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X-RAY STUDIES OF LATTICE DYNAMICS
BY
MARTIN VICTOR HOLT
B.A., Rice University, 1998
THESIS
Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Physics
in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2002
Urbana, Illinois
ii
ACKNOWLEDGEMENTS
I’ve never really understood the purpose of this section. After all, if I for some reason took it seriously then
its length could easily dwarf the entire rest of my thesis (then again, so could the average daily issue of the USA
Today ... with more color figures). This work is a product of the support and inspiration I’ve received over my
entire life, both personally and professionally. There are a few easy targets that I’ll take care of first, and then I’ll
just send up some shout-outs. I’d like to thank my dad for showing me the more you teach, the more you learn, my
mom for the immense therapeutic value of keeping happy meal toys on your desktop, my brother for a little friendly
competition (after all, it’s all fun and games until someone catches on fire), and my sister for letting me know that
coffee is sweeter late at night at Denny’s. Thanks also to Jennifer Gibas and other members of the Gwendolyn
Brooks co-operative for their kind hearts and callipygous pulchritude. Props also to the second floor study lobby of
Brown College, Rice University for, if not the inevitability of socialism, at least the inevitability of socialist
propaganda. This would also not have been possible without my cats. Yow.
Science is in large part how one looks at the world, and that’s not an easy thing to quantify. Thanks go to
members of Chiang National Lab past and present, Shashank Misra, Tim Kidd and my advisor T.-C. Chiang for
policing the fine line between the scientific method and applied wishful thinking. I’ve also been privileged to work
with UNICAT scientists and users who are dedicated to building a research community, and along the way are doing
some outstanding science. No matter the problem, there are really only three things to do – i) decide on the correct
thing to do, (ii) do it correctly, or (iii) waste your time.
Speaking of which, did you know that the UNICAT facility at the Advanced Photon Source (APS) is
supported by the University of Illinois at Urbana-Champaign, Materials Research Laboratory (U.S. DOE, the State
of Illinois-IBHE-HECA, and the NSF), the Oak Ridge National Laboratory (U.S. DOE under contract with UT-
Battelle LLC), the National Institute of Standards and Technology (U.S. Department of Commerce) and UOP LLC?
Amazingly enough, the APS is still supported by the U.S. DOE Basic Energy Sciences Office of Science under
contract No. W-31-109-ENG-38. This author would also like to acknowledge a discussion within a seminar given at
McGill University as the initial motivation for the investigation of critical phonon scattering in Chapter 5. The
Plutonium images were taken in collaboration with Dr. Joe Wong of Lawrence Livermore National Laboratory, who
also provided the samples. Dr. Howard Birnbaum is also acknowledged for providing the excellent samples of
Niobium.
iii
TABLE OF CONTENTS
Chapter 1: Introduction ................................................................................................................... 11.1 X-ray Diffraction 11.2 Lattice Dynamics 21.3 Thermal Diffuse Scattering of X-rays 2References 3
Chapter 2: Background .................................................................................................................... 42.1 Theory of X-ray Diffraction 42.2 Theory of Lattice Dynamics 72.3 Thermal Diffuse Scattering of X-rays 92.4 Structural Phase Transitions and Phonons 12 2.4.1 Charge Density Waves and TiSe2 15 2.4.2 Antiferroelectric Ordering in SrTiO3 18References 21
Chapter 3: Experimental Methods ................................................................................................. 243.1 Synchrotron X-rays and Beamline Components 24 3.1.1 The Undulator 25 3.1.2 The Monochromator 263.2 The Diffractometer and Experimental Setup 273.3 X-ray Detectors 29References 31
Chapter 4: Determination of Phonon Dispersions ......................................................................... 324.1 Silicon 324.2 Niobium 37References 43
Chapter 5: Observation of Phonon Softening ................................................................................ 455.1 Titanium Diselenide 455.2 Strontium Titanate 50References 59
Chapter 6: Summary and Oulook ................................................................................................... 616.1 Summary 616.2 Outlook 61
X-ray Studies of Lattice Dynamics
Martin Victor Holt, Ph.D.Department of Physics
University of Illinois at Urbana-Champaign, 2002Tai-Chang Chiang, Advisor
Analysis of thermal diffuse scattering of x-rays (TDS) has recently shown great promise as a tool for phonon
studies, complementing more traditional neutron scattering techniques. Quasi-elastic x-ray scattering from
thermally populated phonon modes has been used to study lattice dynamics and phonon phase transitions in several
systems. Several experiments are reviewed - the viability and accuracy of this method was established by
establishing phonon dispersion curves (PDCs) in elemental systems including Si and Nb while the applicability to
more complex materials was demonstrated by studying phonon softening in charge density wave (CDW) TiSe2 and
antiferrodistortive (AFD) SrTiO3. This thesis work helps to develop analysis of x-ray thermal diffuse scattering as a
viable method for studies of lattice dynamics.
1
Chapter 1:Introduction
1.1 X-ray Diffraction
Since the discovery of x-rays by Röntgen in 1895, the ability to create and detect high energy photons has
had enormous impact on science. This light has also revolutionized a wide range of related fields from medicine to
airport security. These common uses of x-ray sources (Fig. 1.1) are based on the transmittance of x-ray light being
strongly dependent on the density of the illuminated material. A separate but related phenomenon is x-ray
diffraction. The principle of diffraction deals with the coherent reflection of x-rays by well-ordered solids.
Beginning with the first observation of diffraction (or Roentgenstrahlinterferenzen) in 1912 by Max von Laue, x-ray
probes have been widely used to examine structural properties of solids [1].
Diffracted x-rays used in solid state physics are fundamentally probes of long range order. Due to their
low interaction rate with matter as compared to visible light, significant scattering of x-rays is only achieved when
many identical units of matter are illuminated simultaneously. The intensity reflected is directly related to not only
the number of units illuminated, but how similar they are. The structure and arrangement of these units can then be
determined by the distribution of scattered intensity. One could picture throwing a handful of pebbles in a pond and
attempting to reconstruct the landing spot of each pebble by measuring the combined waves at the shoreline. One
gets a sense that the problem is quite complicated and that closed solutions depend on regularity of sources and
simplifying assumptions.
Figure 1.1. Common uses of X-rays.
2
1.2 Lattice Dynamics
Investigation of the solid state is in some sense defined by the concept of spatially localized atoms. Most
macroscopic properties of solids fundamentally depend on the identity and arrangement of their constituent atoms.
However, for any non-zero temperature these atoms are in motion and will be found away from their equilibrium
positions. An understanding of the thermal motion of atoms in crystals is therefore of central importance to many
areas of solid state physics.
The fundamental quantum of elastic lattice vibrations within a crystal is the phonon (after the photon for
EM waves). These phonons are both the simplest constituent vibrational waves within the crystal and the basic unit
of heat transfer. Phonons play a critical role in phenomena such as superconductivity and many types of phase
transitions, and are the basis for the acoustic, thermal, elastic, and infrared properties of solids [2]. However, not all
phonons are created equal - sending sound waves through a crystal in different directions will yield very different
results. This is because vibrational waves of different wave-vectors within a crystal encounter what appears to them
to be very different solid media, due to the strength and arrangement of the bonds between the atoms. For example,
if we - for the moment - represent phonons of a fixed wavelength by sound waves in air, depending on the direction
of their travel they seem to be going through either a trumpet or a tuba with very different results to the ear. This
dependence of the frequency of a wave on its wave vector is known as the dispersion relation. The dispersion
relation gives a fundamental description of harmonic phonons within a lattice and is determined traditionally
through neutron scattering [3] or, more recently, through inelastic x-ray scattering.
1.3 Thermal Diffuse Scattering of X-rays
Historically, analysis of x-ray thermal diffuse scattering (TDS) is among the oldest experimental methods
that deal with the dispersion of phonons in crystals. This method was first proposed by Laval [4] in 1938 but
remained impractical and was almost immediately abandoned with the advent of neutron scattering, in part due to
poor data collection rate and a difficult fitting process. Neutron scattering has remained the traditional vehicle for
phonon studies as it boasts both energy resolution and an energy-momentum relationship well matched to materials
studies, at the cost of a technically demanding experiment with frequently poor statistics. The situation has changed
3
recently due to advances in synchrotron radiation instrumentation and computational power. Undulator beams at
third-generation synchrotrons, such as the Advanced Photon Source, now yield a brightness about 8 orders of
magnitude higher than a conventional laboratory source. The combined improvements of photon flux and
computational accessibility of the fitting process make it possible to carry out such analyses with a high degree of
precision and efficiency.
This thesis is a study of lattice dynamics in various materials based on intensity patterns produced by X-ray
scattering from thermally populated phonons. It is in essence a collection of four recent results in crystals of varying
complexity all dealing with x-ray studies of phonon dispersion. Before the experimental topics, there will be a brief
mathematical introduction to x-ray diffraction, the subject of phonons, and the effect of atomic motion on x-ray
scattering. We will then discuss the determination of complete phonon dispersion curves in single crystal silicon
and niobium based on x-ray transmission scattering. The final subject is the observations of phonon softening near
the structural phase transitions of TiSe2 and SrTiO3.
References
1. X-rays 100 Years Later, Physics Today (special issue) 48, 1995.
2. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).
3. S. W. Lovesey and T. Springer, Eds., Dynamics of Solids and Liquids by Neutron Scattering (Springer-
Verlag, New York, 1977).
4. J. Laval, C. R. Acad Sci. Paris, 207 169 (1938).
4
A
B
C A:
B:
C:k
ε
Chapter 2:Background
2.1 Theory of X-ray Diffraction
To build a complete classical picture of x-rays interacting with matter we begin with the simplest building
block; x-ray scattering from a single electron. This result will then be used to formulate x-ray scattering from a
single atom, which will in turn be used to predict the scattered intensity from a regular array of atoms in a crystal.
We will see that scattered radiation is the square of the Fourier transform of electron density of the solid - this is the
essence of x-ray diffraction. The math to come is straightforward and often stated without proof. Similar treatments
of x-ray diffraction from perfect crystals can be found in many places, in particular the books by Warren [1] and
Als-Nielsen [2].
The classical picture of photon-electron scattering is shown in Fig. 2.1. Here an electromagnetic (EM)
wave is incident on a single electron. The incoming z-directed plane wave is represented by
0 ˆ( , ) E ikz i tz t e e ω−=E ε (1)
where E0 is the amplitude of oscillation, ε̂ is the direction of polarization, k defines the period in space and ω the
Figure 2.1 - An EM wave is incident on a single electron. The amplitude of sinusoidal motion appears different forobservers A, B, and C - this motivates the polarization factor of cos(φ) discussed in the text.
5
period in time. The resulting sinusoidal acceleration of the electron is given by
( ) ( 0, )( )e e
t e z ttm m
− ⋅ == =F Ea . (2)
This motion radiates light in nearly all directions. Taking into account the speed of this radiated light, an observer a
large distance R from the electron would be “seeing” the motion at a previous time
observedRt tc
= − . (3)
Consider also viewing this sinusoidal motion from different places - the frequency remains locked with the incident
wave, but the apparent amplitude of motion decreases as the viewer moves perpendicularly away from k̂ towards ε̂
(observers A, B, and C in Fig. 2.1). This polarization factor modifies the simple spherical wave and helps motivate
the following expression
( ) ( ) ( )2 ( )
0electron observed2 2
0 0
0 in
E ˆ, cos4 4
cos( )
Ri tc
e
ikR
e eR t t ec R c R m
erR
ωϕ
πε πε
ϕ
− −
−
−= = ⋅ ⋅
= − ⋅
E a
E
ε(4)
where r0 is the Thomson scattering length for a single electron defined by
Figure 2.2 – The path length difference of scattered radiation between two objects at separation r.
6
2
0 204 e
erm cπε
= . (5)
If instead the same light is incident on an entire atom, the outgoing (scattered) wave from the Z electrons
localized near the atom would be the coherent sum of each electron’s contribution. Therefore we must take into
account a path length difference shown explicitly in Fig. 2.2, and integrate over where each electron is likely to be
(introducing ( )nρ r - the charge density distribution for the nth electron orbital). This gives us
( ) ( ) ( )Z
( ) 3atom n electron
1, , i
nR t e R t dρ′− ⋅ −
==��
r k kE r E r . (6)
Now consider the same light incident on a perfect solid crystal - the outgoing radiation is a coherent sum of
intensities from a regular array of atoms.
( )crystal atom
all atoms
mim
m
e ′⋅ −= �r k kE E (7)
For clarity, we collect atoms of different types into a basis ( mb ) generated by each point of an underlying lattice
( lR ) as shown for the case of NaCl in Fig. 2.3.
Figure 2.3 – NaCl – an underlying face-centered cubic lattice with two basis atoms, creating a simple cubic lattice.
7
( ) ( ) ( ) ( ) crystal
lattice basis atoms basis atoms lattice
lattice
l m m l
l
i i im m
l m m l
i
l
e e e
F e
′ ′ ′+ ⋅ − ⋅ − ⋅ −
⋅
= =
=
� � � �
�
R b k k b k k R k k
R q
E E E(8)
From this, we see that F (the structure factor) contains all the information about the identity and
arrangement of atoms in the unit cell. The underlying lattice structure generates strong, sharp peaks of intensity at
certain wave-vectors
( ) 2l nπ′− ⋅ =k k R (9)
where lR is a lattice vector and n is any integer. This result is Bragg’s law for crystal diffraction - which
completely describes classical x-ray scattering from a perfect crystal with stationary atoms. At any non-zero
temperature however, the atoms in a solid are not stationary due to thermal vibrations. Even at zero temperature, the
quantal zero-point motion of the atoms must be considered. Therefore, the next step in quantifying the effects of
this on x-ray scattering is to reasonably describe atomic motion.
