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Digital Comprehensive Summaries of Uppsala Dissertations
from the Faculty of Science and Technology 1344
Thermodynamics of Hydrogen in
Confined Lattice
XIN XIAO
ACTA
UNIVERSITATIS
UPSALIENSIS
UPPSALA
2016
ISSN 1651-6214
ISBN 978-91-554-9473-5
urn:nbn:se:uu:diva-275629
Dissertation presented at Uppsala University to be publicly examined in Häggsalen,
Ångströmlaboratoriet, Uppsala, Thursday, 17 March 2016 at 10:15 for the degree of Doctor
of Philosophy. The examination will be conducted in English. Faculty examiner: Professor
Astrid Pundt (University of Göttingen).
Abstract
Xiao, X. 2016. Thermodynamics of Hydrogen in Confined Lattice. Digital Comprehensive
Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1344. 59 pp.
Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9473-5.
Three of the most important questions concerning hydrogen storage in metals are how much
hydrogen can be absorbed, how fast it can be absorbed (or released) and finally how strongly the
hydrogen is bonded. In transition metals hydrogen occupies interstitial sites and the absorption
as well as desorption of hydrogen can be fast. The enthalpy of the hydride formation is
determined by the electronic structure of the absorbing material, which determines the amount
of energy released in the hydrogen uptake and the energy needed to release the hydrogen.
This thesis concerns the possibility of tuning hydrogen uptake by changing the extension of
the absorbing material and the boundary conditions of extremely thin layers. When working
with extremely thin layers, it is possible to alter the strain state of the absorbing material, which
is used to influence the site occupancy of hydrogen isotopes. Vanadium is chosen as a model
system for these studies. V can be grown in the form of thin films as well as superlattices using
MgO as a substrate. Special emphasis are on Fe/V(001) and Cr/V(001) superlattices as these
can be grown as high quality single crystals on a routine basis. The use of high quality samples
ensured well-defined conditions for all the measurements.
In these experiments the hydrogen concentration is determined by the light transmittance
of the thin films. By changing the temperature and the pressure of the hydrogen gas, it is
possible to determine the thermodynamic properties of hydrogen in the samples, from the
obtained concentrations. Measurements of the electrical resistivity is used to increase the
accuracy in the measurements at low concentrations as well as to provide information on
ordering at intermediate and high hydrogen concentrations. The thermodynamic properties and
the electrical resistivity of VH are strongly affected by the choice of boundary layers. For
example, when hydrogen is absorbed in V embedded by Fe, Cr or Mo in the form of superlattices,
both the thermodynamic properties and the changes in the resistivity are strongly influenced.
The critical temperature and H-H interactions of hydrogen in thin V(001) layers are
found to increase with thickness of the thin films and superlattices. The observed finite size
effects resemble same scaling with the thickness of the layers as does the magnetic ordering
temperature. The results were validated by investigations of isotope effects in the obtained
thermodynamic properties. Close to negligible effects are obtained when replacing hydrogen by
deuterium, with respect to the thermodynamic properties. These observations are rationalised
by an octahedral occupancy in the strained layers, as compared to tetrahedral occupancy in
unstrained bulk. The octahedral site occupancy is found to strongly alter the diffusion coefficient
of hydrogen in thin V layers.
Keywords: thermodynamics hydrogen superlattice
Xin Xiao, Department of Physics and Astronomy, Materials Physics, 516, Uppsala University,
SE-751 20 Uppsala, Sweden.
© Xin Xiao 2016
ISSN 1651-6214
ISBN 978-91-554-9473-5
urn:nbn:se:uu:diva-275629 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-275629)
Dedicated to all hard-working
doctoral students at Uppsala University
List of papers
This thesis is based on the following papers, which are referred to in the text
by their Roman numerals.
I
II
Finite-size effects: hydrogen in Fe/V (001) superlattices
X. Xin, G. K. Pálsson, M. Wolff, and B. Hjörvarsson
Physical Review Letters 113, 046103 (2014)
Hydrogen in V: site occupancy and isotope effects
X. Xin, R. Johansson, M. Wolff, and B. Hjörvarsson
Submitted to Physical Review B
III
Inﬂuence of boundary on hydrogen absorption in V
X. Xin, S. Droulias, M. Wolff, and B. Hjörvarsson
In manuscript
IV
Proximity effects on H absorption in ultrathin V layers
G. K. Pálsson, A. K. Eriksson, M. Amft, X. Xin, A. Liebig, S.
Ólafsson, and B. Hjörvarsson
Physical Review B 90, 045420 (2014).
V
The inﬂuence of site occupancy on diffusion of hydrogen in
vanadium
L. Mooij, W. Huang, S. A. Droulias, R. Johansson, O. Hartmann, X.
Xin, H. Palonen, R. H. Scheicher, M. Wolff, and B. Hjörvarsson
To be submitted to Physical Review Letters
Reprints were made with permission from the publishers.
Contents
1
Introduction
2
Background and theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Energy of hydride formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Phases of vanadium hydride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
Electrical resistivity and ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Optical transmission and hydrogen concentration . . . . . . . . . . . . . . . . . . . . . . .
2.5
Statistical mechanics of MH system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Chemical equilibrium of hydride formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7
H-H interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Critical behavior of hydrogen in metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9
Hydrogen diffusion in metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1
Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2
The combined optical and electrical measurements set-up . . . . . . . . . 29
4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Thermodynamics analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Finite size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
Isotope effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
32
35
37
5 Review and summary of hydrogen in Fe/V, Cr/V, Mo/V and W/Nb
superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
Enthalpy dependency of hydrogen in superlattices . . . . . . . . . . . . . . . . . . . . . .
5.2
Thermodynamic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3
Effect of local strain ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4
Dependency of critical temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
41
43
46
49
6
51
Conclusion
..................................................................................................
.................................................................................................
Summary in Swedish
14
15
17
19
20
20
24
25
26
27
.......................................................................................
52
.............................................................................................
55
........................................................................................................
56
Acknowledgment
References
9
1. Introduction
Fossil fuels (oil, natural gas and coal) will still be the major source of the
primary energy in the coming years. However, these conventional energy resources including nuclear fuel, are non-renewable and need to be replaced in
the near future. Although the current energy policy of most nations will not
be changed in a short time, the environmental problem drives all these policies
toward decreasing the consumption of hydrocarbon.
Hydrogen is the most abundant element in the universe, even by mass fraction. Hydrogen is also very abundant element by atom fraction on the earth
surface in the form of compounds. It has become one of the most promising candidate for energy storage during the 21th century. In modern industry,
hydrogen is commonly produced from petroleum (hydrocarbon) and water.
To generate hydrogen in the ﬁrst place, one starts with breaking a chemical
bond. Let us take water as an example, one way to produce hydrogen is via
electrolysis. This reaction can be activated or catalyzed using certain metals
or complexes. About 5% of industrial hydrogen is produced by this method
[1]. There are many other methods to produce hydrogen from different resources, such as water dissociation at very high temperatures (around 2500 ◦ C
[2]) with catalysts, photocatalytic water splitting, hydrocarbon (or hydrogen
compounds) stem reforming and thermolysis etc. Therefore we believe the
hydrogen economy is in a great demand and could provide a new pathway to
changes in energy infrastructure. However more efﬁcient and safe technologies are still required to store hydrogen.
The most efﬁcient way to use hydrogen is a direct use [3, 4]. The chemical
energy of hydrogen combustion is many times higher than any other chemical
fuel and the only end product is water. A possible application for hydrogen
is in the sector of urban mobility and transportation. Today transportation
consumes about a quarter of world energy use, majority of which is based on
fossil fuels. This gives rise to local pollution which is of concern for residents’
health and environmental issues. Hydrogen, however, can be used in fuel cells
to produce electricity without harmful exhaust gases. Promoting the use of
hydrogen as a fuel is a way to improve pollution from vehicles. Using combustion engine or generating electricity through the fuel cell is possible method
to utilized hydrogen. To practically utilize hydrogen, problems on storage and
transports need to be solved ﬁrst. Compressed and liqueﬁed hydrogen storage are not ideal, due to low volumetric energy density and potential security
problems. Therefore materials which can absorb and release large quantities
of hydrogen rapidly are needed.
9
The technology for hydrogen storage can be based on physical (physisorption, compressed or liqueﬁed hydrogen) or chemical (compound) storage, depending on whether hydrogen stay molecular or form ionic bonds with the host
material. Metallic materials are most common for chemical storage. These
form binary hydrides, such as LiH [5], or more complex compound hydrides
like NaAlH4 [6] and LaNi5 H6 [7]. Beside the ionic hydrides, hydrogen has
been found to penetrate through different metals, without forming strong or
localized bonds, early in 1860s [8, 9]. Actually hydrogen dissolve into almost
all metals at some conditions, especially transition metals like vanadium, niobium, zirconium and tantalum crystals absorb large amounts of hydrogen [10].
Palladium can absorb up to about 900 times of its own volume of gaseous hydrogen at standard condition (room temperature and one atmosphere pressure)
[11]. In 1974 a MH storage system was constructed at Brookhaven National
Lab, with a total weight of 564 kg with 400 kg Ti (Fe, Mn) type hydride
alloy. Hydrogen is up to 1.6 weight percentage (6.4 kg hydrogen) could be
stored in this system. The ﬁrst transition metal hydride carrier was made
in 1976 [12]. Afterwards many large scale containers were built but market
demand was still weak. Metal hydrides (MH) have been applied in varying
ﬁelds. The department of energy in the United States started research in battery and ultra-capacitor technologies in 1976, which leads to the development
of Nickel Metal Hydride (NiMH) batteries, used in Hybrid Electric Vehicles.
Since 2000 a transition metal based AB2 -type alloy family has been chosen as
a new candidate for solid hydrogen storage. A ﬁber-wrapped hybrid container
with heat exchanger and MH power system was designed and a Prius vehicles
was demonstrated to run with hydrogen. The storage system comprised of two
vessels which each has a capacity up to 1.8 kg hydrogen. Hereafter updated
vessel with 3 kg hydrogen capacity was built, it has the same alloy under a
large charging pressure. Gravimetric density is mostly demanded for hydride
vehicles. Many containers with higher capacity of hydrogen were built but
using higher pressure is also required. Low gravimetric density is not an obstacle in many other ﬁelds, such as mechanical handling equipment, forklift,
stationary backup power and etc.
