H is off axis enough, all the domains are aligned..
Thus,-what one observes is the net resistance of many small resistors
whose value depends upon its magnetic state.
Once, all of the resistors
have the same value (the same magnetic .state and o r i e n t a t i o n ) , t h e samplebehaves as a single, resistor (domain). ' This is exactly what is shown in '
Figure 19«
The family of curves produced as the magnetic field strength
.is varied.also supports this interpretation, of the results.
At low fields
the •sample is essentially ,demagnetized and hence no variation in
A -V
is observed; as the field strength is increased the effect.is to align
a greater percentage of the domains and hence a greater "sharpness" of
change in
AV
when the magnetization of the sample switches at H^..
-48-
0. = Sin-1 Hc/HA p p l .od
Figure 20.
Hematite M - H Loops.
-49-
'
It has been found experimentally that the temperature at which
the Morin transition occurs is a function of the applied magnetic field.
Increasing the magnetic field lowers
This is shown in Figure 21.
in a nearly linear relationship.
It is thus expected that the change in
A V
exhibited in Figure 16 should occur at a different temperature as the
external magnetic field is changed.
The temperature dependence of
densities is shown in Figure 23«
This is observed i n .Figure 22.
A V
for three different current
Two interesting results a p p e a r :
(l) apparently the Morin temperature may be changed by varying the cur­
rent density;
(2) the activation e n e r g y ■(or temperature coefficient) is
observed to change sign at a certain current density.
The first result may be a result of the experimental procedure.
The sample has a resistance of about one kilohm.
A change of current
from 2 to 2.5 milliarnps would result in an additional
being d i s sipated.in the sample.
2.25 milliwatts
This additional power dissipation could
raise the temperature of the sample a few degrees thereby causing an
apparent shift in the Morin temperature with increasing- current.
It is difficult to imagine any factor attributable to .the experi­
mental techniques used which would produce a change in the slope-of the
curves,
(i.e., a change in. the activation energy), as 'the current
through the- sample is v a r i e d , .
It is possible that the change in the slope is a result of the
electric field having an effect on the transition probabilities of the
hopping electrons.
For example, referring to Figure 11, at a given
hIEEB !-J-
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Shift of the Morin Transition as a Function of the Magnetic Field.
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Figure 22.
X ff=
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Shift of the Transverse Voltage as a Function of the Applied Magnetic Field.
j
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Figure 23.
T
H
-U 4
—I—I— j—I—-I—}—I—*—j—4—I —f - i—;—t —I—j—-U j-
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I
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T C d H ^ ^ C C T
Transverse Voltage as a Function of the Current Through the Sample.
Etr' '
E tS
I
I
V
rui
-
i
53
-
value of electric field conduction by the transition indicated by Vj
may-be occurring but the electric field strength may not be sufficient
to produce a significant contribution from V^.
electric field the contribution from
Upon increasing the
may. become appreciable=
The
result is'an increase in the number of possible hopping sites' available
for the conduction process,
’
( R )
Holstein and Friedman ^
mention that the Hall mobility,
yLtg,
is given by
Zu
H = c/H
c rX
/
where the conductivity tensor
tensor.
c r XX =
CT is
C /B
M x y z
xx.,■
M
(39)
the inverse of the resistivity
They further state that mobility along the primary direction
of current flow,
/06
-•
-EZlcrP
, is predominantly of the activation form e z
while the transverse mobility,
f-T^ ^ may vary as
l/T and hence the
Hall mobility may exhibit a negative activation energy,
Equation 39 would- allow both
ylX
and
f^-xx
varJr as e
and under proper conditions- the ratio would produce a negative tem­
perature coefficient.
The above discussion is not intended to be rigorous but only to
■give
a plausible explanation of. .the.factors that would contribute to-
an electric field dependent temperature coefficient for a material in
which conduction occurs by hopping.
CONCLUSIONS
This work was intended to be an experimental study to see if
materials possessing a magnetic state have electrical conductivity
properties which are influenced by the magnetic properties of the
m a t erial«
Hematite was chosen for the study because of its precisely
defined transition (Worin transition) in which it goes from a weak
ferromagnetic to an antiferromagnetic state.