2.2 Theory of Lattice Dynamics
The traditional approach to atomic motion within a solid stems from work by Born and von-Karman which
describes lattice vibration in terms of inter-atomic force constants. The starting point is to consider the total energy
of the crystal as a function of the atomic positions 1 2E( , ,..., )Nr r r . For simplicity we will assume a monatomic
basis. Typically, atoms in a solid are found near their equilibrium positions, so we expand the crystal energy in
terms of displacement of each atom from equilibrium ( n n n= −u r R ).
2
1 2 1 2, ,
E 1 EE( , ,..., ) E( , ,..., ) ...2
i i j
N N i i ji i ji i j
∂ ∂= + ⋅ + ⋅ +∂ ∂ ∂� �
R R R
r r r R R R u u ur r r
(10)
As the derivatives are evaluated at equilibrium, the first non-zero correction to the equilibrium energy is the
second term. By neglecting all subsequent terms we restrict ourselves to the harmonic approximation. This is
related but not equivalent to the assumption of small oscillations, that is the displacement of atoms from their
equilibrium sites is small. It is important to note that even when the approximation of small displacements is valid,
8
many observable equilibrium properties of crystals cannot be predicted without the use of anharmonic terms in Eqn.
(10) [3]. A measure of the strength of these anharmonic terms is the width of the single-phonon scattering peaks in
neutron energy spectra. A comparison of phonon peak widths from available neutron data for the systems of Si [4],
Nb [5], TiSe2 [6] and SrTiO3 [7] show that they are all quite similar, and on the order of 1meV. This width suggests
that the harmonic approximation is essentially valid for the purposes of the experiments to follow [4]. Importantly
the linewidth of the R25 soft mode in SrTiO3 remains roughly constant to within 1.3 K of the transition temperature,
well within the critical region to be discussed later.
We identify from the second term in Eqn. (10) the inter-atomic force constants , , ,i jDα β
2
, , ,, ,
Ei j
i j equil
Dr rα βα β
∂≡∂ ∂
(11)
which is the force on ion i in cartesian direction α resulting from the displacement of ion j in direction β. Within the
harmonic approximation, these force constants specify all features of phonon dynamics. The N ions (of mass M) in
the lattice obey the following 3N equations of motion.
, -i i j jj
M = �u D u�� (12)
Assuming a plane wave solution to the atomic motion of ion i with polarization vector ε̂ ,
( ) ( )ˆ ii ti t e ω⋅ −= k Ru ε (13)
and defining the dynamical matrix D(k) as the fourier transform of the Di,j ,
( ) 0( ),0
iii
i
e− ⋅ −= ⋅�k R RD k D (14)
we are led to the following eigenvalue problem, solutions to which describe all harmonic phonon states.
2ˆ ˆ( ) Mω⋅ =D k ε ε (15)
Therefore, by defining suitable inter-atomic force constants , , ,i jDα β , usually limited to near neighbors, we
can calculate the eigenvalues and eigenvectors of Eqn. (15) yielding both the dispersion relation ( ),ω λk and the
polarization vector ( )ˆ ,λkε for a phonon of wave-vector k and mode λ.
9
2.3 Thermal Diffuse Scattering of X-rays
The following section will be a development of the effects on the diffraction of X-rays in crystals by the
atomic motion described in Sec. 2.2. The result will be a prediction for the first-order thermal diffuse scattering
intensity that may also be found in Warren’s book [1] and elsewhere. This expression is central to our data analysis.
We consider a slightly distorted crystal structure illuminated by monochromatic x-ray radiation. Each atom
re-emits this radiation similarly to the discussion Sec. 2.1, but the outgoing intensity pattern is made vastly more
complicated by interference from the small distortions of the lattice. So we will essentially calculate a time-
averaged diffraction pattern of a classical EM wave outgoing from a vibrating crystal. It is worthwhile to discuss at
the outset several underlying assumptions of this simple model. First, to a good approximation the scattered x-rays
have the same energy as the incident x-rays (elastic approximation). This is a consequence of the fact that typical
photon energies used in the following experiments (~10 keV) are six orders of magnitude greater than typical
phonon energy levels (~10-2 eV). The only other source of inelasticity is the electron recoil (Compton scattering),
which will be accounted for later as a smoothly varying background [1]. This allows us to treat the scattering semi-
classically, as in this picture the only relevant quantum aspect of lattice vibrations is that they obey Bose-Einstein
statistics. Secondly, we again employ the harmonic approximation – that all atomic motion can be reasonably
described in terms of phonon eigenstates developed in Sec. 2.2 (ref Sec. 2.2 for discussion). Thirdly, before the end
of this calculation we will neglect multiphonon processes (although these contributions may be calculated explicitly
via this method). The intensity arising from these terms – while it cannot be absolutely neglected – is a slowly
varying function of momentum transfer and will be represented by a smooth background in the sections to come.
The results of this section may also be obtained under the particle picture of light by developing a scattering
equation based on Fermi’s golden rule and then applying the above assumptions [3].
We first consider a beam of monochromatic X-ray radiation of wave-vector k incident on a perfect crystal
of stationary atoms. In Sec. 2.1 we showed the outgoing wave ψ collected at some wave-vector ′k is given by
( )
lattice
li
l
F e ′⋅ −⋅ �R k kψ = . (16)
10
To introduce atomic motion, we allow each atom n to be found away from its equilibrium position by nu .
For simplicity, we are again considering the case of a lattice with a single atom in the basis, so as before
i i i= −u r R . We can then write down the collected intensity at ′k (where ′= −q k k ),
( ) ( )* 2
,
i j i ji i
time avgi j
I F e e⋅ − ⋅ −= = �q R R q u uψψ (17)
and use the fact that the iu are small to approximate the time average (here F is the structure factor from Eqn. (8)).
( ) ( )( )212 2
,
i ji ji
i j
I F e e− ⋅ −⋅ −= �
q u uq R R(18)
Next, we can express the atomic motion as a superposition of plane waves (using the phonon eigenstates from Sec.
2.2 as a basis)
( ), ,, ,
,
ˆ ii ti a e λ λω δ
λ λλ
⋅ − +=�k kk R
k kk
u ε (19)
where ,a λk is the amplitude, ,ˆ λkε the polarization vector, ,λωk the frequency, and ,λδk the random phase
associated with a phonon of wave-vector k and mode λ - allowing for the fact that there are no phase relations
between thermal waves. Using this one can show the following
( )( ) ( ) ( ) ( )( )
( ) ( ) ( )( )
2 22 2 2, , , ,
,
22, ,
,
1 1 ˆ ˆ cos2 2
1 ˆ 2 cos2
m n m n
m n
a a
M a
λ λ λ λλ
λ λλ
� �⋅ − = ⋅ − ⋅ ⋅ −� �
� �= − ⋅ ⋅ −� �
�
�
k k k kk
k kk
q u u q q k R R
q q k R R
ε ε
ε(20)
where ( )M q is related to be the Debye – Waller factor, representing a zeroth-order measure of thermal phonon
activity along q . Placing this result into the predicted scattered intensity from Eqn. (18) we have
( ) ( ) ( ) ( )( )222 2, ,
, ,
1 ˆexp cos2
m niMm n
m nI F e e a λ λ
λ
⋅ −− � �� �= ⋅ ⋅ ⋅ −� �� �� �� �
q R Rqk k
kq k R Rε (21)
The amplitude of oscillation 2,a λk is again small, so we may expand the second exponential in a power series,
retaining the first order contribution of thermal motion to the scattered intensity
11
( ) ( )
( ) ( ) ( ) ( )( )
22
,
222 2, ,
, ,
0
1 ˆ cos2
. . . . . .
m n
m n
iM
m n
iMm n
m n
TDS
I F e e
F e e a
I I
λ λλ
⋅ −−
⋅ −−
=
� �+ ⋅ ⋅ −� �� �
+= + +
�
� �
q R Rq
q R Rqk k
k
q k R Rε (22)
Here the first term is exactly the result of Sec. 2.1 for crystalline reflections, but the intensity is reduced by the
Debye Waller factor of ( )2Me− q . Most of the missing intensity is found away from the diffraction condition in the
second term, which is the first-order thermal diffuse scattering. The subsequent terms explicitly represent the
contribution higher-order phonon processes. As we will see, the first-order TDS intensity is not just an isotropic
‘fuzzing out’ of the crystal diffraction but has a rich structure derived entirely from the phonon dispersion in the
crystal. To elucidate this, we perform the sums over m, n and k within the second term
( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )( )
22, , ,
, , , , ,
,,
,
1 ˆ cos( )2
m n m n m ni i im n
m n m ne a e eλ λ λ
λ λ
λλ
λλ
γ
γ
γ
⋅ − ⋅ − ⋅ −� �⋅ ⋅ − ∝ +� �� �
∝ δ ±
∝
� � �
�
�
q R R q+k R R q-k R Rk k k
k k
kk
g
q k R R
k q + B
ε
(23)
here ,λγk is shorthand for the terms depending on k and λ, B is the nearest reciprocal lattice vector to q , and g is
now the reduced wave vector (following = +q g B ). All this simplifies Eqn. (22) into the following.
( ) ( )222 2, ,ˆM
TDSI F e a λ λλ
−∝ ⋅�q
g gq ε (24)
Next, by making use of Bose-Einstein statistics to calculate the average thermal energy
,
,2 2, , , , ,
1 1 1 coth2 2
1BBk T
E m ak T
eλ
λλ λ λ λ λω
ωω ω ω
� �� � = = + = � � −� �
g
gg g g g g�
�� � (25)
we arrive at the first order contribution of thermal lattice vibrations to scattered x-ray intensity.
( ) ( ) ( )22 ,2,
,
1 ˆ cothMTDS
B
I F ek T
λλ
λ λ
ωω
− � �∝ ⋅ � �
� ��
q gg
g
q q�
ε (26)
12
This form emphasizes that TDS intensity may be completely described by predicting the phonon dispersion ( ,λωg )
and polarization ( ,ˆ λgε ) over the first Brillouin zone. The scattering in general then is a repeating pattern simply
modified by the factors that vary slowly with q . This expression can be generalized to the case of more than one
atom per unit cell [8].
( ) ( )2
3, ,
1 1,
ˆ1 coth n n
s sM i
TDS nnB n
I F e ek T m
λ λ
λ λ
ωω
− − ⋅
= =
⋅� �∝ � �
� �� �
g q g q τ
g
� ε(27)
This takes into account s atoms per unit cell, where nF , ( )nMe− q , and nτ are the atomic scattering factor, the
Debye-Waller factor, and the position vector for an atom n in the basis. This equation is the basis of our analysis
of x-ray intensity patterns from TDS.
2.4 Structural Phase Transitions and Phonons
Structural phase transitions in solids pose a vast array of complex and often subtle physical problems.
Even the simplest of examples must deal with a coupling of a lattice to the spin, charge, thermal or other exotic
degrees of freedom of the constituent elements. It is a topic well deserving of the available books and reviews that
the interested reader is directed to [9-11]. This author is fortunate in that the effect studied – phonon softening
leading up to a periodic lattice distortion - is largely independent of the actual microscopic mechanisms responsible
for the transition and can be well described within a general phenomenological framework. With this in mind, it is
appropriate to introduce phase transitions and phonon softening in the context of a phenomenological Landau
theory, and then briefly survey the theory specific to the two systems studied – TiSe2 and SrTiO3.
As an eminent scientist recently reminded this author, it is worthwhile to pose the question “why do phase
transitions – any phase transitions – happen?” [12]. A system in equilibrium acts to both minimize its total energy
and maximize its entropy (a measure of the total number of states available to the system). This balance, written as
F U TS= − (28)
(here F is the free energy, U the internal energy, T the temperature and S the entropy) when minimized defines the
equilibrium state of the system. For example, at room temperature it would still be energetically favorable for water
to appear in solid form, but it has vastly more states available to it in its observed liquid state (and as the number of
13
accessible states diminish on cooling, this balance shifts to the solid state). Similarly, structural phase transitions in
solids are motivated by the crystal lattice interacting with some other degrees of freedom (conduction charge density
in the case of TiSe2 and atomic dipole moments in the case of SrTiO3) to produce a distorted ground state for the
system which is then ‘discovered’ at lower temperatures.
For a simple example, we consider a one-dimensional chain of atoms with some coupling to other degrees
of freedom. Following Fig. 2.4, this model lattice by some mechanism undergoes a periodic lattice distortion at
temperature Tc to a state with a doubled real-space periodicity, one in which every other atom has moved. In
general these motions are very small in absolute terms, yet any non-zero displacement breaks the symmetry of the
original lattice and is readily identified for this. The changing equilibrium position of this atom in the new unit cell
is a natural defining quantity of the progress of this phase transition, and we will call it the order parameter, η - the
displacement from equilibrium. It is an assumption of Landau theory that the change in free energy of the crystal
near the transition temperature is dominated by this order parameter and so may be written as an integer power
series
nn
nG A η=� (29)
where each coefficient Ai may be a smooth function of temperature. The minimization of G at each temperature
determines the value of the order parameter.
0G
ηη∂ =∂
(30)
We expect this minimizing value to be zero above the transition temperature and move away from zero to approach
the true ground state value below Tc. Also, we expect ( ) ( )G Gη η= − , and η itself to be small. All these
assumptions may be applied to the power series, yielding
Figure 2.4. A periodic lattice distortion in one dimension.