Using metal hydrides for hydrogen storage has many advantages. Hydrogen can be absorbed and released relatively fast and reversibly as it stays in
many metals. The operating temperature and pressure are also in a wide range.
Volume density of H in MH can be up to 1.5-2.0 times higher than liquid hydrogen depending on different metals. Hydrides can be classiﬁed as
1) Ionic hydride. Hydrogen atoms bound to most of the extremely electropositive metals of s-block on the periodic table as ionic hydrides, which is
also known as pseudohalide. These hydrides are commonly non-volatile and
non-conducting crystals, such as LiAlH4 , NaBH4 and LiH are mostly used for
hydrogen storage and as catalysts.
2) Interstitial hydride. Besides the hydride gap regions, d-block and f-block
elements can form metallic (interstitial) hydrides. These could be metals or
10
alloys with metallic bonds. The Metal-Hydrogen bond can in principle cleave
to produce a proton, hydrogen radical or hydride [13]. They are usually conductors and non-stoichiometric and H are relatively free to diffuse in these
materials. These hydrides are usually used in synthesis and fuel cells. But
these interstitial hydrogen atoms can cause embrittlement during application.
3) Covalent hydride. P-block elements predominantly bind as covalent hydrides. They are mostly in the form of discrete molecules and some are polymers. The stability of these hydrides increases with increasing electronegativity. The covalent hydrides are widely used in organic synthesis.
At standard temperature and pressure pure hydrogen forms diatomic gas
as H2 molecules. When metals absorb hydrogen atoms, it requires the cleavage of the molecular bond. The energy required to break the chemical bond
is called bond dissociation energy. It is the difference between the bond energy of H2 and the zero-point energy of the molecule: Ed = E0 (4.748 eV)-EV0
(0.372 eV)=4.376 eV [14]. This energy needs to be either applied to dissociate
H2 , an energy demanding process, or it is avoided by the using catalysts. The
dissociation is certainly isotope dependent since the frequency contributes to
the zero point energy. Comparing with other elements hydrogen has smallest
radius [15], smallest nuclear mean mass and medium value of electronegativity, which also makes it form various chemical bonds with many elements.
For transition MH, the 1s orbitals of the hydrogen atoms hybridize with the
d-orbitals of the host metallic atoms and small H atoms can enter narrow interstitial sites and move around. In a MH valence electrons concentrate around
the proton (H+ ), the electronic reactions produce both short-range and longrange interactions between the dissolved hydrogen. The fact is that the hydrogen atoms are effectively neutral allowing the high mobility. In a MH system,
vibration energy of hydrogen must be considered as it affects the enthalpy of
solution. This energy Ev of interstitial H atoms is much higher (50-200 meV)
than that of metal lattice itself. This energy inﬂuences the phase stability and
the preference of the type of site occupancy.
Many interstitial metals (including some inter-metallic compounds and solidsolution alloys) can quickly absorb and release certain amounts of hydrogen
(up to several times of the number of its host atoms) at room temperature and
atmospheric pressure. The most interesting question for these hydrogen storage materials is how to promote the amount of uptake, which is known as the
enthalpy in thermodynamics. The left panel of Fig. 1.1 shows enthalpy of
hydrogen for d-band metals (interstitial hydrides). A linear relation between
the enthalpy and binding energy density of metals is found [16]. This relation
is also part of the hydrogen curve in the right panel of Fig. 1.1, which also
shows the energy for ’solution’ (embedding energy) for several light elements
with respect to electron density for the given model. From this ﬁgure we ﬁnd
hydrogen uptake is strong exothermic under medium electron density (oxygen
forms strong ionic bond with metals.) and it is an ideal model for its simplicity
in theoretical calculation.
11
Figure 1.1. (Left) Relation between heat of solution for hydrogen and calculated binding energy density [16]. ΔH̄∞ is the enthalpy at inﬁnite dilution, ΔE is the band structure energy and Wd is the width of the d band of the host metal, and R j is the distance
separating a given hydrogen atom from the atom j in the ﬁrst nearest neighbour metal
atom shell. (Right) Embedding energy of different element in a homogeneous electron
gas with electron-gas density [17].
In order to obtain the information of capacity and the cycling time of the
hydride, both the thermodynamics and kinetics of this process need to be understood. For example Mg can store large amounts of hydrogen but the absorption takes very long time. In addition MH causes signiﬁcant volume change,
heat effect, thermal driven compression, oxidation and etc. These side effects
as well as surface material (catalyst or protection) should be considered when
choosing a material. To explore routes for obtaining better performance and
with respect to hydrogen uptake, laser ablation, vapour condensation, sputtering and ball milling have been used [18]. Reducing the grain size can dramatically change their physical and chemical properties. It is found that not
only the size of host materials but also the strain ﬁeld (compressive or tensile)
and intermediate materials can affect the type of occupancy, electrical as well
as optical scattering and effective medium approximating (For example, thin
metallic ﬁlms are more sensitive to boundary conditions than bulk.). Growing
single or bilayer thin ﬁlms on substrate can change in plane strain state, restrict
the occupancy and change the induced expansion due to hydrogen. This can
eventually change the electronic density of metal and the enthalpy of hydrogen solution. For example, by growing Mo/V or Fe/V superlattice, hydrogen
solubility can be alternated to be more or less exothermic comparing with bulk
V. These artiﬁcial materials could also have different kinetic properties. A material with lower solubility but better mobility or thermal conductivity is often
12
needed. However to perform research on this topic, the access to thin ﬁlms
of high quality and low defect density are of utmost importance. At least we
always seek for a safe, compact and technically ﬂexible hydrogen storage. In
this thesis thin ﬁlms and superlattices are used as model system to investigate
the inﬂuences of conﬁnement, strain state etc. This is relevant to the future use
of hydrogen storage.
13
2. Background and theory
H
Gas
Sub
Pd
V
ΔK
Surface
0
H2
physisorption
chemisorption
]
Figure 2.1. Interpretation of hydrogen absorption procedure
To better understand the process of H2 absorption metals, we can take the
H2 -palladium-vanadium system as an example and set the potential of H2 as
the reference of energy level. As illustrated in Fig. 2 the process can be divided
into several steps
1) Physisorption: H atom in the vacuum has the highest potential level. It
can be trapped in the physisorption at surface.
2) Chemisorption: An dissociation energy (Ediss =0.45 eV) is needed to
break the covalent bond of the trapped H2 and it can diffuse forward. H2 splits
spontaneously at surface of palladium, nickel and platinum [19]. The energy
of chemisorption for hydrogen in Pd is -0.87 eV [20]. The chemisorption is
more energetic favorable than physisorption.
3) Subsurface site: The energy barrier of subsurface in Pd is 0.65 eV and
heat of solution is 0.52 eV [20].
4) Diffusion through palladium: It is more energetic favorable for hydrogen
to reside in V as compared with Pd. Diffusion barrier in bulk palladium is 0.49
eV (heat of solution is -0.20 eV) [20].
14
5) Diffusion in vanadium: Diffusion barrier in vanadium is 0.05 eV [14] and
the heat of solution is -0.31 eV [21]. V is more energetic favorable than Pd in
this case as mentioned above. At lower pressure hydrogen is predominantly
found in the V layers while it is much less in Pd.
2.1 Energy of hydride formation
In contrast to kinetics, thermodynamics only depends on the initial and ﬁnal
states and it is independent of any transition state. When the hydrogen absorption procedure is viewed as a chemical reaction: M + 0.5xH2 → MHx , the heat
of solution can be described as the change in energy [22]
1
(2.1)
ΔE = EMHx − EM − xEH2
2
where EMHx , EM and EH2 are the total energy of MH, host metal and the
dissociation energy of hydrogen molecules respectively. This is based on a
one-electron consideration: approximating EMH − EM = ΔE, the difference
between the summation of the band energies of the metal hydride and the host
metal. For a stoichiometric hydride x = 1, it can be written as [22]
MH M MH M ) − εLB ) + nMH
− εd ) + εFM
(2.2)
EMH − EM = 2( εLB
d ( εd
M
MH are the lowest bands of hydride and metal, εdMH and εdM
εLB and εLB
represent the shift of the average energies of the occupied d states lying above
lowest band in the metal and the hydride, nMH
is the number of d electrons in
d
the hydride and εFM is the approximation of the contribution of the additional
electron per unit cell due to the H atom which is added at the Fermi energy.
Because εFM is based on the absolute energy scale, EMH − EM depends on the
choice of zero point energy of host metal.
Effective Medium Theory is another method to describe enthalpy of hydride formation. It is a simpliﬁed method to calculate the total energy of low
symmetry systems [23, 17, 24]. In this method host metal has been replaced
with homogeneous electron gas. The electron density of the host metal is used
as the only primary parameter for: heat of solution, activation barrier of diffusion, trapping to interstitials, relaxation of metal atoms surrounding the H
atoms, H-H interactions, conﬁguration of hydrogen in metal (at surface and
open defects). Again the heat of solution is deﬁned as [23]
ΔE = ΔEhom (n0 ) + ΔEcore + ΔEhyb + ΔEinh
(2.3)
ΔEhom (n0 ) is the change in energy depending on the density of homogeneous
electron gas n0 . It is shown in Fig. 2.2 as the whole process of a hydrogen atom
entering the lattice. The rest of this equation are the correction terms: ΔEcore is
the repulsive interaction of hydrogen with the metal ion core electrons. ΔEinh
15
Figure 2.2. Schematic illustration shows how a hydrogen atom interacts with metal
[23]. The path of a hydrogen atom travels from outside the surface until the inside of a
lattice is shown in (a). The electron density corresponding to the full path is shown in
(b). The effective energy, calculated based on this electron density, is shown in (c). It
has to be pointed out that the minimum in this energy is outside the surface where the
electron density is not the lowest. The small peak corresponding to the lowest electron
density is caused by the vacancy in the lattice.
16
is the inhomogeneity of the metal. ΔEhyb is the hybridization between the
hydrogen state s and different metal states for example d electrons. Effective
Medium Theory connects electron density to the enthalpy of hydrogen uptake.
Changes in volume, strain, interface regions, etc. can all be discussed in the
context of changes in electron densities.