This is believed to be
the first study of this kind undertaken using single crystals.
The results indicate that the electrical properties'are altered
at the Morin transition for current flow in the
Qlll]
direction
but do not change when the 'current flows within the (ill) p l a n e .
These results are interpreted as being a consequence of double exchange
The data also indicates that the conduction process, which
occurs by hopping, is not linearly dependent upon the electric field ■
applied to the sample.
this behavior using
A plausible explanation is put forth to explain
concepts related to conduction by hopping.
It also appears that even though the sample is a single crystal
its conductivity is affected by the domain structure of the sample.
Further studies certainly seem appropriate.
Perhaps most in­
teresting would be a theoretical study in which an attempt is.made to
determine why there is no linear Hall effect but only a second order
dependence upon the applied magnetic field. .
I
-
55
-
'
.
Also of interest would be 'calculations for', the' temperature
'
x :'rV
dependence of the electrical conduction, in the various directions
within the crystal.
There is more need for experimental studies both to more com­
pletely characterize the conductivity tensor and to observe the effects
of other dopants.
A study correlating the resistance and domain structure could
also prove useful
I
APPENDIX A
DlSCEIPTION OE THE EXPERIMENT:
The experimental configuration used to measure the transverse
voltage of the sample is shown in more detail in Figure 24.
The.
temperature of the sample is determined ."by the use of a thermistor.
The thermistor is a YSI-Components precision thermistor # 44005 having
a 25° C value of 3000 ohms.
The temperature is determined by passing
100 microamps through it and measuring the voltage .produced by the use
of a differential amplifier.
The output of this- amplifier supplies the
x-axis input for a Mosley model 135A x-y recorder.
This is the source
of the temperature scale for Figures l 6 , 17, and 23.
.
Knowing the current
through the the r m i s t o r , the amplifier gain and having a calibration of
the resistance vs temperature of the thermistor enables one to determine
the temperature of the thermistor.
■ To obtain a voltage proportional to the angle of the sample holder
with respect to the applied magnetic field direction, a potentiometer was
connected to the sample rod.
This voltage supplies the input to the x-y
recorder to provide the 9 values in Figures 18 and 19.
The transverse voltage is obtained by the use of a differential
voltmeter which- supplies the input to the y-axis. of the x-y recorder.
This provides the y-axis readings on Figures l 6 , 17, 18, 19, 22, and 23.
The differential voltmenter consists of an integrated circuit operational
amplifier produced by Fairchild ( # yuA 709 )■
It. is used in a single
---- - V V —
ended non-inverting mode as shown below:
^
(3m .
R‘
i
-
Leads for current
sources and
—
differential
voltmeter
57
-
P
Sample Holder
Dewar-filled with
acetone to which
dry ice is added
to lower the
temperature of the
sample
Magnet pole piec
Enlarged viewapproximately
3 times actual
size
Thermistor
Tygon tubing.
,Sample holder
Sample holder
mide
Dewar
wall
Soldering
iron
Hematite
sample
Printed circuit
board on which
sample is mounted and leads for sample attached
Figure
2h.
The Experimental Configuration.
In this mode and for the values of the gain obtained (' 3 and 10 ) the.
amplifier has a worst case input impedance of approximately fifty megohms.
A typical run in which the temperature, of the sample is varied is
carried out as follows:
The value of the current through the sample is
chosen as are the magnetic field strength and direction.
Dry ice is then-
added to the acetone in the dewar until the lowest temperature desired
is obtained.
The system is then allowed to "warm-up" by itself.and it
is during this time that the' data is obtained.
A typical warm-up rate
is approximately one degree centigrade every three m i n u t e s .
It was found, during the initial experiments, that because, of the
small diameter of the dewar in the region containing the hematite sample
that the temperature would fluctuate quite rapidly.
Adding the Tygon
tubing and a soldering iron operating at a low temperature, as illustrated
in Figure
2k, caused a small stream of bubbles to pass upward through
the dewar stem and had the effect of keeping the liquid stirred and at
a more uniform temperature.