14
2 40 2 4
0 {for i odd}iA
G A A Aη η=
≅ + +(31)
where A0 represents all things unchanged by the phase transition. The behavior of A2 and A4 as a function of
temperature near Tc must satisfy the above requirements, giving to leading order [11]
2
4
( )0
cA T TA
∝ −>
. (32)
Remarkably, this is an important result for phonons above the transition temperature. To elucidate this
relationship we can view the periodic lattice distortion of Fig. 2.4 as a “snapshot” of a sound wave within the crystal.
That is, any dynamic displacement within the lattice can be expressed as a superposition of phonon modes. A static
displacement is distinguished from this when the relevant normal coordinate becomes time independent and non-
zero (the relevant “spring” is permanently “bent”). So by expressing the free energy in terms of normal coordinates
(as a function of all possible lattice displacements) we have [9,13]
( ) 2 22 2,
,
2 2
1 1, ... ...2 2
...
G Q Qλ η ηλ
η
ω λ ω
ω η
= + = +
∝ ⋅ +
q(33)
which makes use of the fact that the appropriate normal coordinate, as the time-averaged amplitude of the distorting
phonon, is proportional to the order parameter (the static displacement from equilibrium). By direct comparison to
Eqn. 31 we see that the frequency of this phonon has a specific leading order dependence on temperature.
( ) ( )2cT T Tηω ∝ − (34)
From this, we see that the frequency of the participating phonon mode changes from its room temperature value
(T>Tc) to zero at Tc. This is known as phonon softening and is evidence of a progressive lattice instability leading
up to the periodic distortion. This is made clear by considering the generalized lattice restoring force of the
participating phonon mode, viewed as a simple oscillator.
2F Qη η ηω= − (35)
Thus the generalized lattice restoring force must go to zero at Tc to allow for a static distortion, then below Tc it
becomes slowly positive increasing the magnitude of this distortion. As well the participating phonon mode
becomes fully damped at Tc and below this the harmonic frequencies are then imaginary. This means the crystal is
15
stabilized by anharmonic terms in the self-energy, which we have not considered – they determine the frozen
amplitude of the static mode below Tc (but not its temperature dependence, which is given by Landau theory). So,
considering all results of this section, we can see that the dispersion of the participating phonon mode is intimately
related to several key features of the phase transition.
It is important to note that the above predictions rest on certain mean-field assumptions, and so are
expected to hold true only where the system to be studied is sufficiently mean-field in character. In particular the
expansion of the free energy (Eqn. 31) depends only on the average value of the order parameter – fluctuations of
the system away from this value are neglected. However, for phase transitions in any three-dimensional system
fluctuations become increasingly important near the transition temperature and this mean-field assumption breaks
down. The near-transition region of temperature in which this theory is not valid is called the critical region, the
width of which is estimated by the Ginzburg criterion [11, 14]
( )
42
04
d
c Bd
c
T T kT C ξ
−
� �− =� �⋅∆ ⋅� �
(36)
where d is the dimensionality of the phase transition (in this case d=3), C∆ is the discontinuity in heat capacity at
the transition, and 0ξ is the correlation length far from the transition (essentially an estimate of the range of
interaction of the microscopic driving forces). From this we see that Landau theory works well for higher
dimensional systems, and for systems where the range of interaction ( 0ξ ) is large compared to the length scale of
the fluctuations. The size of this critical region - where deviations from the predictions of Landau theory are
expected - will be estimated in later sections for both systems to be studied.
In summary, by developing the harmonic phonon picture within a reasonable set of phenomenological
assumptions we have arrived at a basic, yet illuminating, picture of second order displacive phase transitions within
a lattice. This helps motivate the following experiments in which a key feature – phonon softening of the
participating mode as predicted by Eqn. (34) - will be observed over a wide temperature range via x-ray thermal
diffuse scattering (Sec. 5).
2.4.1 Charge Density Waves and TiSe2
16
A charge density wave (CDW) system is found in metals where the conduction electron density and lattice
structure are coupled so as to produce a periodic distortion in both at lower temperatures. This is perhaps best
thought of by considering a one-dimensional lattice weakly coupled to an electron gas of one electron per atom
displayed by Fig. 2.5. Here we see that if atomic motion is allowed, the electron density of the normal state is no
longer the ground state of the system – a charge density wave and accompanying periodic lattice distortion develop
at lower temperatures. This is driven by the conduction electrons – a weak periodic potential due to the lattice
distortion opens up a gap at the Fermi level lowering the energy of all occupied electronic states – offsetting the
gain in energy from the static displacement of the lattice. In this 1-D case we see the Fermi surface is perfectly
nested, that is one vector can span all points on the Fermi surface (both of them). This mechanism of CDW
formation in a normal metal is known as a Peierls transition.
Charge density waves have been studied extensively for decades, and the most widely investigated systems
are the numerous layered transition metal dichalcogenides (TMDC) [10,15,16]. They provide a rich testing ground
for basic concepts in electron-phonon coupling, electron correlation, and structural phase transitions, which impact
our quest for understanding and design of modern complex functional materials. While much is known about these
transitions, some important effects remain unexplored due to the lack of appropriate tools. TiSe2 is a layered semi-
metal TMDC with a unit cell shown in Fig. 2.6. The crystal structure consists of alternating sheets of Se-Ti-Se
Figure 2.5. A charge density wave in a one-dimensional lattice.
17
weakly held together with Van der Waals forces. This structure’s quasi – 2D nature plays a role in many aspects of
this system. Upon cooling to ~200 K a CDW develops, and the accompanying lattice distortion exactly doubles the
size of the unit cell (the new lattice of 2x2x2 periodicity is known as a commensurate superlattice) at low
temperatures [15]. This displacement is the static “snapshot” of the lower transverse phonon mode L1− at the
Brillouin zone boundary, which presumably softens to this state as temperature decreases [17-20]. In section 5 we
will report on a detailed x-ray scattering study of this system and show that the frequency of the L1− phonon mode
indeed decreases gradually to zero at the transition temperature. This work thus confirms the soft mode concept,
which is central to the theory of the charge density wave phenomenon in this type of material.
The simplest model for charge density wave formation involves Fermi surface nesting resulting in an
enhanced electron-phonon coupling that drives a lattice distortion, as mentioned above (also [2]). TiSe2, however,
does not fit into this conventional model because its Fermi surface does not nest. A number of other mechanisms
Figure 2.6. Real and reciprocal space structure of TiSe2. The charge density wave-vector from Γ to L is shown.The accompanying periodic lattice distortion doubles the size of the real space unit cell.
18
have been proposed and discussed in the literature, including exciton formation due to electron-hole coupling across
a narrow gap [21-23], an antiferroelectric transition suppressed by carriers [24], and a band Jahn-Teller effect [10,
25]. To this date, this subject remains highly controversial, mostly because of a disagreement on the electronic
structure. The most detailed microscopic theory in existence for this system suggests a band Jahn-Teller effect as the
likely explanation [10]. A key feature of this theory is that the L1− phonon frequency, renormalized by a
temperature-dependent electronic susceptibility, softens towards the transition temperature.
It is appropriate now to estimate the size of the critical region for TiSe2. We will first note that no
deviations from mean-field predictions have been observed previously [18], and the transition is widely held to be
essentially mean-field in character [10]. Unfortunately the experimental data necessary to directly evaluate the
Ginzburg relation (Eqn. 35) is not available. To partially remedy this, we will calculate the Ginzburg interval (GI)
for the closely related CDW compound TaSe2. As this system has some experimentally observable non-mean field
behavior near the normal-incommensurate CDW transition at Tc = 122 K [26], we can use the predicted width of the
critical region in TaSe2 as an upper bound for possible critical behavior in TiSe2. Following the methods of Ref.
[27], Eqn. (35), and using the values of 0ξ = 10 Å and C∆ = 4 J/mol K [28], we calculate a critical region of width
TG = Tc ± 1.05 K. In the present case of TiSe2 we have taken experimental data no closer than Tc ± 2 K to the
transition temperature, so we conclude that the temperature dependence of this observed softening should be
essentially mean-field in character.
2.4.2 Antiferroelectric Ordering in SrTiO3
Ferroelectric (FE) ordering within a crystal is an alignment of atomic electric dipole moments producing a
net dipole in exactly the same sense that a ferromagnet orders local spins. Similarly, an antiferroelectric (AF)
ordering is a long-range alternating anti-alignment along some axis producing no net dipole moment. Importantly, it
is distinct from both the disordered case as well as the ordered FE case. In the case of SrTiO3 the assignment of AF
ordering to the low temperature state is essentially a formal one – that is, any symmetry breaking periodic lattice
distortion that does not affect the original charge distribution of the lattice will produce an effective dipole ordering
of the same symmetry as the distortion. This classification does not really shed light on possible underlying
microscopic mechanisms; so we will skip a detailed discussion and move on to describe the system.
19
Periodic Lattice Distortion
Figure 2.7. The real and reciprocal space structure of SrTiO3. The wave-vector of the structural phase transition andthe doubling of the real space unit cell are shown. The order parameter corresponds to the angle of rotation of theoxygen octahedra, viewed from along the c-axis
20
Strontium titanate is a metal oxide insulator popular as a growth substrate for complex crystals. It has a cubic unit
cell shown in Fig. 2.7 which – much like TiSe2 - is unstable against a lattice distortion corresponding to a zone
boundary phonon mode [29]. This lattice undergoes a second-order displacive phase transition upon cooling below
~105K, creating a commensurate (2x2x2) unit cell [9,29,30]. This involves an alternating rotation of the oxygen
octahedra as shown in Fig. 2.7, and the condensation of the lowest lying phonon mode at the R point in reciprocal
space. The proposed microscopic mechanism is phonon interaction with the electronic susceptibility, in particular
the lattice interaction with the oxygen dipole moments. The assignment of AF ordering is based on the symmetry of
the distorted structure. A subtlety of this transition is that SrTiO3 is an incipient ferroelectric material; it is also
unstable against a zone center optical phonon mode that softens monotonically from its room temperature value to
plateau near absolute zero, never reaching the phase transition. It is believed that the weak AF ordering of the 110K
structural transition is sufficient to frustrate the FE transition that would occur at finite temperature otherwise (and
does in similar metallic oxides) [31].
This phase transition has been well studied by both theory and experiment, as it is for the most part a
prototypical second order displacive phase transition fitting well into Landau theory. However, within a critical
region near the transition temperature the behavior of the system deviates significantly from mean-field predictions.
This region has been estimated to be Tc = 105K ± 10K [32], agreeing well with experimental observations [33].
The correct prediction of this critical behavior was an early success of renormalization group (RG) theory [34,35].
The RG approach is generally based on two assumptions – (i) that the critical behavior of a system is characterized
by pure power laws, and (ii) these power laws are universal, that is they depend only on certain key features of the
phase transition (such as symmetry and dimensionality) and not on microscopic details. A thorough treatment of the
renormalization group is well beyond the scope of this thesis, and the interested reader is directed to Ref. [11].
Much work has been done in developing sufficiently advanced RG techniques and phenomenological mean field
theories to predict strain, critical exponents, order parameter fluctuations, and mode frequencies of SrTiO3 [9,30,36].
To see how the critical region affects phonon softening we return to our harmonic view of the lattice
restoring force (Eqn. 35). This linear restoring force is in balance with the polarization of the crystal driven by a
temperature dependent electronic susceptibility (χ).
2 1 1
2 1
F Q E P Qη η η η η η
η
ω χ χ
ω χ
− −
−
= − ∝ ∝ ∝
∴ ∝(37)
21
where Eη is the electric field due to the polarization Pη - which in this case is the degree of antiferroelectric
ordering, our order parameter. Importantly this equation is a harmonic statement and not a mean field statement so
remains true within the critical region, until quite near the transition (following the arguments in Sec. 2.2).
Therefore we expect the behavior of the phonon softening in this region to be
2 1 - c
c
T TT
γ
ηω χ− � �∝ ∝ � �
� �(38)
where γ is a critical exponent. Well away from the critical region this exponent follows the predictions of mean-
field theory (γ = 1.0, recovering Eqn. 34), whereas within the critical region the exponent crosses over to γ = 1.375,
a value predicted by renormalization group theory for the universality class of a Heisenberg antiferromagnet [34].
The universality class of the 105K phase transition in SrTiO3 has been the subject of some debate, being
variously attributed to an XY model [37], and more recently to a 3D Heisenberg model [32,34]. The fundamental
issue at stake is how to represent the fluctuations of the R25 phonon mode. This triply-degenerate mode represents
independent rotations of the oxygen octahedra about each cubic axis, and so in principle fluctuations may occur in a
3D continuum of states, supporting the Heisenberg viewpoint. However, as below the transition temperature a
single axis is selected (in each domain) such that counter-rotations of octahedra about that axis define the distorted
unit cell, it seems that the relevant fluctuations should be confined to the other two possible axes, supporting the XY
viewpoint. It rests with sufficiently precise experimental measurements of relevant static critical exponents to
resolve this debate.
So we see that the softening of a zone boundary phonon mode plays a central role in much of the theory
surrounding the phase transition, a verification of which helps to motivate the TDS experiment that will be discussed
in Sec. 5.
References1. B. E. Warren, X-Ray Diffraction (Dover, New York, 1969).
2. J. Als-Nielsen and D. McMorrow, Elements of Modern X-Ray Physics (Wiley, New York, 2001).
3. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).
4. G. Dolling, Inelastic Scattering of Neutrons in Solids and Liquids (IAEA Vienna, 1963).
22
5. F. Guthoff, B. Hennion, C. Herzig, W. Petry, H. R. Schober, and J. Trampenau, J. Phys. Condens. Matter 6,
6211 (1994).