2.2 Phases of vanadium hydride
Interstitial MH exists over extended ranges of nonstoichiometric compositions. The interstitial sites are randomly occupied at high temperatures and
in the form of certain regular ways at lower temperatures. Metal lattices undergo structural changes to accommodate a larger number of H atoms. Therefore several phases exist in a MH binary system. These kinds of MH systems
can be understood analogous to the H2 system with gaseous, liquid and solid
phases [26].
The phase diagram of hydrogen in transition metals such as V (BCC), Pd
(FCC), and Ti (HCP) etc. are usually presented as 2D projections of temperature versus composition. Usually these phase diagrams are spinodal decomposition systems including both single and coexist phases, have many ordered
phases at low temperatures, are isotope dependent and have different melting
points than the pure metal.
Fig. 2.3 shows the phase diagrams of V-H (D) systems. As described in
[14], there are ﬁve phases in vanadium: (α) disordered solution, probably
tetrahedral; (β ) V2 H, monoclinic, octahedral Oz , c/a = 1.1; (γ) VH2 , FCC dihydride phase, a = 0.42 nm; (δ ) V3 H2 , low temperature ordered phase, monoclinic, O occupancy; (ε) V2 H, tetrahedral, 1/2 probability that all 4Oz are
ﬁlled. ε+β =β ’.
Type of hydrogen occupancy (T or O) in the lattice is determined by the
electron density in the lattice. Growing a thin ﬁlm on a substrate is one way to
change the strain state in order to affect electron density in material. This can
be obtained in superlattice as illustrated in Fig. 2.4. The right graphs show the
location of two possible sites. The BCC crystal can accommodate up to 6 or
Figure 2.3. Phase diagram of VH2 and VD2 [25].
17
Figure 2.4. Schematic of the superlattice Fe1 V7 design [27]. 25 bilayers are grown as a
single crystal on MgO substrate, with Pd as the capping for protection and activation.
The right side graphs show two types of interstitial occupancies of hydrogen (red
spheres): Oz and Tz . Light blue and dark blue spheres symbolize the V atoms and Fe
atoms, respectively.
3 H atoms at T or O interstitial sites per unit cell respectively. T and O can
be divided into three groups, depending upon the direction of the rotational
symmetry axis. In Fe/V (001) superlattices, the V layers are under bi-axial
compressive strain and makes Oz site more favorable than T site [28]. This
occurs even at low concentrations while in bulk V the similar β phase exists at
higher concentrations. This change in site occupancy also results in different
thermodynamic properties.
Phase of hydrogen in metals such as Nb, V or Ta can be recognized as diagrams topologically similar attractive as gas-liquid-solid (g-l-s) or gas-isotropic
liquid-anisotropic liquid [26, 29, 30]. The elastic interaction of hydrogen via
long range displacements has been identiﬁed as the cause of the phase transition of MH. In most cases the triple point Tt is lower than the critical points Tc .
In MH the shift of Tt beyond the Tc could be understood as the gas condenses
into an anisotropic liquid. Comparing MH phase diagram with solid-liquidgas diagram, α phase corresponds to the gas, α to the liquid and β to the
solid phase. The only difference between α and α phase is the H concentration. Phase diagrams for Nb-H, Ta-H, V-H and are all experimentally available, these are different with respect to the ordering and critical temperature
Tc . For example Tc of VH and TaH are below the triple point Tt , while it is
above Tt for NbH [31, 32, 33, 34].
It is seen that in vanadium hydride the critical behaviour takes place below triple point and the ordered β phase may exist at lower concentrations.
Vanadium is recognized as an ideal material for the study the thermodynamics and critical behaviour. This is partially based on the different site occupancy in order and disorder phase, and the possibility to alter the site occu18
pancy with strain state. Most phase diagrams of transition metal hydrides
are complicated. However H-Pd (fcc) only has α and α phases in the diagram, it is simpler than that of the bcc metals. Thus H-Pd has been studied
as a model system for MH. One way to characterize phases of MH is to analyze pressure-concentration and electrical resistivity-concentration isotherms.
These isotherms have plateau region at the triple point in Pd-H and Nb-H,
while for Ta-H and V-H the temperature of critical isotherm is lower than the
triple point: Tc is hidden below α-β or α -β phase transition.
2.3 Electrical resistivity and ordering
At non-zero temperature, atomic vibrations (phonon) cause signiﬁcant change
in the atomic distance. Phonons as quasiparticles interact with and scatter
electrons. Higher temperature increases this scattering and reduce the mobility
of electrons. According to Matthiessen’s rule, the electrical resistivity of metal
with impurities is given by
ρ=
1
ne
1
1
+
μH μlattice
(2.4)
where n is the number of electrons, e is elementary charge and μ is the mobility. H atoms can be treated as impurities in a MH system, thus the mobility
and concentration of hydrogen affect the resistivity of MH.
It is reported that in MH the electrical scattering strength is dependent of
concentration [35]. This makes it possible to quantify the concentration dependence of resistivity. However at low concentrations (inﬁnite dilute), disordered interstitial H scatters conduction electrons, which is the only contribution to the excess resistivity. At higher concentrations, H become more and
more ordered due to the blocking of nearest neighbours, thus resistivity possibly decreases. In other words, hydrogen atoms disturb the periodic variation
potential within the lattice. A simple model is used in some cases (lower concentrations in bulk V) [36]: Δρ = Kc(cmax − c), where Δρ is the excess of
resistivity and c is the atomic ratio H/V . When c → 0, the excess resistivity
becomes linear to concentration. In Nordheim’s theory, the electrical resistivity depends on the number of conduction electrons, the lattice structure and
the temperature. It is determined by the nature of ordering of the atoms. The
electrical resistivity of H in metal not only denotes the degree of ordering but
also reveals the phase transitions. The phase boundary can be inferred from
the inﬂection point of the curve of resistivity versus temperature at a given
concentration. This point is reported to be the initiation in the solid solution
during cooling and a clear thermal hysteresis is found on the curve [37].
19
2.4 Optical transmission and hydrogen concentration
When light travels through a material, the intensity of the transmitted beam is
reduced by reﬂection, scattering, and absorption. The total intensity reduction
(transmission) ΔI is therefore expressed as
ΔI = ΔIabs + ΔIre f + ΔIsca
(2.5)
In this equation the reduction of light mainly depends on the absorption of
the material at normal incidence. The absorption of electromagnetic radiation
within the material can be understood as the energy of photons taken up by the
absorber, or mostly by its electrons. The light can be absorbed only when the
incident energy of the beam equals the energy difference between the ground
and the excited state of the material. The absorption of light in wave propagation is known as attenuation and the optical coefﬁcient is material dependent.
For a metal without hydrogen, according to Beer-Lambert’s law, the relation
between material property and optical intensity is given
I0 = I eδ ρm l
(2.6)
where I0 and I are the transmitted and incident intensity, δ is the attanuation
cross section, ρm is the concentration (density) of the metal and l is the thickness of the substance. Similarly when this substance is loaded with hydrogen
under the same environment, we obtain
Ic = I eδ ρmh l
(2.7)
where the concentration of the hydride ρmh = ρm + ρm c. Combine the two
equations above, we obtain a relation between optical transmittance and hydrogen concentration c
Ic
δ · ρm · l · c = ln( )
(2.8)
I0
This approximation simply extracts the number density for the metal hydride,
although the induced H changes the electronic structure of the hydride. It is
observed that for a given wavelength the optical transmittance changes monotonically and continuously with the applied hydrogen partial pressure [38].
These constants can be determined by comparing the optical measurements
with isotherms. Wavelength with relative large change with hydrogen pressure
should be chosen to optimize the signal for a measurement. Although volume
change per hydrogen is constant in many interstitial MH, the absorbance α
may affected by hydrogen concentration and wavelength of the light.
2.5 Statistical mechanics of MH system
The total energy of a MH system is found to be classiﬁed into the H-H interactions and potential energy (binding energy) between H and host metallic
20
atoms. The Hamiltonian of this MH writes [14]
H=
n
1 n n−1
u
+
i
j
∑ pi
∑∑
2 i=1
j=1
i=1
(2.9)
where ui j is the elastic H-H interactions i and its effective neighbour j, pi is
the potential energy for this atom and n is the total number of H atoms in the
system. According to Mean Field Approximation, the energy of solution for a
single H atom can be expressed as
hi =
n−1
1
ui j + pi ≈ nui j + pi
2
2
(2.10)
This expression helps to understand and analyze the enthalpy, including binding energy and elastic interactions, as well as phase transition and stability.
The total energy of MH system can be also described as the summation of
energies of different modes. Such as conﬁgurational energy which is the portion of energy associated with the various attractive and repulsive interactions
between atoms, translational energy which is the portion of energy associated
with the motion of the centre of the atom etc. If we assume all the modes are
completely uncoupled and uncorrelated, which means all of them are independent, the partition function can be written as
Z = ∏ Zi = Zc Zv Ze Zr Zn Zt
(2.11)
where Zc , Zv , Ze , Zr , Zn and Zt are the partition function of conﬁgurational, vibrational, electronic, rotational, nuclear and translational mode. To introduce
the relation between chemical potential μ and the number of H atoms (concentration c (H/M)), we use the grand canonical ensemble for Z. In the conﬁgurational mode, the MH system is assumed to be composed of fermions, there
are only two microstates for the orbital according to Pauli Exclusion Principle: c = 0, with lowest energy E0 = 0 or c = r with the total E1 = ε, ﬁnally
the conﬁgurational partition function results in
1
μc c/r − E
μc − ε
= 1 + exp
(2.12)
Zc = ∑ exp
kT
kT
c/r=0
where c is the ratio of hydrogen atoms to the metal atoms, r is the ratio of available of interstitial sites to metal atoms, k is Boltzman constant and μc is the
conﬁgurational chemical potential. Thus the relation between concentration c
and μc is described as
−1
μc − ε
∂ ln(Zc )
(2.13)
= exp −
+1
c/r = kT
∂μ
kT
The conﬁgurational entropy is easily written. In the grand canonical ensemble
Z can be understood as probability and this could also give the same result for
21
S [meVK-1 H atom]
0.5
0.0
-0.5
0.0
0.1
0.2
0.3
0.4
Concentration [H/V]
Figure 2.5. Entropy of Fe1 V7 superlattice (open circles) and its ﬁtting based on equation 2.14: Δs = k ln((r − c)/c) + const.
sc
r−c
sc = k ln
c
N!