’
'
.
-
59
-
LITERATUBE CITED
■
1.
E, M. Pugh and N. Hostoker, E e v s « Modern P h y s » 25, 151,
2.
A. H. M o r rish 1 "The Physical Principles of Magnetism," John.
Wiley- & Sons, Inc., (1955), New York.
Chapter 8 .
3.
F» Jo Morin, "Semiconductors", Edited by N. B. H a nnay 1 Eeinhold
Publishing Corporation, (1959), New York.
4.
T. Holstein, Ann. Phys.,
5«
C. Zener, P h y s . Eev., 8 2 , 403,
6«
P. W. Anderson and H. Hasegawa,
7.
P.-G. de G e r m e s 1 P h y s . Eev., 1 1 8 l4l,
(N.Y.)
(1953)•
8_, 325, (1959)«
(1951)«
P h y s . Eev., 1 0 0 , 6?5,
(1955)«
(I960),
8 . F. J. Morin, P h y s . Eev., ? 8 , 819, (1950).
9«
P. J. Flanders and J. P. Eemeika, Phil. Mag.,
1 1 ,. 1271,
(1965)*
83, 1005, (1951).
10.
F. J. Morin, P h y s . Eev.,
11.
E.F.G. Gardner, F. Sweett and D. W. Tanner, J . P h y s . Chem. Solids,
24, 1175, (1963),
12.
E.F.G. Gardner, F. Sweett and D. W. Tanner,' J. P h y s . 'Chem. Solids,
24, 1183, (1963).
■ "
13«
T. Holstein,
14.
L. Friedman and T. Holstein, Ann, Phys., 2 1 , 494,
15«
T. Holstein- and L. Friedman, P h y s . Eev., 165,
Ann. Phys.,
8_, 343, (1959)«
(1963).
I 65, (1968).
(1958).
lb.
I. Dzialoshinski, P h y s . and Chem. Solids, 4, 241,
17«
T. Moriya, P h y s . Eev., 1 2 0 , 91,
18.
J. 0. Artm a n 1 J. C. .Murphy, and S. Foner, P h y s . Eev., 138,
A912, (1965).
. —
19«
S. T. Lin, J. A p p l . Phys., 30,
(i960).
306 S, (1959).
- 60LITERATURE. CITED (continued)’
20.
S. T, Lin, J . A p p l , ;P h y s .■* 31, 273'S , (i960).
21.
S. T. Lin, J. A p p l * Phys., 32, 394 £, (1961).
22.
Personally obtained data,
23»
C. W. S e a r l e .
24.
A. Messiah, "Quantum Mechanics", vol. I I , North Holland P u b l .
Company, Amsterdam, page 533»
25«
A. Messiah, "Quantum- Mechanics", vol. I I , North Holland P u b l .
Company, Amsterdam, page 546.
26.
A. Messiah, "Quantum Mechanics", vol. I I , North Holland P u b l .
Company, Amsterdam, page 736,
27.
P. Besser.. Thesis, University of Minnesota.
28.
Foner Vibrating Sample Magnetometer, manufactured byPrince tori "Applied. Research, Princeton. Nev/ Jersev..
29.
The phrase "double exchange" was originally coined by Zener^ ^
to refer to+the'transfer of an electron from one Na ion to the
adjacent Na
ion as- the transfer of an electron, say, from the
left,Na atom to the central Cl ion simultaneously with the+transfer of an' electron from the central Cl ion to the right N=I ion,
i.e. the system before the transfer consists of Na Cl Na and' ,^ \
after the transfer consists of Na Cl Na.
Anderson and Hasegawa^
later relaxed the definition to include a simple transfer of an
electron between two ions, for example, before the transfer
Mn Mn and after the transfer Mn Mn .
Thesis, University of Minnesota.
9
t
I
l^^^Ransoni, G. L.
R174
Effect of the Morin
cop.2 transition upon electron
^liransport in magnesium
^^■^^doped single crystal
hematite.
H a M e AN o ADDHtas
=%s
fsMi