6. D. E. Moncton, F. J. DiSalvo, and J. D. Axe, in Proceedings of International Conference on Lattice
Dynamics, edited by M. Balkanski (Flammarion, Paris, 1978), p. 561.
7. S. M. Shapiro, J. D. Axe, G. Shirane, and T. Riste, Phys. Rev. B 6, 4332 (1972).
8. M.Y. Chou and M. Choi, Phys. Rev. Lett. 83, 3317 (1999).
9. A.D. Bruce and R.A. Cowley, Structural Phase Transitions (Taylor & Francis, London, 1981).
10. Structural Phase Transitions in Layered Transition Metal Compounds, edited by K. Motizuki (D. Reidel,
Boston, 1986).
11. N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Perseus, Reading MA,
1992).
12. P. M. Goldbart, Preliminary examination of this author.
13. R. A. Cowley, Phys. Rev. 134, A981 (1964).
14. V. L. Ginzburg, Sov. Phys. Solid State 2, 1824 (1960).
15. Charge Density Waves in Solids, edited by L. P. Gor’kov and G. Grüner (North Holland, New York, 1989).
16. J. A. Wilson and A. Yoffe, Adv. Phys 18, 193 (1969).
17. W. G. Stirling, B. Dorner, J. D. N. Cheeke, and J. Revelli, Solid State Comm. 18, 931 (1976).
18. F. J. DiSalvo, D.E. Moncton, and J. V. Waszczak, Phys. Rev. B 14, 4321 (1976).
19. D. E. Moncton, F. J. DiSalvo, and J. D. Axe, in Proceedings of International Conference on Lattice
Dynamics, edited by M. Balkanski (Flammarion, Paris, 1978), p. 561.
20. N. Wakabayashi, H. G. Smith, K. C. Woo, and F. C. Brown, Solid State Comm. 28, 923 (1978).
21. J. A. Wilson, Phys. Stat. Sol. (b) 86. 11 (1978).
22. W. Kohn, Phys. Rev. Lett. 19, 439 (1967).
23. Th. Pillo, J. Hayoz, H. Berger, F. Lévy, L. Schlapbach, and P. Aebi, Phys. Rev. B 61, 16213 (2000).
24. R. M. White and G. Lucovsky, IL Nuovo Cimento, 38 B, 280 (1977).
25. H. P. Hughes, J. Phys. C 10, L319 (1977).
26. D. E. Moncton, J. D. Axe, and F. J. DiSalvo, Phys. Rev. B 15, 801 (1977).
27. J. A. Aronovitz, P. M. Goldbart, and G. Mozurkewich, Phys. Rev. Lett. 64, 2799 (1990).
23
28. W. L. McMillan, Phys. Rev. B 16, 643 (1977).
29. P. A. Fleury, J. F. Scott, and J. M. Worlock, Phys. Rev. Lett. 21, 16 (1968).
30. Structural Phase Transitions, edited by K.A. Muller and H. Thomas (Springer-Verlag, Berlin, 1981).
31. O. E. Kvyatkovskii, Physics of the Solid State 43, 1401 (2001).
32. P. Sollich, V. Heine, and M. T. Dove, J. Phys.: Condens. Matter 6, 3171 (1994).
33. K. A. Muller and W. Berlinger, Phys. Rev. Lett. 26, 13 (1971).
34. R. A Cowley and A. D. Bruce, Journ. Phys. C 6, L191 (1973).
35. P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977).
36. M. A. Carpenter, A. I. Becerro, and F. Seifert, American Mineralogist 86, 348 (2001).
37. H. E. Stanley, Structural Phase Transitions and Soft Modes pp 271-89 (Oslo: Universitetsforlaget, ed G. J.
Samuelson et al, 1971)
24
Chapter 3:Experimental Methods
3.1 Synchrotron X-rays and Beamline Components
The production of x-rays has been a field of exponential growth since the cathode-ray tube of Röntgen in
1895 [1]. Improvements in the brilliance, focus, and energy selection of x-ray sources have made entire new fields
of experimental observation possible. A defining moment for modern x-ray science came when it was discovered
that relativistic electrons stored in storage rings for high-energy physics experiments could be used to generate x-
rays. From then on, source brilliance was limited mainly by the flux of stored electrons, eventually resulting in
third-generation synchrotron sources such as the Advanced Photon Source. The interested reader may find several
reviews of the history of x-rays [2,3]. This section will focus on a basic description of the synchrotron x-ray source,
insertion device, and monochromator used in these experiments.
All experiments to be discussed were performed at the undulator beamline of Sector 33, operated by the
University, National Laboratory, and Industry Collaborative Access Team (UNICAT), at the Advanced Photon
Source, Argonne National Laboratory. The Advanced Photon Source is a high-flux electron storage ring dedicated
Figure 3.1. The Advanced Photon Source at Argonne National Laboratory
25
to the production of synchrotron x-rays. The electrons travel roughly in a circle (clockwise from top) with the beam
motion and focus provided by externally generated magnetic fields. The actual electron beam control and motion
will be left at the “aerial photograph” level of description, but its path is generally made up of several long straight
sections alternating with bending sections. Interior to the straight section at the 33-ID beamline (UNICAT) the
electrons encounter an undulator – an insertion device consisting of powerful, closely alternating magnetic poles.
This causes rapid sinusoidal acceleration of the electrons resulting in an intense, focused beam of x-rays traveling
tangentially outward from the storage ring. This undulator light is then monochromated – it encounters a component
that selects x-rays of a certain wavelength - and is then guided by mirrors to the target.
3.1.1 The Undulator
Even a casual description of the workings of an undulator will involve some relativity, so we will begin by
considering an electron moving in a circle. Viewed edge on (Fig. 3.2), it appears that the electron spends most of its
time at the endpoints of the circle, where the velocity perpendicular to the observer is the smallest. If the electron is
moving near the speed of light, this effect is even more pronounced – one may liken it to a cartoon character moving
feverishly between two equally important activities. Nearly all radiation is given off at these points, where the
observed acceleration is tremendous. Using this, an undulator can capture essentially all forward emitted radiation
of this hypothetical circularly moving electron by instead creating a series of small amplitude motions in a forward
moving electron (that is, taking the top and bottom parts of the circle).
An equally important concept to the undulator design is the selection of a fundamental wavelength.
Consider a single electron traveling through the undulator (Fig. 3.3) – at each magnet it emits a flash of
Figure 3.2. The relativistic doppler effect. A relativistic electron in circular motionviewed edge on appears to spend most of its time at its endpoints.
26
polychromatic radiation. Depending on the time the electron travels between each flash, one particular wavelength
of light (and integer multiples of it) will add up coherently with the previous wave-fronts and be correspondingly
more intense than all other wavelengths. Looking at Fig. 3.3, we see that the time the electron takes to reach the
next node depends on the path traveled and hence on the external magnetic field. Therefore, by changing the
magnetic field strength (usually by adjusting the gap between the sets of magnets), one can emphasize a certain
wavelength of light.
3.1.2 The Monochromator
Figure 3.3. The alternating poles of an undulator create a series of small amplitude oscillations in a forward movingelectron, which emits x-rays at each pole. A certain fundamental wavelength is selected by the path length Ltraveled – this is determined by the gap d.
Figure 3.4. A double-crystal monochromator. Polychromatic x-ray light is incident on the first set of lattice planes–essentially only one wavelength of this light fulfills the two successive Bragg conditions necessary to be wellreflected from the second crystal.
27
For many experiments, however, one needs to know the wavelength of light with far greater accuracy. At
the 33-ID beamline this is done with a double-crystal Si[111] monochromator. This device depends on two
successive applications of Bragg’s law for crystal diffraction (Sec. 2.1).
2 sin( )d nλθ = (1)
Here -following Fig. 3.4 - by choosing a certain lattice spacing of crystal presented to the beam, different
wavelengths of incident light are reflected at separate outgoing angles. This spatially separated polychromatic beam
is presented to a second crystal – there essentially only one wavelength is at the correct incident angle to fulfill its
Bragg condition. So, as only light of one wavelength is well reflected, outgoing from the second crystal one
achieves a virtually monochromatic beam.
3.2 The Diffractometer and Experimental Setup
Collecting diffracted x-ray radiation from a crystal requires that all degrees of freedom be controlled with a
high degree of accuracy and reproducibility. To this end, we use a layered set of circular goniometers controlling
both sample and detector degrees of freedom known as a diffractometer. The Newport Kappa diffractometer used in
this experiment is shown in Fig. 3.7. The six circles of motion overdescribe the possible diffraction conditions and
Figure 3.5. Eulerian four-circle scattering geometry. Three sample degrees of freedom and one detector degree offreedom are shown.
28
so one can access all possible reciprocal points using a conventional reflection geometry and a limited set of motor
motions. This results in a virtual Eulerian four-circle geometry described in Fig. 3.5 that has one detector and three
sample degrees of freedom. One collects several reflection points to align the sample. This uniquely determines a
linear solution that interprets the Euler angles of the sample in terms of motor positions.
Two distinct experimental geometries were used (Fig. 3.6). The first case is a standard reflection geometry
in which the momentum transfer is a point-like probe of reciprocal space determined by the sample and detector
Figure 3.6. Diagram of the two types of experimental setups used – a standard reflectiongeometry and a transmission Laue geometry.
29
degrees of freedom. The second case is a transmission geometry, in which the area detector positioned behind the
sample collects scattered radiation. The momentum transfer is a planar projection of the detector area onto the
Ewald sphere, which is a simultaneous collection a spherical surface in reciprocal space.
3.3 X-ray Detectors
This section will have a brief description of the three types of detectors used in this experiment, the
scintillation detector, the charge-coupled-device (CCD), and the image plate (IP). Each is based on distinct
fundamental principles and fulfill complementary roles in the measurement of scattered x-ray intensity. Further
Figure 3.7. Newport Kappa Diffractometer
Figure 3.8. Sketch of a scintillation detector.
30
details may be found in the book by Saleh and Teich and elsewhere[4].
A scintillation detector is based on the interaction of x-rays with crystals containing high Z elements. A
scintillation event occurs when an x-ray is absorbed by a high Z atom, which then ejects a core electron from its
shell. The electronic transitions necessary to fill this hole from outer atomic shells emit photons in the visible
spectrum. This flash of light indicates a single x-ray has been detected.
Following the sketch shown in Fig. 3.8, the light from a scintillation event in the front end crystal (in our
case NaI) is caught and amplified by a photomultiplier tube. One pulse of visible light from the scintillator creates a
current in a photosensitive material at the top of the photomultiplier tube. These electrons are cascaded down a
chain of amplifiers each biased in parallel to a chain of resistors. The output is a current pulse, which is then
counted externally. There are a few important things to note about this detector: (i) the wavelength of x-rays
generally determines the current of the output signal (as higher energy x-rays may eject more tightly bound
electrons, creating a larger cascade and hence more signal per photon) – this can be used to separate an elastic x-ray
signal from inelastic background, (ii) two x-rays scintillating simultaneously will be erroneously counted as a
single pulse (that is, the multiplication process takes an appreciable amount of time - known as the rise time of the
detector - and so two x-rays striking within this time will increase the current of the pulse, but be counted as a single
event), and (iii) a high incident photon flux lowers the overall voltage scale (this is known as detector “sag”, as the
reference biases of the dynodes are overcome by an appreciable current through the photomultiplier). All of these
features indicate this detector is best used in low flux rate situation where distinguishing single photon events is
important.
Figure 3.9. Schematic of a charge-coupled-device (CCD) camera
31
The CCD is a two-dimensional detector based on the internal photoeffect in which incident photons
generate electron-hole pairs within a photodiode. Following the schematic in Fig. 3.9, an x-ray encounters a
phosphor coating at the front end of the CCD camera. This produces optical light, which is coupled by a fiberoptic
taper onto individual pixels on a CCD chip. The optical photon hits a Schottky-barrier photodiode, exciting an
electron up to the conduction band and a hole below the Fermi level. The electron is of sufficient energy to
overcome the barrier and enter the Si substrate, leaving a residual positive charge on the diode. The accumulated
charge on each pixel is read out by raster scanning the array onto a single read-out collector on one corner. Two
things to note about this detector: (i) overexposure results in accumulated charge breaking the barriers between
pixels (“charge drip”, which may create horizontal and vertical bars due to insufficient de-charging of intense points
during read-out), and (ii) the detector is limited by convoluting both the quantum efficiency of the phosphor screen
AND the efficiency of the photodiode.
An image plate detector is essentially a reusable x-ray sensitive film. The IP itself is a photo-stimuable
phosphor powder (in this case BaFBr0.85I0.15:Eu2+) deposited on a flexible plastic substrate. The incident x-rays
create a latent phosphorescent image within the powder. It is the logarithmic response of the fluorescent excitation
process to x-ray flux that dictates fundamentally that data is collected on a logarithmic scale. This image is
recovered by scanning a laser across the surface of the plate. The induced photoluminescence at each laser spot is
recorded by an optical detector, making use of the wavelength difference of the emitted light to filter out reflected
laser light. The resolution is given by the grain size and is at least equivalent to a conventional CCD camera, with a
much higher usable detection area. That is, weak features are recorded with excellent statistics, and intense
features are less likely to saturate the detector, ensuring a wide dynamic range. However, the read out time of this
detector is much longer than an optimized CCD set up, and if placed manually the position of the image plate is not
reproducible.