= k ln
n!(N − n)!
(2.14)
Fig. 2.5 shows the measured entropy (open circles) versus concentration.
As seen in the ﬁgure the data is ﬁtted by s = k ln((r − c)/c) + snc (equation
2.14), when the other entropies are considered not concentration dependent.
Once hydrogen concentration is measured, sc can be well analyzed. Another
very useful mode of partition function Zν should be introduced when we quantitatively discuss the isotopic dependence for thermodynamics. The vibrational motion can be uncoupled from other degrees of freedom of the system,
the partition function Zν would therefore only depends on the vibrational degrees of freedom
∞
n
hν (l + 1/2)
Zv = ∏ ∏ ∑ exp −
(2.15)
kT
i=1 j l
where ν is vibrational frequency of hydrogen, h is the Planck constant, j is
the index of the mode for harmonic oscillator and l is the quantum number for
each energy level of the j mode. To simplify this expression, the vibrational
partition function for a single hydrogen atom of independent three dimensional
harmonic oscillators, in a bosons composed system, which results in
⎞(3c)
hν
⎜ exp − 2kT
⎟
⎟
Zv = ⎜
⎝
hν ⎠
1 − exp −
kT
⎛
22
(2.16)
Phase
H [meV]
D [meV]
T [meV]
Gas [39]
535
387
309
516
371
340
Gas
α(Tet)
1802 × 128
1272 × 90
1042 × 74
V X0.012 (300K)
1702 × 106
1202 × 75
982 × 61
V X0.5
1802 × 113(498K) 1232 × 82 1042 (425K) × 65
β (Oct)
542 × 220
382 × 156
312 × 127
2
2
V X0.5 (295K)
54 × 220
47 × 155
362 × 127
2
2
V X0.5 (80K)
50 × 223
56 × 230
392 × 164
3
3
γ
165
117
953
Table 2.1. Vibrational energy of hydrogen as molecules and atoms. It may be classiﬁed into soft and hard mode in the MH system [14].
From this we obtain the expressions of chemical potential and entropy
∂ (ln Zv ) 3
hν
μν = −kT
= hν + 3kT ln 1 − exp
(2.17)
∂c
2
kT
∂ (ln Zv ) 3hν
=
+
∂ c∂ T
2T
1
3hν
(2.18)
hν
T
exp
−1
kT
however the partial molar entropy is the difference between that of MH and
the standard entropy of the molecular hydrogen in the gas phase: Δs = Δssol −
Δs0gas , therefore a reference of s0gas need to be chosen. Table. 2.5 summarizes
some hydrogen (including its isotopes) frequencies at different condition. ν
decreases with isotopic mass and it depends on temperature, occupancy type
and concentrations. Occupancy type (octahedral or tetrahedral) also affects
the vibrational mode and degrees of freedom.
Finally, chemical potential can be expressed as [40]
hν
3
c
μH = h0 +uc+ V̄H p+ hν +3kT ln 1 − exp
+kT ln
−T snon
2
kT
r−c
(2.19)
where h0 can be considered as the binding energy between MH, u is the effective H-H pair interactions (u = f (c)), V̄H is expansion vector, and snon is the
remaining entropies corresponding to electronic, rotational, nuclear and translational ones: snon = se + sr + sn + st . In a phase diagram of MH system, in the
vicinity of the critical point, chemical potential does not change with concentration anymore and it must be inﬂection point. The critical point should fulﬁl
this criteria [40, 14]
∂ 3μ
>0
(2.20)
∂ c3
sv = kT
∂ 2μ
∂μ
=0
=
∂c
∂ c2
(2.21)
23
1
μ α = μ β = μ gas
(2.22)
2
We combine equation 2.21 and 2.19, the expression of critical temperature is
obtained
uc(r − c)
(2.23)
Tc = −
2kr
This approximation indicates that the critical temperature only depends on the
H-H attractions.
2.6 Chemical equilibrium of hydride formation
The formation of MH takes place in the system including H2 gas phase and
the lattice. If the gaseous hydrogen is considered as an ideal gas, its chemical
potential can be expressed as
p
(2.24)
μH2 = kT ln
− Ediss
p∗ (T )
where Ediss is dissociation energy of H2 and p∗ is the reference pressure. When
a H2 -MH system reaches a chemical equilibrium, chemical potential of H atom
can be obtained from that of H2 in the gas phase: μH = 0.5μH2 . To avoid Ediss ,
we introduce the change in chemical potential
1
1
1
p
∗
ΔμH = ΔμH2 = (μH2 − μH2 ) = kT ln
(2.25)
2
2
2
p∗
At a given temperature, hydrogen concentration is monotonically changing
with hydrogen partial pressure. In equation 2.19, when the concentration c
is very low, many terms such as H-H interactions u, volume vector V̄ and so
forth are approximated to be zero. Thus we have
T Δs − Δh
p
exp
(2.26)
c=
p∗
kT
In this case of inﬁnite dilute, enthalpy Δh and entropy Δs are constant as well.
√
Therefore concentration has a linear relation with p, it is also known as
Sievert’s law. This relation holds for many MH formations.
With increasing hydrogen concentration, H-H interactions turn from attractive to repulsive, this is found at the minimum in enthalpy vs. concentration
plot in many metals such as Pd, Nb, Ta, V, Y and Sc [14]. Heat of solution
depends on both MH binding energy and the inter-atomic interactions [41].
This interaction includes elastic and electronic contributions
u = uela + uele
24
(2.27)
Any two H atoms are not able to get closer than 0.21 nm [42]. This also
indicates that the pairwise interaction must have optimum point at a certain
concentration. Located H atoms start to block nearest sites mutually, then it
extends to the second then to the third and goes on. With the increasing of concentration this repulsive interactions makes the MH system become ordered.
2.7 H-H interactions
In order to accommodate H atoms, a structural change has to take place in the
lattice. This change can cause a displacement of nearest neighbor atoms and
can even distort the structure several unit cells away. The local displacement
ﬁeld created by hydrogen decreases as 1/r2 . This displacement is determined
by the symmetry of both crystal structure and type of interstitial occupancy.
In a certain lattice, the elastic dipole tensor Pi j quantiﬁes the strength and the
symmetry of the long range displacement ﬁeld, as analogous to the electric
dipole moment in an electric ﬁeld. In a elastic displacement contour, Pi j characterizes the displacement vector ui j . Interaction between two H in a long
range elastic ﬁeld is understood as elastic dipole-dipole interaction. Higher
multi-pole interactions only exist in short range ﬁelds.
Elastic dipole-dipole interaction is more difﬁcult to measure than the mean
interaction energy. First we use a non-directional dependent P to describe
Pi j by a δ function Pi j = Pδi j . Then we use εiloc
j to describe the local strain
ﬁeld, which results from all the dipoles except the one considered. Thus the
interaction energy ui j of one elastic dipole with this strain ﬁeld is given as
[43, 44]
ui j = −Pi j ∑ εiloc
j
(2.28)
The local strain ﬁeld εiloc
j is smaller than the mean strain ﬁeld for a homogeneous distribution of hydrogen atoms. It is the difference of the change in
volume between ﬁnite and inﬁnite elastic ﬁeld
∑ εilocj = ∑ εifjin − ∑ εi j∞ =
ΔV
V
−
f in
ΔV
V
(2.29)
∞
Finally strain ﬁeld is related to a volume change ΔV /V , which also depends
on the dipole tensor constant value P
ΔV
V
=
f in
ρP
κ
(2.30)
25
where atomic density per cell ρ = 2c/a3 (c is hydrogen concentration a is
lattice constant) and κ is the isothermal compressibility. Therefore we have
the expression of H-H interaction
u=
γP2 c
κa3
(2.31)
γ is the factor for the relation between local ﬁeld and mean ﬁeld and depends
on the elastic constant. This equation only holds for BCC metal due to the
expression of atomic density.
The elastic interaction contributes to the heat of hydride formation. The
interstitial H atoms produce lattice dilatation, which interacts with the strain
ﬁeld of each H atom, and reduction of the heat of solution is proportional to
hydrogen concentration [45]. From the whole process of derivation we can
ﬁnd that the mean interactions u depend on the boundary (surface) condition
but not the local conﬁguration of H atoms, as it is called mean ﬁeld contribution. For a repulsive interaction, the elastic ﬁeld becomes more and more
negative. Elastic interaction dominate the attractive interaction of hydrogen
atoms in metal, this determines the phase transition. This elastic interaction
plays an important role in other solid state systems or magnetic ﬁeld.
2.8 Critical behavior of hydrogen in metals
The free energy can be expanded in a Taylor’s series around the critical point,
so the chemical potential is given by
c − cc T − Tc
c − cc 3
μ − μc
=A
+B
+ ...,
kTc
cc
Tc
cc
(2.32)
where μc is the chemical potential at the critical concentration for different
temperatures, A and B are constant parameters. If the mean ﬁeld approximation is appropriate for this system, the general expression based on the mean
ﬁeld exponents
(μ − μc )/(kTc )
=f
[(T − Tc )/Tc ]α+β
(c − cc )/cc
[(T − Tc )/Tc ]β
(2.33)
where γ=1 and β =0.5, the static scaling relations can be deduced. It means all
the isotherms should fall on one curve. By extracting these parameters from
experimental data, one can determine the universality class of the transition
and assign the system to an accompanying lattice gas model.
26
H-H interaction is crucial for understanding both thermodynamics and kinetics of a MH system. It can be linked to the chemical potential and compressibility κ of MH [43]
∂p
−1
2∂μ
(2.34)
= ρm c
κ = ρm
∂ρ
∂ c T
where ρm is the number of metal atoms per unit volume. As we know the
compressibility follows critical behaviour with an exponent γ. We can link
∂ μ/∂ c to the critical temperature [43]
T − Ts γ
∂μ
(2.35)
∝
∂c
Ts
Where Ts is spinodal temperature. From this expression one obtains T = Ts
if ∂ μ/∂ c = 0. Thus by plotting (∂ μ/∂ c)1/γ vs. T , the spinodal temperatures
can be extracted.