References
1. W. C. Röntgen, Sitzungber. Phys.-Med. Gesellschaft Würzburg, 132 (1895).
2. P. P. Ewald, Fifty Years of X-ray Diffraction (International Union of Crystallographers, Utrecht, 1962).
3. J. Als-Nielsen and D. McMorrow, Elements of Modern X-Ray Physics (Wiley, New York, 2001).
4. B. E. A. Saleh and M. C. Teich, Photonics (Wiley, New York, 1991).
32
Chapter 4:Determination of Phonon Dispersions
4.1 Silicon
Central to this thesis is the claim that x-ray thermal diffuse scattering measurements using synchrotron
radiation has sufficient accuracy for realistic studies of lattice dynamics in solids. To support this we will determine
the force constants and phonon dispersions for crystalline silicon via a least-squares fit to the measured transmission
scattering intensity.
The experimental details are as given in Sec. 3. A transmission Laue geometry was employed, in which a
28 keV beam was sent at normal incidence through commercial Si wafers with a thickness of 0.5 mm. An image
plate positioned behind the sample was used to record the images with an exposure time of ~10 s each. The incident
beam was polarized in the horizontal direction. Data were taken with the sample in air, in a helium atmosphere, or in
vacuum with similar results. The data shown below were taken with the sample in air.
Figures 1(a) and 1(b) are experimental pictures shown on a logarithmic scale of Si(111) and Si(100),
respectively. Figures 1(c) and 1(d) are the corresponding model calculations to be discussed below. The differences
between experiment and calculation are too small to be visible to the eye, except for the shadows of a circular beam
stop at the center of each experimental picture and a supporting post. At the center of the shadow of the beam stop is
a tiny bright spot caused by the direct beam passing through the 2-mm molybdenum in the beam stop. It is a
measure of the beam size and the picture resolution. By virtue of the wavelength selected, the Bragg condition is
never satisfied over the entire area of detection. Thus, none of the bright spots are caused by crystal diffraction. One
can readily see the symmetry of the pattern. The picture is three-fold symmetric for the (111) sample, and four-fold
for the (100) sample.
Each point in the picture corresponds to a planar projection of a unique momentum transfer q on the Ewald
sphere. Because each unit cell of Si contains two atoms, there can be up to six distinct phonon modes (three optical
modes and three acoustic) at each q contributing to the scattering intensity. The bright spots in each picture are
points on the Ewald sphere that are closest to neighboring reciprocal lattice points, where the thermal acoustic
phonon population is high. The center of the picture is dark despite the high acoustic phonon population because of
33
Figure 4.1. Thermal diffuse scattering (TDS) of x-rays from Si. Data is displayed on alogarithmic scale. Panels (a) and (b) are from Si(111) and Si(100) respectively. Panels (c) and (d)are the corresponding theoretical images from a simultaneous pixel-by-pixel fit to the data.
(a) (c)
(b) (d)
34
a q2 dependence in the cross section. The diffuse lines connecting the bright spots are associated with high
symmetry directions where the phonon is soft and has a high density of states. The rich structure of the picture is a
direct consequence of the use of a logarithmic intensity scale. This is analogous to the use of a logarithmic scale for
x-ray truncation rod analysis, making it possible to detect the contribution from one surface atomic layer within a
spectrum dominated by bulk contributions [1]. In the present case, the logarithmic function largely compensates for
the thermal population factor, and thus, the phonon modes at different energies contribute to the intensity in an
approximately linear manner. Without this logarithmic conversion, the acoustic phonon contribution near reciprocal
lattice points would be the only features seen.
The theoretical pictures and phonon dispersions are derived from a force-constant formalism of the lattice
dynamics, also known as the Born-von Karman model (Sec. 2.2, also [2-4]). Force constants up to the sixth nearest
neighbors are included in a harmonic lattice model, and diagonalizing the dynamic matrix results in the phonon
eigenmodes. The intensity of scattering by an unpolarized incident x-ray at a given momentum transfer q is given by
Eqn. (2 . 3 . 27), a sum over the contributions from the six phonon branches:
( ) ( )2
6 2, ,
01 1,
ˆ1 coth Si nM iSi
nB Si
I F e ek T m
λ λ
λ λ
ωω
− − ⋅
= =
⋅� �∝ � �
� �� �
qk k q τ
k
� ε. (1)
In this equation, FSi is the atomic scattering factor, M is the Debye-Waller factor for Si [5], ω is the phonon
frequency, ε̂ is the polarization vector of the phonon mode, k is the reduced wave vector, kB is the Boltzmann
constant, T is the sample temperature (300 K), n the index and nτ the position vector for the two Si atoms in the
unit cell and λ is the index for the six phonon branches. The hyperbolic cotangent function in this equation
represents a sum of the Bose-Einstein distribution function and the zero-point mode occupancy. The calculated
intensity in each pixel, on a logarithmic scale, is given by
( ) ( ) ( )3 2 2 20log cos 2 ( , ) sin cos cos 2{ ([ ] ) }theoryI D T A I B Cθ θ φ φ φ θ= ⋅ ⋅ ⋅ + ⋅ ⋅ + + .(2a)
In this equation, φ is the azimuthal angle between the plane of polarization of the incident beam and the scattering
plane, and 2θ is the scattering angle. The expression within the brackets containing these angles accounts for the
linear polarization of the incident beam. The quantity A is an intensity normalization factor, B represents a constant
background from higher order and defect scattering from the sample, C is an overall constant background, and D is
35
an overall scaling factor related to the image plate response function. One factor of ( )θ2cos on the left hand side
of the formula is for solid angle conversion associated with planar projection, and the additional ( )θ2cos2 factor
allows for the inverse square distance dependence of the scattering intensity from the sample to different points on
the detector plane. The sample attenuation, given by ( )φθ ,T , calculates the distances traveled by the incident and
scattered beams depending on the Euler orientation of the sample.
( )
( ) ( )
( )
( )
coscos
i f
i f i
( , )cos( ) 1cos
ˆ ˆcos( ) cos( )
2 sin cos cos2 2
00 2 sin cos sin
2 21
fi
tt
i
f
i f
T e e
t
k k
k k
λλλθ ϕ
θ θ ϕ
θ θ ϕ
−−ΘΘ
� �� �= ⋅ −� �� �Θ � �� �⋅ −
� �Θ� �
⋅ ⋅Θ = Θ =
� � � �⋅ � � � �� � � �
� �� � � � = = + = ⋅ � � � � � � � �
� �
k n k n
k k q k i
2
ˆ ˆˆ proper
2 sin2
ˆ ˆz y x
θ
� �
+ � �− ⋅ � � � �� �
= ⋅ ⋅ ⋅
k
n R R R n (2b)
In this equation λ is the attenuation length of the sample, t is the thickness, and Θi and Θf are the polar angles of the
incident and scattered momentum vectors ki and kf with respect to the surface normal n̂ in the lab frame - which is
shown to depend explicitly on the Euler angles of the sample orientation by some rotations R from the perfectly
aligned case (where i properˆk n� ).
Equations (1) and (2) are used to generate theoretical patterns, which are compared to the experimental
ones. The shadows of the beam stop and post are excluded from this comparison by using a mask function. A least-
squares algorithm (of specifically (ln(theory)-ln(experiment))2 ) is employed for a simultaneous pixel-by-pixel fit to
both experimental pictures shown in Fig. 1. The fitting parameters include the force constants, A, B, C, and D in Eq.
(2), three Euler angles specifying the exact crystallographic orientation of the sample in each case, and the exact
distance from the sample to the detector. The best fits are shown in Fig. 4.1, and the corresponding phonon
dispersion curves generated by the best-fit force constants are shown as solid curves in Fig. 4.2. The stability and
36
Crystal Momentum
Freq
uenc
y (
THz)
0
2
4
6
8
10
12
14
16
Γ ∆ Χ Σ Γ Λ L
NN Theoretical Neutron X-ray
1(1,1,1) α1 = -0.5470 β1 = -0.3880 α1 = -0.5123 β1 = -0.3677 α1 = -0.5204 β1 = -0.3657
2(2,2,0) µ2 = -0.0350 ν2 = -0.0290 µ2 = -0.0338 ν2 = -0.0563 µ2 = -0.0330 ν2 = -0.0510
γ2 = 0.0170 δ2 = 0.0680 γ2 = 0.0795 δ2 = -0.0231 γ2 = 0.0784 δ2 = 0.0037
3(1,2,3) µ3 =0.0070 ν3 = -0.0100 µ3 = -0.0097 ν3 = -0.0081 µ3 = -0.0094 ν3 = -0.0025
γ3 = 0.0080 δ3 = 0.0010 γ3 = -0.0112 δ3 = 0.0143 γ3 = -0.0104 δ3 = 0.0140
4(0,0,4) µ4 = -0.0130 δ4 = 0.0180 µ4 = 0.0102 δ4 = 0.0100 µ4 = 0.0094 δ4 = 0.0086
5(3,3,1) µ5 =-0.0020 ν5 = -0.0010 µ5 = 0.0043 ν5 = -0.0035 µ5 = 0.0054 ν5 = -0.0096
γ5 = 0.0040 δ5 = 0.0300 γ5 = -0.0289 δ5 = -0.0018 γ5 = -0.0286 δ5 = 0.0019
6(2,2,4) µ6 =-0.0020 ν6 = 0.0060 µ6 = -0.0067 ν6 = 0.0047 µ6 = -0.0011 ν6 = 0.0029
γ6 = -0.0000 δ6 = 0.0010 γ6 = -0.0051 δ6 = 0.0000 γ6 = -0.0052 δ6 = 0.0003
Figure 4.2. Phonon dispersion curves of Si. Open circles are neutron scattering data from Ref. [6] (error barsshown), and closed circles are neutron data from Ref. [7] (error on the order of the size of the circle). Solid curvesare derived from a best fit to the x-ray scattering intensity patterns shown in Fig. 1. Dotted curves are obtained froman independent fit to the neutron data using the same lattice dynamics model, and can be regarded as an interpolationof the neutron data.
Table 4.1. Interatomic force constants of Si (105 dyn/cm). Coordinates of neighbors are expressed in units of a/4with a being the cubic lattice constant. The force constant matrix elements are in conventional notations (Ref. [2]).The first column of values are theoretical results from a first-principles calculation using the pseudopotential methodand the local-density approximation (Ref. [4]). The second column shows best-fit values obtained by a fit ofpredicted phonon frequencies to the neutron data points shown in Fig. 4.2 [6,7]. The third column shows best-fitvalues obtained by a fit of predicted x-ray TDS images to the experimental images in Fig. 4.1.
37
uniqueness of the fit were established by randomly generating new starting points within a ten percent Gaussian
error of the starting adjustable parameters, and the final parameters return to the same best set.
The agreement with available neutron data, presented in Fig. 4.2 as circles, is excellent, thus validating the
present method as a technique for determination of phonon dispersions. The dotted curves in Fig. 4.2 are obtained
from an independent fit to the neutron data points using the same lattice dynamics model, and can be regarded as a
k-space interpolation of the neutron data. The two sets of curves, solid and dotted, are very close. The small
differences, mostly no larger than the error bars associated with the neutron data, suggests that the accuracy of the
present approach is similar to that of neutron scattering, as was previously claimed.
4.2 Niobium
The results of the previous section suggest that it is possible to carry out detailed quantitative phonon studies
via x-ray TDS measurements. The method is appealing for several reasons: (1) x-rays can be focused down to
submicron or even smaller spots, thus allowing investigations of very small crystals; (2) the method is highly
efficient, and high quality data can be collected within seconds; and (3) the deep penetrating power of x-rays allows
in situ studies under various environments. However, Si is one of the most perfect crystalline materials available,
and the natural question is whether or not the TDS method can be applied, with comparable accuracy, to other "more
common" materials with significant structural defects that can affect such analysis. To address this issue, we have
chosen to investigate Nb as a test case.
The choice of Nb is also motivated by a long history of interest in the lattice dynamics of this material [8-
12]. Its phonon dispersion exhibits a number of pronounced anomalies caused by a strong electron-phonon
interaction involving long-range forces. This strong interaction is the underlying reason for the unusually high
superconducting transition temperature compared to all other elements in the periodic table. The phonon anomalies
have attracted much theoretical interest, and attempts have been made to relate these to the 4d electronic structure of
Nb [13]. These issues have also spurred the neutron scattering community to carry out detailed measurements of the
phonon dispersion curves [8-12], which are thus available for comparison with our TDS results. As shown below,
our TDS measurements yield phonon dispersion curves that are very close to the neutron results. The agreement,
however, is not quite as good as in the case of Si, and the reasons for this difference will be discussed.
We should add that inelastic x-ray scattering has also become a competitive method for phonon
measurements [14]. This method, like neutron scattering, has the advantage of energy resolution and is capable of
38
direct mapping of the dispersion curves. However, the required instrumentation is much more complex, and the
measurement is generally limited to one point in k space at a time. So far, only a limited number of systems have
been examined by this method, and Nb is not only of those.
The experimental setup is similar to the previous section and as described in Sec. 3. A transmission Laue
geometry was again employed, in which an 18.7 keV beam was sent through a crystalline Nb foil of thickness 0.1
mm. The foil crystal was provided to us (Acknowledgements) and prepared by prolonged vacuum annealing of a
high-purity polycrystalline foil, resulting in recrystallization over much of the area to form a single crystal oriented
with the [110] direction normal to the foil surface with typical grain size on the order of centimeters. A charge-
coupled device was used as an area detector, positioned behind the sample to record the transmission scattering
images with an exposure time of 5 s each. The incident beam was polarized in the horizontal direction. Data was
taken with the sample in air.