2.9 Hydrogen diffusion in metals
Understanding diffusion of hydrogen is essential to hydrogen storage. Large
efforts have therefore been made to study the diffusion of hydrogen in metals
[46, 47, 48, 49]. The diffusion of interstitial hydrogen is interpreted using
transition state theory and it is understood as thermal activated hopping. If it is
assumed that H vibrations are decoupled from phonon of solid, the diffusivity
can be expressed using standard statistical mechanical arguments [47]
hνi
3
∏i=1 νi f
Δh − T Δs
2kT
exp −
(2.36)
D=
hνtr, j
kT
2
∏ j=1 νtr, j f
2kT
where f(x)=sinhx/x and νtr and νi are the H vibrational frequencies for the
interstitial and transition state respectively. The diffusion coefﬁcient is isotope dependent due to the difference in mass. Finally the kinetics of MH is
dominated by the zero point energy (ZPE) of H atoms, and it is affected by
the interstitial occupancy, lattice spacing and the electronic structure. When H
concentration c, position x and time t are measured, the diffusivity D can be
obtained [50]. This makes it possible to study hydrogen diffusion by watching
light transmission.
The diffusion is usually understood as the discrete jumps between interstitial sites in the lattice. Neutron scattering and NMR have been used to
determine hydrogen diffusion. Hydrogen concentration proﬁle can be use to
model the diffusion in materials. Light transmission has been used to watch H
diffusion in thin ﬁlms [50].
27
The ﬁlm surface is made impermeable to hydrogen by depositing a layer of
amorphous aluminum oxide, leaving a small opening rectangle as a diffusion
window. Besides the oxide, the design of the sample is seen in Fig. 2.4. More
details about the sample can be found elsewhere [50, 51, 52]. The whole set up
is similar to the one of thermodynamic measurements [27]. The transmitted
light is registered by a CCD camera allowing the registration of H concentration, time and position. The diffusion is mathematically described by error
function. Given the boundary condition in this case, the diffusivity is obtained
[50]
∞
c(x,t) = c0 ∑
k=1
2(k − 1)L + x
2kL − x
√
√
erfc
+ erfc
(−1)k−1
2 Dt
2 Dt
(2.37)
The initial concentration c0 and the distance from the end of the active area L
are measured independently. With the diffusion coefﬁcient activation energy
can be obtained: Ea = kT ln(D0 /D), in order to understand the barriers among
the interstitials.
Diffusion in superlattices are expected to be different than that of bulk
metal. The change in site occupancy, interfaces and electron density may affect the diffusion coefﬁcient and activation energy. For example the activation
energy in thin ﬁlm is close to that in bulk at lower concentrations while it becomes larger than that in bulk as the concentration increases [52]. Hydrogen
diffusion in Fe/V superlattice is found to be vastly lower than thin ﬁlms due to
the different site occupancy and lattice distortion. The inﬂuence of the change
of isotope on the diffusion can also be readily determined with this method
and has been recently done in Fe/V superlattices. [51].
28
3. Experimental
3.1 Sample preparation
In the course of this project various thin ﬁlms are used, including monocrystalline V thin ﬁlms and Fe/V and Cr/V superlattices. All samples have
been grown with an in-house, multi-target, magnetron sputtering setup. The
growth chamber has a base pressure of 1 × 10−8 Pa and uses high purity argon
(> 99.9999% Ar) as sputtering gas as well as high purity targets. A detailed
description of the used growth process can be found elsewhere [53].
All thin ﬁlms are grown on the highest possible purity, single-crystal, MgO
(001) substrates of 10 × 10 × 0.5 mm3 [54]. The substrates for optical measurements are double-sided polished to allow optical transmission of light
through the entire sample. MgO is selected as it exhibits a lattice parameter
thus a distance d between the planes in the (110) diof aMgO =0.421 nm and √
rection of a110 = aMgO / 2=0.298 nm [55]. This is close enough to the lattice
parameters of vanadium (aV =0.303 nm), iron (aFe =0.287 nm) and chromium
(aCr =0.288 nm) to allow epitaxial growth of these metals on MgO [27].
To prevent post-growth oxidation each sample is covered by a 5 nm thick
palladium capping layer. In addition to the protection from oxidation, palladium is catalytically active for hydrogen dissociation at room temperature
and above. It also has a signiﬁcantly higher enthalpy of solution compared to
vanadium, which results in it having negligible effects on the outcome of the
experiments. Post-growth and prior to hydrogenation all samples have been
thoroughly characterized by means of x-ray scattering techniques. A description of the procedure can be found elsewhere [56].
3.2 The combined optical and electrical measurements
set-up
We utilized an in-house built setup, called STORM, for simultaneous optical
transmission and resistance measurements of thin-ﬁlm samples. A detailed
description can be found elsewhere [57]. It has been further modiﬁed as part
of this thesis work. The setup was designed to allow the determining of the
conﬁguration and composition of H in thin ﬁlms by the scattering of electrons
and photons respectively.
Fig. 3.1 shows the experiment setup. A combination of a backing pump and
a turbo molecular pump can lower the pressure to 1 × 10−7 mbar ultra-high
29
B
A
C
C
E
E
B
D
A
D
F
G
H
I
J
Figure 3.1. (Left) Optical devices and chambers. A: LED light source (driven by
a DC power supply with incoming TTL pulse signal). B: Detector for optical intensity monitoring (Thorlabs PDA10A ampliﬁed silicon detector). C: Detector for
transmission (Thorlabs PDA10A). D: Detector for reﬂectivity (Thorlabs PDA10A).
E: Mounted sample, in contact with a four-terminal resistance sensor and a thermocouple (K-type). (Right) Controlling terminals. A: Power supply of the heating units
(Delta Electronica 030-10,measuring temperatures of three spots of the set up). B:
temperature controller (Lakeshore 331 ). C: Vacuum controller (Pfeiffer PKR251)and
a cold cathode (CC) pressure gauge. D: Controller for pre-vacuum. E: Lock-in ampliﬁer for LED controlling (Stanford SR844). F: controller for auto pressure valve. G:
Source-meter for four-terminal sensing. H: Keithley 2002 multi-meter for 1000-Torr
gauge readout of Inﬁcon SKY Capacitance Diaphragm Gauges (0-1.33 mbar). I: Multimeter (Keithley 2002) for 1-Torr gauge readout of Capacitance Diaphragm Gauges
(Inﬁcon SKY, 1-1333 mbar). J: Lock-in ampliﬁer for detectors (SR830).
30
vacuum, UHV. A residual gas analyzer (RGA) is connected before the turbo
pump to detect and quantify impurities in the experimental setup. A gas supply
bottle ﬁlled with Fe-Ti powder is linked to a motorized valve. H2 is ﬁltered
through a NUPRO puriﬁer before it is absorbed by the powder. The hydrogen
loading is provided directly through a 1-L buffer chamber for safety and it is
limited to 1.3 bar. This motorized valve is controlled either automatically by
the acquisition software or manually. A bellow, set between the pumps and the
sample chamber, minimizing the transfer of vibrations. The rest of the tubes
and chambers are made of stainless steel. The sample environment includes
a machinable glass-ceramic (Macor) sample holder inside a quartz tube. This
tube is connected to the chamber and a wire feed-through via a T-manifold.
The hydrogen pressure is measured in the range of 10−3 − 103 mbar. The
temperature can be controlled from room temperature up to 280 ◦ C. The controller displays three values of temperature: (i) the surface temperature of the
sample read by a K-type thermocouple, (ii) the temperature between the quartz
tube and the 30 Watt columnar heating jacket, (iii) the target value set on the
computer. Four-terminal sensing wires (Four Probe Resistivity detection) directly contact the surface of the sample. 1 mA current is applied through the
two outermost wires while the inner two wires measure the voltage drop. The
synchronized optic devices are shown in left panel of Fig. 3.1. Frequency of
the incoming beam is set on computer. The signals of the transmitted intensity
and the monitor intensity are read by the two lock-in ampliﬁers. There is a
spectra mode, LED and detectors can be replaced by Thorlab OSL1 halogen
white light source and spectrometer (AvaSpec 2048) respectively. Under this
mode, transmitted spectra as a function of temperature and pressure can be
observed (with or without H2 ) to choose the optimal wavelength (which gives
best change in transmittance due to hydrogen) for each sample. A wavelength
of 639 nm is chosen for the measurements of the Fe/V superlattices, the relative difference of intensity is up to 35% between the non-hydride and hydride
sample (pH2 = 6.7 kPa) [38].
LabVIEW software is used for collecting, displaying and saving data. The
isotherms can be measured under absorption or desorption of hydrogen. The
whole process is controlled by the motorized valve which is well controlled by
the program. Real time data such as temperature, pressure, resistance, optics
and the averaged values at chemical equilibrium are saved into separate ﬁles.
Equilibrium is deﬁned in this setup as the point when the pressure, temperature
and the resistance ﬂuctuate with less than the set values.
31
4. Results
4.1 Thermodynamics analysis
>H9+@
>H9+@
.
.
.
.
6LHYHUWV/DZ
S
F>+9@
7>.@
V>PH9.+@ K>H9+@
F>+9@
9DQ
W+RIIDQDO\VLV
6LHYHUWV/DZ
9DQ
W+RIIDQDO\VLV
6LHYHUWV/DZ
ILWWLQJ
F>+9@
Figure 4.1. (Upper left panel) Chemical potential as the function of concentrations
for Fe/V 1/7. The intercept between the solid black lines and the measured isotherms
are used for the Van’t Hoff analysis shown in the upper right panel. (Upper right
panel) Van’t Hoff analysis of the interpolated data using linear relation. (Lower left
panel) Using Sieverts Law to determine experimental isotherms at low concentrations.
(Lower right panel) Enthalpy and entropy from Van’t Hoff analysis (black circles) as
well as the ones obtained by Siverts’ Law for low concentrations (red squares). The
solid line is the ﬁt considering conﬁgurational entropy: Δsc = f (c).