Figures 4.3(a)-4.3(c) are experimental images of Nb which are, respectively, taken at near normal incidence
to the crystal, with the crystal rotated 200 about an arbitrary axis, and with the crystal rotated 400 about the same
axis. The total data acquisition time for the entire experiment is about 15 s. These images are displayed on a
logarithmic intensity scale in order to bring out weak features that are otherwise not visible to the eye. Figures
4.3(d)-4.3(f) are the corresponding theoretical images to be discussed below. The intensity pattern in each case is
due to x-ray scattering from thermally populated phonons. Figure 4.3(a) exhibits a twofold rotational symmetry and
reflection symmetries about the horizontal and vertical axes, while the other two patterns reveal no apparent
symmetries due to the arbitrary rotations. As with the Si patterns, none of the bright spots are caused by crystal
diffraction. The circular shadow of a suspended beam stop is visible at the center of each experimental image, and a
shadow due to a suspension post is in the upper right corner. Each point in a TDS picture corresponds to a unique
momentum transfer q determined by the scattering geometry and the x-ray wavelength employed in the experiment.
In the case of Nb with just one atom per unit cell there are three phonon modes at each q contributing to the
scattering intensity. The bright spots in each picture are points on the Ewald sphere that are closest to neighboring
reciprocal lattice points, where the acoustic phonon population is high leading to a high scattering rate. Due to the
close proximity of the detector to the sample, a large scattering angle is achieved at the edge of each picture. The
maximum momentum transfer is 8 Å−1 in our experiment, which is significantly larger than the Brillouin zone size
of Nb (the distance between the zone center and the H point on the zone edge is only 1.9 Å−1). Pictures taken at
39
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.3. Transmission x-ray scattering images taken from Nb(110) (a) at near normal incidence, (b) with thesample rotated 20o about an arbitrary axis, and (c) with the sample rotated 40o about the same axis. In each case, theshadows of a beam stop in the center of the image and a suspension post in the upper right hand corner are evident.The corresponding calculated images based on a best simultaneous fit are shown in (d)-(f)
40
different sample rotation angles correspond to the same Ewald sphere intersecting the crystal Brillouin zones at
different angles. Thus, the three pictures in Fig. 4.3 together yield a substantial sampling of the reciprocal space.
The theoretical pictures shown in Figs. 4.3(d)-4.3(f) are derived from a force constant formalism of lattice
dynamics as described in Sec. 2.2 and the previous section. Force constants up to the eighth nearest neighbors are
included in a harmonic lattice model. The intensity of scattering by an unpolarized incident x-ray at a certain
momentum transfer q is given by a sum over the contributions from the three phonon branches, and is otherwise
treated exactly as in the previous section. This model was used to generate theoretical images, which were
employed in a least-squares, pixel-by-pixel fit to all three experimental images simultaneously. The areas in the
images around the beam stop and the support post in the upper right hand corner were removed from the fit via a
mask function. The fitting parameters include the three Euler angles determining the sample orientation, the exact
distance from the sample to the CCD screen, the x and y position of the screen relative to the beam, and the Born-
von Karman force constants.
The fitting of calculated images to the data can essentially be thought of as numerically solving Eq. (2) to
reproduce the phonon frequencies as a function of momentum transfer. The images from the best fit are shown in
Fig.4.3. These are very close to the corresponding experimental images. The detailed shapes of the bright spots,
some of which exhibit a "star" pattern, as well as the subtle variations of the less intense, diffuse regions are all well
reproduced. There are differences on a small subset of pixels, however, such as the faint streaks in the experimental
pictures extending from the brightest spots upward to the edge of each picture. This is particularly noticeable for the
bright spot on the left in Fig. 4.3(c). These streaks are not seen in the calculation and are caused by the nonideal
behavior of the charge-coupled device as outlined in Sec. 3.2. In most applications, this streaking is not noticeable.
In the present case, the use of a logarithmic intensity scale to bring out weak features accentuates the problem (c.f.
Fig. 4.4). While this does not spoil the picture in any major way, the "noise" introduced can potentially affect the
accuracy of the fitting parameters.
The phonon dispersions along high symmetry directions obtained from the best fit are shown in Fig. 4.5(a)
by the solid curves. These compare fairly well with the circles, which represent results deduced from neutron
scattering. The agreement is very good at low frequencies, but at high frequencies, there are noticeable differences
in certain regions in the Brillouin zone, with the maximum deviation up to about 10%. The overall fit is certainly not
quite as good as in the case of Si previously. This might be due to a confluence of several factors that affect the
41
Figure 4.4. Quasi-contour plot of TDS images. The panel layout is identical to Fig. 4.3, however the scale allowslow-intensity features to be clearly visible. Low intensity noise in the experimental images is now apparent, as wellas small vertical streaks caused by the non-ideal behavior of the CCD readout. The rich underlying structure at lowintensity is due entirely to dispersion of high-frequency phonons.
42
Phonon Dispersions for NiobiumFr
eque
ncy
(TH
z)
0
1
2
3
4
5
6
7
8
Crystal Momentum
Freq
uenc
y (T
Hz)
0
1
2
3
4
5
6
7
ΓΓ H P N
(a) NeutronTDS
NeutronTDS
(b)
Figure 4.5. Phonon dispersion curves of Nb. Open circles with error bars are neutron scatteringdata from Ref. [12]. Solid curves are derived from a best fit to the TDS patterns based on a latticedynamic calculation using (a) an eighth nearest neighbor Born von-Karman model and (b) a thirdnearest neighbor Born von-Karman model modified to account for charge fluctuation.
43
analysis. First, due to the anomalies in the dispersion very long range forces are needed to obtain a reasonable
description of the phonon dispersion curves [8-12]. It is known that even the eighth nearest neighbor Born-von
Karman model cannot reproduce all the features seen in the neutron data when fit directly. Thus some discrepancies
are expected. Secondly it has been suspected that the shortfall of the force constant model might get worse as one
moves away from the high symmetry directions probed in the neutron studies. Since the bulk of our TDS data are
taken from regions in the reciprocal space off the high symmetry directions, the limitations of the model can be
exacerbated when the comparison is made only along high symmetry directions. Thirdly the disagreement is mostly
in the high frequency regions, where TDS intensity is the lowest and noise on a logarithmic scale becomes
correspondingly more important. The streaking problem mentioned above may very well have contributed to this
disagreement. It is noted that Chang and Colella have reported some systemic discrepancies between x-ray and
neutron results for the lattice dynamics of Nb [15]. However, this might have been caused by variations in sample
quality as discussed by Rowe and Magerl [16]. We believe that our sample, grown in vacuum at high temperatures,
is free from the impurities that might have significantly affected the earlier x-ray measurements.
Fitting of the images using the Born-von Karman model was a rather demanding computational job due to
the large number of pixels that need to be calculated and the large number of force constants that need to be
optimized. Simpler models exist, but many of these yield relatively poor descriptions of the dispersion curves [8]. A
very promising model is the charge fluctuation model that involves many fewer parameters at the expense of a
slightly worse description of the phonon dispersion curves [17]. This model is based on a third nearest neighbor
Born-von Karman model, with the important modification that the ionic charges on the atoms are allowed to
redistribute. This model is also employed in our analysis, and the resulting dispersion curves are shown in Fig.
4.5(b). The computation time is very much reduced, but a main drawback of this model is that it cannot reproduce
several observed features such as the "swoop" anomaly in the lowest transverse acoustic branch dispersion near the
Γ point (specifically, compare Figs. 4.5(a) and 4.5(b) near Γ along Γ-H; the dispersion goes up linearly from Γ and
then drops below the linear extrapolation). As a result, the fit in the low frequency regions is compromised, while
the discrepancies in the high frequency regions remain. This example illustrates that the TDS method, as
implemented here, is limited by the model used for fitting. A poor model can lead to significant errors in the result.
In summary, the present work offers a test of the TDS method with fairly satisfactory results on a system
that has been the subject of intensive studies of its lattice dynamics. The "swoop" anomaly that is the focus of
44
numerous previous studies is accurately reproduced. The crystalline quality of our Nb foil sample cannot be
compared with that of the Si wafers used in previous studies, but the slightly worse fit here is unlikely to be sample
related. Rather, the limitations of the force constant model and the imperfections of the charge-coupled device as a
detector appear to be the main issues. None of these is fundamental in nature, and can be overcome as the methods
become further refined.
References:
1. I. K. Robinson and D. J. Tweet, Rep. Prog. Phys. 55, 599 (1992).
2. F. Herman, J. Phys. Chem. Solids 8, 405 (1959). An error in this work was pointed out by C. Patel, W. F.
Sherman, and G. R. Wilkinson, J. Phys. C: Solid State Phys. 17, 6063 (1984).
3. J. P. Wolfe, Imaging Phonons (Cambridge, Cambridge, 1998).
4. S. Wei and M. Y. Chou, Phys. Rev. B 50, 221 (1994).
5. M. Deutsch and M. Hart, Phys. Rev. B 31, 3846 (1985).
6. G. Dolling, Inelastic Scattering of Neutrons in Solids and Liquids (IAEA Vienna, 1963).
7. G. Nilsson and G. Nelin, Phys. Rev. B 6, 3777 (1972).
8. P. H. Dederichs and H. Schober, Landolt Bornstein NS Group III 13a p. 96 (Springer-Verlag Berlin, 1981).
This article contains a detailed listing of references and a review of the lattice dynamics models and the
neutron scattering results.
9. Y. Nakagawa and A. D. B. Woods, Phys. Rev. Lett. 11, 271 (1963).
10. R. I. Sharp, J. Phys. C 2, 421 (1969).
11. B. M. Powell, A. D. Woods, and P. Martel, Neutron Inelastic Scattering, Vol. 43 (IAEA Vienna, 1972).
12. F. Guthoff, B. Hennion, C. Herzig, W. Petry, H. R. Schober, and J. Trampenau, J. Phys. Condens. Matter 6,
6211 (1994).
13. K.-M. Ho, C.-L. Fu, and B. N. Harmon, Phys. Rev. B 28, 6687 (1983); B 29, 1575 (1983).
14. See, for example, F. Occelli, M. Krisch, P. Loubeyre, F. Sette, R. Le Toullec, C. Masciovecchio, and J.-P.
Rueff, Phys. Rev. B 63, 224306 (2001).
15. S. S. Chang and R. Colella, Phys. Rev. B 15, 1738 (1977).
16. J. M. Rowe and A. Magerl, , Phys. Rev. B 21, 1706 (1980).
17. N. Wakabayashi, Solid State Commun. 23, 737 (1977).
45
Chapter 5:Observation of Phonon Softening
5.1 Titanium Diselenide
Based on the results of the previous section, we claim that suitable analysis of diffuse x-ray scattering from
thermal phonons can provide quantitative, accurate information on lattice vibrations in solids. However, beyond
demonstration of TDS analysis as a vehicle for phonon studies, we must now substantiate its applicability to more
complicated systems. To this end, we will present measurements on two systems – TiSe2 and SrTiO3 – whose
structural phase transitions involve the temperature-dependent frequency change (softening) of a participating
phonon mode. This success of TDS analysis in quantifying temperature dependent phonon dispersion in correlated
systems will illustrate its applicability to systems of interest in materials studies.
This section reports a study of TiSe2, a prototypical charge density wave system with the 1T structure
which undergoes a CDW transition at ~200 K to form a simple commensurate (2×2×2) superlattice at low
temperatures (for further details and a review of theory see Sec. 2.4.1). At room temperature, neutron scattering
measurements of the phonon dispersion relations show already the presence of a dip at the Brillouin zone boundary
for the lower transverse phonon mode L1−, which presumably softens as the temperature decreases [1-4]. However,
this softening behavior has never been observed directly due to a very low neutron cross section. We have
performed a detailed x-ray scattering study of this system and show that the frequency of the L1− phonon mode
indeed decreases gradually to zero at the transition temperature. This work thus confirms the soft mode concept,
which is central to the theory of the charge density wave phenomenon in this type of material.
In general, the experimental methods are outlined in Sec. 3.1. A single crystal of TiSe2 was prepared
following standard methods [5] and attached to the copper cold finger of a closed-cycle helium refrigerator by
thermally conductive grease. The sample assembly was enclosed in a vacuum shroud equipped with a hemispherical
Be dome for the x-ray measurements. The temperatures reported below were the readings from a thermocouple
attached to the copper cold finger. Incident radiation was set at 8.1 keV, and the quasielastically scattered radiation
was collected using a four-circle diffractometer in reflection geometry. The incident radiation was sufficiently
intense to cause sample heating, and calculations suggested a local temperature rise of about 10 K relative to the
46
TiSe2Scan along A-L-Aq = (1, q2, 7/2)
-0.6 -0.5 -0.4
190
185
185
190200
210
230
250
270
290
Nor
mal
ized
Inte
nsity
q2
187.5
-0.46-0.54T (K)
T (K)
Figure 5.1. Linear scans of x-ray thermal diffuse scattering along an A-L-A line in reciprocal space. q2 = −1/2corresponds to an L point. The sample temperature is indicated for each scan. The inset contains additional scanstaken with a finer k resolution and displayed with a reduced vertical scale to show the emergence of a Bragg peak atL below the transition temperature. The solid lines are a single parameter theoretical fit to each temperature – below190K an additional central component was added.
47
cold finger. Thus, the apparent transition temperature as measured by the thermocouple would be about 10 K below
the actual transition temperature.
Figure 5.1 shows typical scans along an A-L-A line in k space. The momentum transfer q ≡ (q1, q2, q3) is
constrained such that q1 = 1, q2 varies linearly between –0.55 and –0.35, and q3 = 7/2 in normalized units. The
observed intensity variation is derived from thermal diffuse scattering (scattering by thermally populated phonons).