The hydrogen concentration is obtained from the change in optical transmission as a function of pressure and temperature. The conversion factor from
optical transmission to hydrogen concentration is derived from the comparison
to the p-c-T isotherms of FeV/V 6/21 by neutron measurements [58], where
32
H concentration c is determined from the observed changes in the integrated
unpolarized neutron reﬂectivity of the ﬁrst order satellite of the FeV/V superlattice structure. The scaling factor changes if a sample of different total
thickness is used. For example, we use c = −4 ln(Ic /I0 ) to calibrate concentration c from optical transmission I for Fe/V 1/7 sample. The relative chemical
potential Δμ can be easily derived from pressure Δμ = 0.5kT ln(p/p∗ ). The
upper left panel of Fig. 4.1 depicts the change in chemical potential Δμ with
c. The black vertical lines shown in the same panel, indicate a set of concentrations that have been chosen to be used in the Van’t Hoff analysis. Since the
chemical potential cannot be evaluated at exactly the same concentration for
different temperatures, the change in chemical potential is determined using
cubic interpolation. Δμ of several chosen concentrations are interpolated as
vertical lines in order to carry on Van’t Hoff analysis (Δμ = Δh − T Δs). This
Van’t Hoff analysis is shown in the upper right panel of Fig. 4.1. Thus by
ﬁtting Δμ over T enthalpy and entropy are obtained: Δμ = Δh − T Δs. The
measurements yield few points for Δμ at low concentration and interpolating
causes large uncertainties. In the low concentration region Sievert’s Law is
√
used: c ∝ p in order to increase the accuracy of measurements. This correc√
tion is shown in the lower left panel of Fig. 4.1 where p is plotted versus c.
Enthalpy Δh and entropy Δs from Van’t Hoff analysis are shown as black circles in the lower right panel of Fig. 4.1. Larger uncertainties are seen at lower
concentrations for both Δh and Δs. The red squares represent the Van’t Hoff
analysis from the analysis by the above mentioned Sieverts Law. There is no
ideal and general expression for concentration dependent enthalpy, but an approximation for entropy is plotted in this ﬁgure. The entropy is interpreted as
the summation of concentration dependent Δsc and concentration independent
Δsn .
Curie-Weiss analysis for Ts is shown in Fig. 4.2. γ ≈ 1 is found in most MH
systems, thus we use γ = 1 in the forth coming analysis. The values of Δμ at
each chosen concentration are determined by the interpolation of the measured
isotherms. An approximation of concentration dependent Δμ is used to ﬁt the
measured isotherms as shown in the upper left panel of Fig. 4.2. The equation
a(log(x/(1 − x)) + bx3 + cx2 + dx + e is used to ﬁt the data points in the whole
concentration region. ∂ μ/∂ c of four selected concentrations are shown in
the upper right panel. A linear ﬁt is applied to ﬁnd the temperature where
∂ μ/∂ c = 0. The ﬁnal results of Ts is seen in the lower panel of Fig. 4.2. The
background colour stands for the values of ∂ μ/∂ c. A cubic polynomial like Ts
is shown with uncertainties from the linear relation analysis, the uncertainties
become slightly larger at very low and high concentrations. The co-existence
region is from 0 to 0.35 H/V.
33
/ c [eV/H]
[eV/H]
c [H/V]
Temperature [K]
T [K]
/ c
c [H/V]
Figure 4.2. Analysis of Ts for FeV/V 6/21: (Upper left) The ﬁtting of interpolated
Δμ versus c. (Upper right) Curie-Weiss analysis of ﬁrst derivative of Δμ at several
concentrations. (Lower panel) The results of Ts versus hydrogen concentration. A
selection of ∂ μ/∂ c vs, T is plotted since every point in the lower panel comes from
such a ﬁtted line. The colormap shows the value of ∂ μ/∂ c of all the interpolated data
points.
34
4.2 Finite size effects
In Fig. 4.3 enthalpy and entropy are plotted as a function of concentration, in
three different Fe/V and one bulk V sample. The thermodynamics values are
found to be completely different to each other in all these V based samples.
Enthalpy at 0 concentration Δh0 (c = 0) for all superlattices are higher than
for bulk V. As described previously the epitaxial growth of thin ﬁlms on substrates can induce strain and stress on the ﬁlm. The strain can cause a change
in preferred interstitial site occupancy of hydrogen as compared to bulk V. It is
reported that H exclusively occupy the out of plane octahedral interstitial sites
(Fig. 2.4) in Fe/V while occupying isotropic tetrahedral sites in the disordered
phase of bulk V, as the electron density of O site is relatively lower [28]. It is
clear that the electron density of the interstitial sites are affected by the initial
strain state of the ﬁlm, for example Fe/V 1/7 has the weakest attractive interaction at high concentration. The slope of enthalpy vs concentration changes
systematically with the number of layers of vanadium, which in turn implies
an increase in the attractive H-H interaction, as can be seen by the slope of
the data in Fig.4.3. Under the almost identical strain state, this interaction in
V layers of superlattices decreases with thickness. The strongest exothermic
process is found in 3/21 at H/V=0.3. For the superlattices the turning points of
enthalpy, from an initially attractive to a repulsive H-H interaction, appear at
concentrations below c = 0.3 (H/V) as compared to about 0.4 (H/V) for bulk
V.
s [meV/KH]
h [eV/H]
-0.2
-0.3
-0.4
-0.3
Fe 1 /V 7
-0.4
Fe 3 /V 1 6
-0.5
bulk V
Fe 3 /V 2 1
-0.6
-0.7
0.0
0.2
0.4
c [H/V]
0.6
0.0
0.2
0.4
0.6
c [H/V]
Figure 4.3. Enthalpy and entropy of Fe/V 1/7, 3/16, and 3/21 superlattices and bulk V.
Electrical resistivity of the samples is measured during the whole process
of hydrogen loading. The change in resistivity ΔR/R0 is plotted versus H
concentration in Fig.4.4. All of the shown isotherms are measured at 443 K.
As seen in the left panel in Fig. 4.4, ΔR/R0 = f (c) almost follows a linear
relation with c in 1/7 at all concentrations. Although when the thickness of
V increases, a maximum point seems to appear at around H/V=0.3. For 3/21,
f (c) = 0 is clearly observed. This indicates that a short range ordering takes
place. We ﬁnd the phase boundary is strongly thickness dependent.
35
0.5
Fe/V 1/7
Fe/V 3/16
Fe/V 3/21
R/R0
0.4
0.3
0.2
0.1
0.0
0.1
0.3
0.5
0.1
0.3
0.5
0.1
0.3
0.5
c [H/V]
Figure 4.4. Change in resistivity as a function of H concentration for Fe/V superlattices of different thickness at 443 K.
0.1
0.12
0.26
320
400
Temperature [K]
200
0
−1 −5
−15
0
0
−10
−5
0.2
1.0
0.3
0.26
0.1
c=0.07
0.2
0.8
0.09
0.1
0.6
0.14
0.14
0.0
0.0
420
460
Temperature [K]
480 500 520
Temperature [K]
0.2
100
0
0.4
dμ/dc [eV/H]
0
−2
300 10
−
5
−1
0.3
0.3 c=0.07
dμ/dc [eV/H]
400
0
Temperature [K]
500
0.5 c=0.07
dμ/dc [eV/H]
dμ/dc [eV/H]
0
−
−3 −40 50
0
−20
−10
600
−2 −2
0 5
−1
0
−5
As discussed in the previous chapters we know that the spinodal temperatures Ts depend on the H-H interaction, concentration, and blocking parameter.
Ts of several superlattices is extrapolated as seen in Fig. 4.5, and we identify Tc
as the maximum value on this spinodal curve: ∂ Ts /∂ c = 0(c = cc ). Once the
shape of Ts is determined, the critical concentration cc and blocking parameter r are obtained. cc of superlattices are found to be smaller than for bulk
V (r=0.4 [34]). This is in agreement with the minimum in Δh which appears
earlier in Fe/V superlattices as compared with bulk. H-H interactions ∂ h/∂ c
behave completely different under the polarized local strain ﬁeld in these superlattices. The octahedral occupancy due to this ﬁeld provides a less negative
heat of solution as indicated by the smaller Δh0 in Fe/V. The more exothermic
absorption is found in the thicker ﬁlms where the average contribution from
the dead layers are relatively small.
0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
Concentration [H/V]
0.1
0.2
0.3
0.4
0 .0
Figure 4.5. Interpolated and extrapolated f (c, T )∂ μ/∂ c in the Fe/V-H phase diagram
is shown as colormap. The samples are (from left to right) 1/7, 3/16 and 3/21. Solid
contour lines are ∂ 2 μ/∂ c2 for determining the critical concentrations. The yellow
circles stands for the Ts results using equation Ts = T |∂ μ/∂ c=0 . The ﬁtting for three
given concentrations are presented in the insets as an example of ﬁnding for Ts . [27]
36
We have analyzed the critical temperatures of the three Fe/V samples, and
we ﬁnd a systematic change depending on the thickness. Thus the hydrogen
free layers are taken into account to quantify this ﬁnite size effect [27]
1 + 2Δn
Tc = Tc,∞ 1 −
(4.1)
n
by ﬁtting the Tc of these three samples we can easily obtain the critical temperature of inﬁnite thick V Tc,∞ and the number of hydrogen free layers Δn.
This ordering dependence on thickness holds universally for magnetic ﬁlms or
water [27].
4.3 Isotope effects
Metal hydride is an ideal model system for investigation of isotope effect due
to its largest relative difference in mass compared with the other elements in
the periodic table. This effect with respect to thermodynamics, diffusion, or
phase conﬁguration could be completely different depending on metals, site
occupancy, or thickness. Notable difference between H and D has been found
in phase diagram of vanadium hydride much larger than observed in other transition metals. Vanadium hydride exhibits much larger isotope effects as compared to niobium and tantalum for example, thus a study of the inﬂuence of
elastic boundary conditions on the isotope dependence can reveal understanding about its nature. Here we investigate the V based superlattices hydride, in
order to explore isotope effect and site occupancy or ﬁnite size effect.
-0.20
h [eV/H(D)]
-0.30
Fe 1 /V 7
-0.35
Bulk
-0.40
s [meV/K H(D)]
-0.1
-0.25
-0.2
-0.3
-0.4
Fe 1 /V 7
-0.5
Bulk
-0.6
-0.7
-0.45
(FeV) 6 /V 21
-0.8
(FeV) 6 /V 21
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
c [H(D)/V]
c [H(D)/V]
Figure 4.6. Enthalpy and entropy for H (black solid) and D (dashed red) in superlattices and bulk are plotted with respect to hydrogen concentration.
Enthalpy and entropy of hydrogen isotopes in superlattices and bulk are
shown in Fig. 4.6. All of the results are obtained by Van’t Hoff analysis using
37
the isotherms above the critical temperature in the disordered phase. Large
differences due to ﬁnite size effect of the samples are found, especially for
the enthalpy. While the shifts in isotopes ΔhH − ΔhD and ΔsH − ΔsD vanishes
when it comes to the superlattices. This is considered to be the contribution
from OZ site occupancy in Fe/V, while hydrogen occupies the isotropic T sites
in bulk V. A theoretical results based on ab initio simulation also indicates that
ΔhH − ΔhD is comparably smaller in OZ site than that of isotropic T. Δh is the
difference between the gas phase and the MH, thus it is determined by the ZPE
in the lattice. Therefore in another word, the shifts due to mass in hydrogen
isotopes is more notable in the steep narrow T site as compare to the broad ﬂat
Oz site.
One way to quantify the contributions to the isotope effect is to ﬁnd the
relation between√thermodynamics and hydrogen vibrational frequency ν, as it
is reduced to ν/ 2 when H is replaced by D. The shift in chemical potential of
interstitial hydrogen isotopes only determined by ZPE as a localized harmonic
oscillator without any rotational contributions. This variation is written as
⎞
⎛
hνH
⎜ sinh kT ⎟
⎟ + aT + b
ΔμH − ΔμD = 3kT ln ⎜
(4.2)
⎝
hνD ⎠
sinh
kT
The right part aT + b depends on the property of hydrogen gas [59]
a
1 0
0
0
0
(H2 ) − S298
(D2 ) + [ST0 − S298
](H2 ) − [ST0 − S298
](D2 )}
+ b = {S298
T
2k
1
[ΔH 0f ,298 (Hgas ) − ΔH 0f ,298 (Dgas )]
+
kT
1
{[HT − H298 ](D2 ) − [HT − H298 ](H2 )}
+
2kT
(4.3)
The result of ΔμH−D from isotherms of Fe/V 1/7 is shown in the left panel
of Fig. 4.7. The decrease with concentration and the temperature dependence
are seen under 323 K to 443 K. ΔμH−D corresponding to ZPE is plotted in
the right panel of Fig. 4.7 using equation 4.2. We ﬁnd this relation is approximately linear especially at low temperatures and with a limited region of
energy. Therefore it indicates ZPE decreases linearly during hydrogen loading.
Hydrogen isotope effects can be unexpected observed in different materials.
For example the relation of the ZPE dependent solubility (pressure) for palladium is: pH < pD < pT and for vanadium is: pH > pD > pT [61]. Here we
use atomic vibrations to describe the observed changes. Hydrogen vibration
in lattices can be approximated as localized Einstein oscillators with threefold
38
0
-5
-10
-15
443 K
423 K
403 K
363 K
353 K
333 K
323 K
100
JDVFRQV
+ '>PH9+@
+ '>PH9+@
150
0
-2
10 K
-4
403 K
50
-6
0
2
4
7>. @
0
2000 K
-50
3000 K
4000 K
-100
-150
0.2
0.4
F>+'9@
0.6
0
50
100
150
200
250
>PH9@
Figure 4.7. (Left) The shift in chemical potential ΔμH−D = ΔμH − ΔμD as a function
of hydrogen concentration in Fe/V 1/7. (Right) The relation between ΔμH−D and ZPE
by equation 4.2. The inset shows the gas constant determined by a/T+b based on
Meuffels’s isotherms [60].
degeneracy mode. Hydrogen occupy octahedral sites in Pd while tetrahedral
sites in V. O type interstitial proves a broad ﬂat proﬁle as compare to the narrow tetrahedral site. Therefore it affects the enthalpies ΔhH and ΔhD , which
is known as the difference in energy potential between gas and lattices phase:
ZPE shift of O sites is smaller than that of T sites. In order to compare the
concentration related ZPE between superlattice and bulk V, we illustrate a 3d
plot which shows how the potential diagram varies with hydrogen concentration in Fig. 4.8. The vertical axis is the out of plane direction. This simple
estimation is analyzed from the averaged values of 3d isotropic harmonic oscillator: ε = mω 2 x2 /2. Hence we see ΔμH−D totally depends on the lattice
spacing. With given expansion of MH is linear with hydrogen concentration,
we ﬁnd the so called Grüneisen parameter γ = ln(Δω/ω)/ ln(ΔV /V ): 0.89
(FeV/V 6/21), 0.77 (Fe/V 1/7), and 2.98 (bulk V). Therefore again we ﬁnd the
hydrogen induced expansion is restricted to Z-direction in Fe/V superlattices
while isotropic in bulk V.
39
(QHUJ\>PH9@
(QHUJ\>PH9@
VXSHUODWWLFH
%XON
F>+9@
SRVLWLRQ>QP@
F>+9@
SRVLWLRQ>QP@
Figure 4.8. ZPE in the potential well as the function of H (upper black lines) and D
(lower red lines) concentration for superlattice and bulk V. The vertical axis corresponds to the out of plane direction in the superlattice.
40
5. Review and summary of hydrogen in Fe/V,
Cr/V, Mo/V and W/Nb superlattices
5.1 Enthalpy dependency of hydrogen in superlattices
Any metal can absorb some amount of hydrogen under certain temperature and
hydrogen pressure. However the enthalpy of absorption is different for each
substance. For example yttrium and lanthanum easily form very stable metal
hydride, the enthalpy of solution is about -1 eV [16]. Even most d-block metals such as vanadium, niobium and tantalum also form hydride exothermically
at room temperature and below 1 bar of hydrogen pressure. This means that
hydrogen can migrate into these metals without any added energy at standard
condition.
Effective Medium Theory is used to calculate the embedding energy of an
atom in a host system like bulk solid, surface or another atom or molecule.
The idea of EMT is to replace a host (low symmetry) by effective host (high
symmetry) consisting of a homogeneous electron gas of a density. The embedding of the particular site ΔE(r) is the energy difference between combining
and separating the atom and host system [17]
ΔE(r) = ΔEhom [n0 (r)]
(5.1)
where ΔEhom (n0 (r)) is the energy for an atom in a homogeneous electron gas
with n0 (r), the host electron density at the siter of the atom. The results of this
local density approximation for several atoms and metals are presented in Fig.
1.1. For the noble gases, ΔEhom is above 0 and linear to the electron density. A
minimum in enthalpy is seen for the other gases. This attraction is determined
by difference in energy between electron gas and the afﬁnity level of the atom
and by the subsequent screening of the extra atomic electrons [17].
Hydrogen in many types of superlattices has been studied during the past
decades. Fig. 5.1 is the comparison of enthalpies at H/V=0 between Fe/V
and Mo/V superlattices of this study and combined with the previous work of
Hjörvarsson [62, 63], Andersson [64, 65, 66, 67, 64], Olsson [68, 69, 70, 71,
72, 73], Eriksson [74] and Stillesjo [75]. Norskov’s curve based on the Effective Medium Theory calculation is plotted among the results as comparison
[24, 17]. It is seen the heat of solution Δh versus estimated electron density
for V based superlattices distribute on both sides of bulk V. Two types of superlattice are shown: left group (red triangles) stands for Mo/V superlattices
and the right group (black circles) represents Fe/V superlattices. This plot
41
Figure 5.1. Illustration of change in enthalpy of hydride formation as function of
estimated electron gas density. Some of the data are taken from literature ignoring the
quality of samples.
includes most of previous publications on superlattice hydride since 1990s,
although experimental technologies and theories have been improved. Thus
some potential uncertainties of enthalpies may be within this result. Some of
the electron density are just taken from the literature if reported, while the
rest are estimated using Poisson’s Ratio for lattice volumes. The Fe/V group
shows less negative Δh on average with higher electron gas density and the
Mo/V group shows more negative Δh relative to Fe/V with lower electron gas
density. These results are in agreement with our understanding: Fe layers in
Fe/V imply compressive in plane strain to V layers, this causes a decrease in
volume (elastic deformation) for V in Fe/V compared with bulk V. Enthalpies
of Fe/V superlattice are more concentrated around Norskov’s curve: the highest enthalpy is about -160 meV for the samples Fe/V 6/13 and Fe/V 6/6. The
most exothermic absorption is found in Fe/V 3/21 sample, Δh exhibits characteristics of enthalpy of bulk at low concentration. All Fe/V (001) superlattices
are grown on MgO substrates while bilayer numbers are between 25 and 40.
For the Mo/V (001) group: Δh of the 4 monolayers Mo/V ﬁlms are not as
lower as expected, but very close to that of bulk V. The most exothermic hydride formation of these thin Mo/V (001) ﬁlms is found in Mo/V 4/4, which
has the thickest tensile Mo layers. The charge transfer affects the two nearest
layers around the interfaces. As there are only 4 monolayers in this sample,
the property of V layers are strongly affected by Mo layers.
Mo/V (001) does not ﬁt well on MgO and potentially creates defect, thus
sapphire substrates are used for Mo/V (110) by Öhrmalm in 1998. Enthalpy
42
Δh for these 6/12 (0.74 nm thick) and 6/14 (1.3 nm thick) Mo/V samples are
analyzed to be -380 and -410 meV respectively. The H-H interaction is repulsive in this (110) sample whereas attractive in (001) samples in the same
concentration range (H/V<0.45 ). The more exothermic absorption is obtained
in Mo/V (110) as its electron density is reduced under the tensile strain. The
deviation of the in-plane lattice parameters lead to the elastic response in the
out of plane direction. In the Mo/V (110) superlattice interstitial T sites are
more favourable at elevated temperatures and low concentrations. The T sites
in (110) Mo/V have a relatively larger volume and tensile V-H distance than
that in the bulk V. While as the O occupancy is more favourable in Fe/V (001)
(probably includes Mo/V (001)), due to the elastic compression V-H distance
and O site volume are smaller than that in the bulk V. These strained T or O
sites may results in a different enthalpy of solution. Moreover Δh is also affected by the difference in electron gas density between the interior and the
interface layers.
5.2 Thermodynamic properties
Table. 5.1 summarizes the samples of present and previous studies on hydrogen in many V based superlattices: Fe/V, Cr/V, Mo/V and W/Nb. All those
samples are (001) oriented grown on MgO, except Mo/V 6/12, 6/14 and W/Nb
samples (110) oriented grown on sapphire substrates. Some information about
the substrates for such as Mo/V 8/14 and 14/14 are not mentioned in the literature, thus these are not shown in the table. The ranges of bilayer numbers
of all the samples are between 25 and 105. The thickness of hydrogen free
monolayers X (Fe, Cr, Mo or W), is less than or equals that of V layers for all
of the samples. Mostly the number of atomic layers are directly from the literature and some are analyzed based on the value of reported thickness. Lattice
parameters of in-plane and out of plane for few samples have been characterized in the literature, the rest are estimated considering substrates and using
Poisson response. X/V is therefore the ratio of hydrogen free atomic layers to
hydrogen absorbing layers (V or Nb).