The line shape is given by (Sec. 2.3)
( ) ( )( )
( )2
9
1
coth2
j j
j j B
FI
k Tω
ω=
� �∝ � �
� ��
q kq
k�
, (1)
where the phonon structure factor F is given by
( ) ( ) ( )3,
1
ˆn n jM
j nn n
F f em
−
=
⋅=�
q ε kq q . (2)
In the above equations, n is an atomic index in a unit cell consisting of one Ti atom and two Se atoms, j is an index
for the 9 phonon branches, and M, m, ω, k, and ε̂ stand for the Debye-Waller factor, the atomic mass, the phonon
frequency, the reduced wave vector and the phonon polarization vector, respectively. The phonon structure factor is
a smooth, slowly varying function. The hyperbolic cotangent function in Eq. (1) represents the phonon population
including the zero-point vibration, which, together with the remaining 1/ω factor, favors heavily the lowest phonon
branch (j = 1). Thus, a large scattering intensity corresponds to a small ω1, and vice versa. It is straightforward to
show that 21−∝ ωI for a small ω1 at the L point.
In Fig. 5.1, one can clearly see a peak at q2 = −1/2, which corresponds to the L point in the Brillouin zone.
This thermal diffuse peak is weak and broad near room temperature. It narrows and intensifies as the temperature
decreases. The inset shows additional scans with a finer k resolution and a reduced intensity scale. As T decreases
below Tc = 189 K, a resolution-limited Bragg peak emerges with an intensity increasing as ( )2/1 cTT− in
accordance with a Landau theory for a normal-to-lock-in phase transition [Sec. 2.4, also 6]. Simultaneously, the
thermal diffuse peak reaches a maximum intensity at Tc, and diminishes at lower temperatures.
48
TiSe2 Phonon Dispersion
A L
Freq
uenc
y (T
Hz)
0
1
2
3
4
5
6
7
8
A L0
1
2
295
270
250
230210
190
T (K)
Figure 5.2. Phonon dispersion curves of TiSe2 as a function of temperature deduced from x-ray diffuse scattering.The lowest phonon branch softens as the temperature decreases towards the transition temperature.
Figure 5.3. The filled circles represent experimentally deduced frequency of the lower L1− transverse acoustic mode
as a function of temperature. The solid curve is a fit using Eq. (3). The open circles are results from the sameanalysis with the Bragg peak ignored. They do not correspond to any physical frequency, and are included here toshow the sharp change at Tc. The dashed curve serves as a guide to the eye.
T (K)150 175 200 225 250 275 300
Freq
uenc
y (T
Hz)
0.0
0.5
1.0
1.5TiSe2
L1- Transverse Acoustic Mode
49
The peaking of the thermal diffuse intensity at L at room temperature is because the dispersion of the
lowest transverse acoustic branch bends downward near the L point. As the sample temperature decreases, TDS
intensity should generally decrease due to a reduction of the phonon population, but the results in Fig. 5.1 are clearly
opposite to this trend. The increased peaking at lower temperatures is due to the gradual reduction of ω1. This mode
softening leads to an increased phonon population, despite the lowering temperature, as well as an increased cross
section through the 1/ω factor. Thus, the measured intensity increase can be directly related to the mode softening
behavior.
The solid curves in Fig. 5.1 are theoretical fits to the data based on Eqs. (1) and (2). In this calculation, a
force constant model is used to describe the lattice dynamics. All force constants have been determined previously
from a fit to the available neutron data at room temperature [7]. One of the force constants, A1, is known to relate to
the softening [4, 7], and this is employed as a fitting parameter in our model for scans at various temperatures. The
only other parameters are a constant background representing higher order, defect, and stray scattering for each scan,
and an overall intensity normalization factor. The fits are excellent. Not only is the intensity variation well
reproduced, so is the lineshape change (narrowing at lower T). All of this is achieved with the adjustment of just one
force constant.
The force constant resulting from the fit yields the phonon dispersion curves shown in the left panel of Fig.
5.2. The upper eight phonon branches are independent of temperature, while the lowest branch, a transverse acoustic
mode, softens near the L point as the temperature decreases toward Tc. The right panel of Fig. 5.2 shows an
amplified view of the variation of this dispersion curve as a function of temperature. At room temperature, a dip is
clearly present, and becomes more pronounced towards lower temperatures. At Tc, the L1− mode frequency becomes
zero, and a static lattice distortion sets in to form a (2×2×2) superlattice. This static distortion increases in amplitude
as T lowers below Tc, giving rise to an increasingly more intense Bragg peak as seen in Fig. 5.1.
Figure 5.3 shows the frequency of the L1− mode derived from our analysis. The solid circles represent data
points above Tc, while the open circles are results obtained from the same analysis of the thermal diffuse peak with
the Bragg peak ignored. These open circles, connected by a dashed curve as a guide to the eye, do not correspond to
any physical frequency, because the lattice is already statically distorted. The system now has a different crystal
symmetry and a different dynamic matrix. We nevertheless include these fitting results in the figure to show the
sharp change around at Tc. The cusp in the curve allows us to pin point the transition temperature, which is in perfect
50
agreement with the observed first appearance of a Bragg peak. The solid curve in Fig. 5.3 for T > Tc is a fit using the
expansion
( ) ( )[ ]2)( ccc TTcTTbaTTT −+−+−=ω , (3)
where Tc = 188.7 K, a = 7.92 × 10−2, b = 2.40 × 10−4, and c = 1.91 × 10−6 are fitting parameters. The cTT −
prefactor is based on a mean field theory for a second order transition, which dominates the temperature dependence
near the critical point [8,9]. The data, however, cover a wide range of temperature, and a second order polynomial
expansion is needed to account for higher-order effects.
As mentioned in the introduction, the soft mode concept is central to many theories of structural phase
transitions. For TiSe2, this predicted soft mode behavior has never been observed directly by neutron scattering
because of the lack of signal strength. With the advent of third-generation synchrotron radiation sources, this phonon
softening phenomenon becomes accessible. Our results clearly show the soft mode behavior, and verify key features
of this phase transition, including the characteristic power laws for the phonon frequency and the intensity variations
near Tc. These results provide confirmation of theoretical concepts that have remained untested until now, and rule
out other possible models such as a mixture of two distinct phases with varying proportions. This work also
illustrates the applicability of the TDS method to phase transitions in complex crystals.
5.2 Strontium Titanate
The previous section reported the first-ever observation of phonon softening in TiSe2 over a wide
temperature range, which was well predicted by a phenomenological theory describing the problem. Whereas a true
confirmation of these results will depend on future work, it is appropriate now to substantiate the accuracy of the
method. This section will describe a similar measurement of phonon softening on SrTiO3, a system that has been
well investigated by both theory and experiment. We will see that for temperatures well above the transition the
softening behavior is predicted by the results of a phenomenological mean-field theory similarly to TiSe2. Near the
transition however, there exists a critical region where the predictions of this theory are not valid and the frequency
change instead follows a power law dependence as predicted by renormalization group techniques. The agreement
of this TDS analysis in both regions with these models and available neutron data argues for the accuracy of the
previous section’s results and further supports the applicability of this method to structural phase transitions in
solids.
51
Strontium Titanate is a metal oxide whose structure, properties, and other background are outlined in Sec.
2.4.2. Upon cooling, it undergoes a structural phase transition at ~105K from its RT cubic structure to a tetragonal
structure via a rotation of the oxygen atoms [10-12]. This has been associated with the condensation of a zone
boundary phonon mode [10,13,14] at the R point in reciprocal space. This triply degenerate mode (R25) softens
from its room temperature value to zero at the transition temperature, where it splits into two optical modes (one
singly degenerate and one doubly degenerate in the new crystal symmetry) that harden in frequency away from the
transition. This section will report an observation of this softening behavior by a collection of x-ray thermal diffuse
scattering at the R point over a wide temperature range.
The experimental setup is given in Sec. 3.1, in which monochromatic x-ray radiation of energy 14 keV was
applied and collected in a standard reflection geometry. The sample was cooled as described in the previous section
by a closed-cycle He displex with the temperature monitored by a Si diode attached to the cold finger. However, in
this case the observed transition temperature agreed well with transport measurements; this implies the beam heating
was negligible in this case, probably due to several small differences in the setup. Primarily, a copper based grease
was used instead of Apiezon vacuum grease to attach the sample at a higher thermal conductivity to the copper
block. Two other contributing differences were the much larger sample volume (~10 x thickness compared to the
TiSe2 sample), which allowed better temperature equilibration between heated and non-heated regions on the
sample, and as well the beam was at a reduced photon flux compared to the previous experiment.
The data is shown in Fig. 5.4. As before, the TDS intensity near the R point increases upon cooling to near
the transition temperature, indicating a frequency drop of the lowest lying phonon mode. The lineshape also
narrows, indicating a change in the near-R-point phonon dispersion that moves continuously with the softening
frequency. At about 15 degrees above the transition temperature, a sharper central component appears above the
broader TDS component. This is due to a near-surface phase transition occurring at an elevated temperature relative
to the bulk, which will be discussed below. Importantly, as is made clear on the logarithmic scale panel at right, this
component is relatively weak and its width is not resolution limited at temperatures greater than ~105K. Below
105K the central component greatly picks up in intensity and the width reduces to the instrument resolution. This
indicates that a true superlattice peak has emerged, and that a static lattice distortion has occurred within the bulk of
the crystal (Fig. 5.5). This jump suggests the transition temperature is between 105K and 102.5K, in agreement with
our fit to the TDS component below.
52
Figure 5.4. TDS intensity of SrTiO3 as a function of temperature near the R-point. The intensity becomes morepeaked and lineshape narrows under cooling to the 105K (commensurate 2x2x2) phase transition. A centralcomponent begins to appear at 120K, due to a near-surface region entering the phase transition at an elevatedtemperature relative to the bulk. The marked intensity increase across 105K in the right panel indicates the bulk hasentered the phase transition. The solid line is a theoretical fit to the data.
(1/2, 1/2, q)1.40 1.45 1.50 1.55 1.60
Line
ar In
tens
ity (a
rb. u
nits
)
(1/2, 1/2, q)1.40 1.45 1.50 1.55 1.60
Loga
rithm
ic In
tens
ity (a
rb. u
nits
)
280 K
120 K
117.5 K
112.5 K
107.5 K
102.5 K
SrTiO3 TDSLinescansM - R - M
53
The solid lines in Figure 5.4 are a theoretical fit to the data. The lineshape is again given by the first order
contribution of thermal diffuse scattering to the scattered intensity (Eqns. 1 and 2). This analysis is nearly identical
to that found in the previous section, except the summation runs over five atoms in the unit cell with the
corresponding 15 normal phonon modes. Another difference was that the phonon frequencies at each momentum
transfer (along the [1/2 1/2 q] high symmetry direction) were given by a direct polynomial interpolation of available
neutron data (Fig. 5.6), avoiding the use of force constant models. The polarization vector for each mode along this
direction was determined by symmetry considerations. This model independent approach gave the scattering for all
temperature independent phonon modes. This phonon mode softening was represented by a two-component
dispersion – a parabolic dispersion near the R point, turning over into a linear dispersion matching the original
interpolation function away from the R point (Fig. 6). This represents the fact that the dispersion at room
temperature is mostly parabolic, whereas the dispersion at the transition temperature must be entirely linear (as the R
point is the new zone center). The two adjustable parameters were the frequency of the lowest lying phonon mode
at the R-point and the position along M – R of the linear-to-parabolic turnover point. These essentially fit the height
and width of the TDS lineshape, respectively. These two parameters determined a softening dispersion near the R
Figure 5.5. Behavior of central (non-TDS) component as a function of temperature. The sudden increase inintensity and narrowing of width below 105K indicates a bulk transition temperature at Tc =104 ± 0.5
Peak
Wid
th (∆q)
10-5
10-4
10-3
Temperature (K)90 95 100 105 110 115 120 125
Peak
Inte
nsity
(cou
nts)
100
101
102
103
104
105 (c)
54
Momentum Transfer (M - R)
0.0 0.1 0.2 0.3 0.4 0.5
Freq
uenc
y (T
Hz)
0
2
4
6
8
10
12
14
16
0.0 0.1 0.2 0.3 0.4 0.50
1
2
3
4
5
280 K
180 K
120 K
105 K
0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.500
1
2
3
Freq
uenc
y (T
Hz)
Figure 6. Phonon dispersion of SrTiO3. The solid lines are a polynomial interpolation of the RT neutron data (opencircles) from Ref [11]. The frequency of the lowest lying phonon mode at the R point was allowed to soften via a twocomponent dispersion (parabolic near R plus linear further away, turnover point marked with triangles) - this wasmatched to the temperature independent interpolation by continuity at q=0.4. The fitting parameters were thesoftened frequency at the R point and the position of the parabolic-to-linear turnover point - all other modes areindependent of temperature. The gray line is the zone center sound velocity of SrTiO3.
55
point ([1/2 1/2 R ± 0.1]) that was matched up to the original interpolation function away from the R point by
requiring simple continuity at [1/2 1/2 R –0.1] and a continuous slope at the interior turnover point (Fig. 5.6). The
frequency of the lowest lying transverse mode at the R point was observed to soften from its room temperature
value. As well the dispersion was found to become increasingly linear leading up to the transition, as expected.
The dispersion of all other modes was assumed to be independent of temperature. This two-parameter fit to the TDS
intensity was employed over a broad temperature range from room temperature to the phase transition, near which
an additional central component modeled as a Voigt lineshape was added. The only other parameters are a constant
background representing higher order, defect, and stray scattering for each scan, and an overall intensity
normalization factor. This method predicts not only the variation in TDS intensity but also the lineshape narrowing
over the entire temperature range.
Various force-constant type models have been developed that predict the phonon dispersion curves of
SrTiO3 [11,13], but all are computationally intensive and offer relatively poor descriptions of available neutron data.
The interpolation method employed by this present work sidesteps this issue by giving the eigenvalues and
eigenvectors for all phonon modes along a high symmetry direction directly from the neutron scattering data without
calculating a dynamic matrix (that would have had to be fit to this same data). Of course, this method is only valid
on the particular symmetry direction (in this case [1/2 1/2 q]) along which the x-ray data was taken, but within this
subset of reciprocal space it predicts the data very well.
The sharp central component has been previously observed in x-ray data at temperatures well above Tc [15-
19], and has been widely attributed to a lattice distortion forming in a near-surface region at elevated temperature
relative to the bulk. The two most conclusive pieces of experimental evidence for this were a depth-dependent x-ray
diffraction measurement indicating that the central component is only present in a surface region approximately
100µm thick, and a direct optical observation of the surface undergoing a phase transition at Tc+45K [20,21]. The
microscopic mechanism remains unclear, but the phenomenon is thought to be related to the high density of
dislocations and defects [22,23], or spontaneous strain [24] in the near-surface region relative to the bulk.
Essentially, a phase transition is thought to accrete above Tc near these areas of broken symmetry within the crystal
where it is easier for the unit cells to undergo a lattice distortion. This then propagates outward from the nucleation
sites until the bulk has undergone the phase transition. Regardless of cause, it is important to note that the central
component is well separated from the x-ray thermal diffuse scattering in both width and intensity, so the two are
56
Temperature (K)100 125 150 175 200 225 250 275 300
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
X-ray dataLandau predictionNeutron data
Mean - Field Region
Temperature (K)105 110 115 120
R25
Pho
non
Freq
uenc
y (T
Hz)
0.000
0.125
0.250
0.375
0.500
RG predictionX-ray Data
Critical Region
Figure 5.7. Phonon softening of the participating R25 zone boundary phonon mode leading up to the 105K phasetransition in SrTiO3. The open circles are the best-fit value of the frequency to the TDS lineshape at eachtemperature. The closed circles are neutron data taken from Ref [12]. The solid black curve is a polynomial fit tothe x-ray data representing a phenomenological prediction of the phonon softening. Within the critical region thistheory fails to predict the data and is replaced by a power law (gray curve - arising from a renormalization groupprediction for critical behavior of the Heisenberg universality class).
57
Temperature (K)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
(R25
Pho
non
Freq
uenc
y)2 (
THz)
2
0.0
0.5
1.0
1.5
2.0
2.5
Reduced Temperature t = (T - Tc)/Tc Tc = 103.9 +/- 0.5 K0.00 0.05 0.10 0.15
(R25
Pho
non
Freq
uenc
y)2/
γ (TH
z)2/
γ
0.000
0.125
0.250
Mean - Field Region ( γ = 1.0 )
Critical Region ( γ = 1.383 )
Figure 5.8. Phonon softening in SrTiO3. The above graphs are identical to those in Figure 5.7, except that eachpanel has been raised to the appropriate power to emphasize the power-law dependence of the frequency change. Asingle power law cannot adequately describe both regions.
58
easily distinguished within this analysis.
The R25 phonon frequency softening is an output at each temperature from a fit to the x-ray TDS intensity,
and is shown in Figs. 5.7 and 5.8. Following these figures we see there are two distinct regions of frequency
change, as expected from the arguments in Sec. 2.4.2. In the region far from the transition temperature, the data is
well predicted by a phenomenological Landau theory for second order displacive phase transitions. The solid curve
is a fit to the data points above the critical region (T > 115 K) using the expression
( )( ) cT a T Tω = ⋅ − (6)
with values of a = 0.0126, and Tc = 109.09. This predicts a higher transition temperature than is actually observed
due to the breakdown of the mean-field approximation in the critical region [25,26]. The x-ray and neutron results
in this region are also very close, with discrepancies at most on the order of the error bars of the neutron data.
Near to the transition fluctuations dominate and the predictions of this mean field theory are not valid. The
size of this region has been estimated to be 105 K ± 10 K [25]. The data points in this region (in the second panel of
Figs. 5.7 and 5.8) were fit using the expression
( )2( ) cT b T Tωγ
= ⋅ − (7)
with best-fit values of b = 0.0604, Tc = 103.70 ± 0.5 K, and γ = 1.383 ± 0.1. This agrees well with the predictions of
renormalization group theory of γ = 1.375 [27] for the expected universality class of the 3D classical isotropic
Heisenberg antiferromagnet [28]. As well this value falls within an experimentally predicted range that can be
estimated by utilizing relationships between other previously observed exponents β = 0.333 ± 0.1[25] and ν = 0.63 ±
0.07 [29], which gives γ = 1.22 ± 0.4. Previous attempts to measure this exponent directly through neutron diffuse
scattering were inconclusive and gave values ranging between 1.21 < γ < 2.5 [30,31]. This was due in most part to
the merging of the phonon branch with an elastic central peak at temperatures close to the transition, although
sample quality was also an issue in these studies (evidenced by depressed, sample dependent transition
temperatures). For these reasons we believe this experiment offers the first direct estimate of the critical exponent γ
for the 105K phase transition in SrTiO3.
In summary, the R-point phonon softening of SrTiO3 leading up to the cubic-to-tetragonal phase transition
as measured by x-ray TDS analysis is found to be in good agreement with previous neutron inelastic scattering
results, and theoretical predictions in both the Landau and critical regime. This experimental agreement across
59
different techniques, experimental conditions, and even samples argues for the accuracy of the previous section’s
claims. As well the validation of theoretical predictions illustrates the potential of this method as a probe of
structural phase transitions. These results help establish x-ray TDS measurement as a valuable complement to
neutron scattering for the characterization of complex systems of interest to materials studies.
References:
1. W. G. Stirling, B. Dorner, J. D. N. Cheeke, and J. Revelli, Solid State Comm. 18, 931 (1976).
2. F. J. DiSalvo, D.E. Moncton, and J. V. Waszczak, Phys. Rev. B 14, 4321 (1976).
3. D. E. Moncton, F. J. DiSalvo, and J. D. Axe, in Proceedings of International Conference on Lattice
Dynamics, edited by M. Balkanski (Flammarion, Paris, 1978), p. 561.
4. N. Wakabayashi, H. G. Smith, K. C. Woo, and F. C. Brown, Solid State Comm. 28, 923 (1978).
5. D. L. Greenaway and R. Ritsche, J. Phys. Chem. Solids 26, 1445 (1965)
6. K. C. Woo, F. C. Brown, W. L. McMillan, R. J. Miller, M. J. Schaffman, and M. P. Sears, Phys. Rev. B 14,
3242 (1976).
7. S. S. Jaswal, Phys Rev. B 20, 5297 (1979).
8. Structural Phase Transitions in Layered Transition Metal Compounds, edited by K. Motizuki (D. Reidel,
Boston, 1986).
9. Charge Density Waves in Solids, edited by L. P. Gor’kov and G. Grüner (North Holland, New York, 1989).
10. P. A. Fleury, J. F. Scott, and J. M. Worlock, Phys. Rev. Lett. 21,16 (1968).
11. R. A. Cowley, Phys. Rev. 134, A981 (1964).
12. G. Shirane and T. Tamada, Phys. Rev. 177, 858 (1969).
13. W. G. Stirling, J. Phys. C: Solid State Phys. 5, 2711 (1972).
14. R. A. Cowley, Phil. Trans. R. Soc. Lond. A 354, 2799 (1996).
15. S. R. Andrews, J. Phys. C: Solid State Phys. 19, 3721 (1986).
16. R. J. Nelmes, P. E. Hatton, and H. Vass, Phys. Rev. Lett. 60, 2172 (1988).
17. D. F. McMorrow, N. Hamaya, S. Shimomura, Y. Fujii, S. Kishimoto, and H. Iwasaki, Solid State Comm.
76, 443 (1990).
18. K. Hirota, J. P. Hill, S. M. Shapiro, G. Shirane, and Y. Fujii, Phys. Rev. B 52, 13195 (1995).
60
19. H.-B. Neumann, U. Rutt, J. R. Schneider, and G. Shirane, Phys. Rev. B 52, 3981 (1995).
20. U. Rutt, A. Diederichs, J. R. Schneider, and G. Shirane, Europhys. Lett. 39, 395 (1997).
21. E. D. Mishina, T. V. Misuryaev, N. E. Sherstyuk, V. V. Lemanov, A. I. Morozov, A. S. Sigov, and Th.
Rasing, Phys. Rev. Lett. 85, 3664 (2000).
22. A. L. Korzhenevskii, K. Herrmanns, and H.-O. Heuer, Europhs. Lett. 45, 195 (1999).
23. Renhui Wang, Yimei Zhu, and S. M. Shapiro, Phys. Rev. Lett. 80, 2370 (1998).
24. Shuichi Doi and Isao Takahashi, Phil. Mag. A 80, 1889 (2000).
25. K. A. Muller and W. Berlinger, Phys. Rev. Lett. 26, 13 (1971).
26. T. Riste, E. J. Samuelsen, K. Otnes, and J. Feder, Solid State Commun. 9, 1455 (1971).
27. M. E. Fisher and D. Jasnow, Phase Transitions and Critical Phenomena 4, C. Domb and M. S. Green ed.
(New York: Academic Press, 1980). This work was summarized by: A.D. Bruce and R.A. Cowley,
Structural Phase Transitions (London: Taylor and Francis, 1981).
28. R. A Cowley and A. D. Bruce, Journ. Phys. C 6, L191 (1973).
29. Th. von Waldkirch, K. A. Muller, W. Berlinger, and H. Thomas, Phys. Rev. Lett. 28, 503 (1972).
30. J. Topler, B. Alefeld, and A. Heidemann, Journ. Phys. C 10, 635 (1977).
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61
Chapter 6:Summary and Outlook
6.1 Summary
Through a review of four recent experiments this thesis demonstrated the development of a method – the
use of thermal diffuse scattering of x-rays as a vehicle for phonon studies. The accuracy of this method in
comparison to neutron studies was first investigated by establishing phonon dispersion in Si. This was confirmed
and the methods were further refined to include the use of data far away from high symmetry directions in elemental
Nb. The applicability of the method to complex systems was illustrated by studying phonon softening leading up to
the structural phase transition in charge-density-wave TiSe2. The accuracy of this type of analysis was substantiated
by comparison to neutron data in a similar study of SrTiO3. Additionally, by studying phonon softening in a critical
near-transition region, it was shown how TDS analysis can experimentally determine critical exponents of phase
transitions In conclusion, based on the results of the completed experiments, we have established quasi-elastic x-
ray scattering from thermally populated phonon modes as a valuable complement to neutron scattering for lattice
dynamics studies. We expect to apply this method to many systems of topical interest to materials research in the
future.
6.2 Outlook
While the development of novel experimental methods is always beneficial in principle, in the present case
of phonon studies via x-ray TDS analysis the scientific motivation is particularly compelling. The future
applications of this method are in a sense defined by these advantages – which lie in the demonstrated efficiency,
precision, and minimum required sample size, as we will discuss below. The single - yet fundamental - drawback of
these experiments is the use of analytic methods to extract energy-resolved information from energy-integrated data.
We hope that the previous sections have convinced the reader that this type of approach is robust and suitable for
use in many areas.
For many systems of current interest to materials science large single crystals are either extremely difficult
to produce (such as the high temperature superconducting (HTS) cuprates, or colossal magneto-resistive (CMR)
manganites), or simply unavailable (as in the case of superconducting MgB2). This limits the effectiveness of
62
conventional scattering techniques, whereas in principle good quality TDS images can be obtained from samples of
much smaller sizes. The efficiency of this method is another clear advantage – for instance the total data taking time
of both experiments in Chapter 4 was less than two minutes. With an optimized CCD setup this efficient parallel
data collection method can be applied to in situ, real-time phonon studies. The precision of this method is perhaps
best illustrated by the phase transition experiments outlined in Chapter 5 - where phonon frequency changes of less
than 0.03 THz were observed with excellent statistics by a relative measure of TDS intensity. In this same near-
transition region where we have the most TDS signal (lowest phonon frequency), studies using neutron scattering
are the most hampered by large contributions to the elastic scattering.
Looking to the future, there are several experiments that would benefit greatly from alternatives to
conventional methods. Continuing with the elemental systems, there are several whose phonon dispersions are
unknown and unreachable by neutron scattering due to a high neutron cross-section. Most recently we have
achieved TDS images of single grains of delta-phase Plutonium within a polycrystalline foil by utilizing beam spots
as small as 5 µm in diameter (Fig. 6.1). Any quantitative information that could be gleaned about phonon
dispersions in Pu from the present method would be of tremendous benefit to actinide science. As well - extending
the work in Ch. 5 - the structural component of many correlated lattice-electron systems (such as CMR or HTS
materials) may now be analyzed over a wide temperature range with excellent statistics using available small single
Figure 6.1 – (a) Transmission TDS image taken of a single grain of δ-Pu (100) within a polycrystalline foil. Thegrain size was approximately 30 µm, and the picture was taken with a 25µm incident beam. (b) Calculation of Pu(100) TDS using a harmonic Morse potential fit to the known sound velocities of Pu.
(a) (b)
63
crystals. Many such crystals are inhomogeneous, and the use of a small probed spot as in the Pu example allows one
to select a single grain or domain of interest. In summary, the results described in this thesis should convince the
reader of the power and scope of x-ray TDS analysis as a method for phonon studies, and it is hoped these results
will also spur wider interest in the development of this promising field.
32