The heat of solution Δh of most superlattices are presented in Table. 5.2,
some are reported in the literature and others are obtained from the isotherms
of this work using Van’t Hoff equation. The blocking parameter r is the maximum number of hydrogen interstitial sites before the ﬁrst phase transition. r is
obtained from the minimum point of the plot of enthalpy Δh versus concentration c, where the derivative of the point is zero. It is also where the H-H interactions turn from attractive to repulsive. However the measured concentration
[H/V], converted from optical transmittance, is the averaged atomic ratio of H
to host metal atoms V. The interface layers, is known as deplete layers, where
no hydrogen exists should be considered for the further analysis. The number
of effective deplete layers of Fe/V and Cr/V are determined to be 1.4 and 1.0
43
Table 5.1. List of the studied superlattice samples (* this work). X is Fe, Cr, Mo or
W layer. ain and aout are the in plane and out of plane parameters respectively. ML
stands for monolayers.
No.
Substrate
Sample
1
2
3
4
5
6
7
8
9
10
11
12∗
13∗
14∗
15∗
16∗
17∗
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
001
001
Sapphire (110)
Sapphire (110)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
MgO (001)
Sapphire (110)
Sapphire (110)
Sapphire (110)
Sapphire (110)
Fe/V
Fe/V
Fe/V
Fe/V
Fe/V
Fe/V
Fe/V
Fe/V
Fe/V
Fe/V
Fe/V
Fe/V
Fe/V
Fe/V
Fe/V
Cr/V
Cr/V
Cr/V
Cr/V
Mo/V
Mo/V
Mo/V
Mo/V
Mo/V
Mo/V
Mo/V
Mo/V
Mo/V
W/Nb
W/Nb
W/Nb
W/Nb
44
Repeat
30
30
30
30
30
40
25
25
25
25
20
8
56
40
105
30
30
30
30
X
ML
V
ML
ain
[nm]
aout
[nm]
X/V
2
3
6
9
13
3
3
3
6
11
13
1
3
3
3
2
3
7
14
8
14
6
6
10
14
1
3
4
3
8
13
18
12
13
13
13
13
10
16
10
6
11
14
7
16
21
24
14
21
14
14
14
14
12
14
10
14
4
4
4
12
12
12
12
0.296
0.298
0.299
0.307
0.17
0.23
0.46
0.69
1.00
0.30
0.19
0.30
1.00
1.00
0.93
0.14
0.19
0.14
0.13
0.14
0.14
0.50
1.00
0.58
1.00
0.45
0.44
1.00
1.00
0.25
0.75
1.00
0.25
0.67
1.08
1.50
0.313
0.307
0.310
0.311
0.233
0.232
0.230
0.229
Table 5.2. List of some analyzed thermodynamics of the studied superlattice samples.
r ,r(in), cc , cc (in) and ctur are the mean blocking parameter, interior blocking parameter, mean critical concentration, interior critical concentration and the concentration
where ∂ h/∂ c = 0 respectively.
No.
1
2
3
4
5
6
7
8
9
10
11
12∗
13∗
14∗
15∗
16∗
17∗
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
r
[H/V]
r (in)
[H/V]
cc
[H/V]
cc (in)
[H/V]
0.04
0.05
0.05
0.06
∂ u/∂ c
c < cc
∂ u/∂ c
c > cc
ctur
[H/V]
Δh0
[meV]
Tc
[K]
240
210
160
220
180
251
316
316
316
316
0.06
0.08
0.03
0.03
20.80
0.09
0.09
0.12
0.04
0.07
0.07
0.33
0.30
0.30
0.37
0.33
0.43
0.17
0.08
0.10
0.09
0.55
0.36
0.35
0.42
0.39
0.48
0.07
0.01
0.03
0.02
0.09
0.08
0.14
0.17
0.17
0.22
0.09
0.03
0.04
0.02
0.15
0.10
0.16
0.19
0.20
0.24
8.88
15.00
20.00
28.87
2.02
10.92
5.72
6.32
10.00
6.11
0.36
-0.07
0.12
0.04
0.10
0.60
0.99
0.95
1.82
1.03
0.15
0.11
0.30
0.17
0.11
0.17
0.31
>0.51
0.61
0.77
0.33
0.38
0.17
0.17
0.20
0.33
0.38
0.41
0.39
0.17
0.19
0.21
0.19
1.00
0.93
0.93
0.62
1.00
0.93
0.93
0.62
0.10
220
250
160
210
230
200
200
300
280
230
110
360
240
360
380
410
300
290
240
260
360
396
390
380
400
170
203
208
244
252
403
445
450
419
456
275
324
389
458
269
45
[ML] respectively from the model of Tc (equation 4.1). The interior blocking parameter r (in) and the interior critical concentration cc (in ) are easily
obtained by the analyzed number of deplete layers. The mean critical concentration cc is concentration corresponding to the maximum of spinodal temperature Ts in the phase diagram. Ts is calibrated using equation 2.19. From the
analysis of Δh versus c, the H-H interaction u is observed to be concentration
dependent: u = f (c). To understand this as simple as possible, we separate
∂ h/∂ c versus c into two linear relations: u = a1 c(c < ctur ), u = a2 c(c > ctur ),
the repulsive and the attractive. At low concentrations, the range of a1 is
between 2.02 and 28.87 [eV/H2 ] depending on samples. The turning concentration ctur are between 0.09 and 0.31 [eV/H2 ]. At high concentrations u
becomes barely concentration dependent, a2 are between -0.07 and 1.82. The
repulsive interaction is not found in Cr/V 3/21 sample even at highest measured concentrations. Critical temperatures are deﬁned as the maximum point
of spinodal temperatures Ts , it is obtained from the isotherms of the literature
using equation 2.23. Such as isotherms or other information of samples are
not reported or not ﬁne enough to analyze thermodynamic, thus some of r, cc ,
∂ u/∂ c, Δh or Tc are not included in the table. Further analysis of these results
are discussed in the following sections.
5.3 Effect of local strain ﬁelds
0
h [meV]
-100
-200
-300
-400
0.2
0.4
0.6
0.8
nFe /n V
1.0
1.2
Figure 5.2. Enthalpy of hydride formation is shown as the function of Fe/V ratio in
atomic layers
According to Effective Medium Theory, the enthalpy of hydride formation
depends on the electron gas density of the host metals. This density is affected
by the change in volume (lattice constant) due to the local strain state in the
46
superlattices. The strain state in the superlattice is determined by the substrate
and the hydrogen free layers such as Fe, Cr, Mo or W. The estimation of
volume is based on the elastic expansion or compression using Poisson ratios
for different metals. Bulk V is used as the reference to deﬁne the electron gas
density, Δh=-300 meV and volume V =0.3033 nm3 . This density of the other
superlattices and thin ﬁlms are analyzed exclusively considering the change in
volume. In Fe/V samples, the ratio of thickness of Fe to V nFe /nV is used as
the contribution to the extra compressive strain in V layers. The ﬁnal results
are plotted in Fig. 5.2. It is seen Δh evenly distribute along the Norskov’s
curve of Effective Medium Theory. Although it seems that grown on the same
substrate, the enthalpy of the MH is inﬂuenced by the hydrogen free layers.
There is still limits of the thickness of bilayers in sample design due to the
risk of defects of mismatch. However these results are analyzed based on
many independent measurements during a very long time of research and other
possible effects are not considered.
25
20
15
10
5
0.0
High concentration
∂ u/∂ c [meV/ H/V]
∂ u/∂ c [meV/ H/V]
Low concentration
0.8
0.6
0.4
0.2
0.0
0.4
0.8
nFe/nV
0.0
0.4
0.8
nFe/nV
Figure 5.3. ∂ u/∂ c at low and high concentration versus Fe/V ratio
A change from attractive-to-repulsive behavior is observed in all Fe/V samples while H concentration increases up to 1 bar. This turning concentration
ctur is found between 0.09-0.31 depending on different samples. However it is
not found in Cr3 V21 . The concentration where the H-H interaction turns from
attractive to repulsive, however, is different between superlattices and bulk V.
The interaction coefﬁcients a = ∂ u/∂ c are distinguished by both sides of the
turning point. These two coefﬁcients a1 and a2 are obtained by the linear ﬁt
on the curve of u versus c. The results are shown in Fig. 5.3 as the derivative
of u versus the monolayer ratio of nFe /nV for the attractive (left panel) and
repulsive (right panel) regions. It is observed that the attractive interaction u
turns to repulsive more rapidly and then increases slightly slowly in the Fe/V
with thicker Fe layers and more compressive strain in V. The out of plane oriented octahedral occupancy is more favourable in these Fe/V superlattices. Fe
affects the band structure of interfaces and compresses V layers to weaken the
47
H-H attractive interactions at the lower concentration limit, while this boundary effect becomes slight for the repulsive interactions.
The H-H distance in metals can not be smaller than 0.21 nm [76]. Therefore
H-H interactions become repulsive when more hydrogen absorbed in lattice.
In the solid solution system, conﬁgurational entropy Δsc , is related to the H
random distribution in all possible interstitial sites. In the process of absorption, the reduction of interstitial sites r causes the decrease of Δsc by the mutual
blocking of H atoms. This is also when the repulsive interactions form ordered
phase, in spite of the transformation of the type of occupancy. However at low
temperatures, attractive H-H interactions prefer to make hydrogen rich phase
precipitate rather than forming ordered phase.
'HSOHWHOD\HU 'HSOHWHOD\HU r
r
nFe /n V
nFe /n V
Figure 5.4. Blocking parameter of H in Fe/V superlattices. d=1.5 ML deplete layers is
used to calibrate the interior concentration as compared with the mean concentration.
In the MH phase diagram the maximum hydrogen concentration where
spinodal curve ends, is deﬁned as r. It corresponds to the turning point between attractive and repulsive H-H interactions or ∂ Ts /∂ c = 0. Fig. 5.4 shows
the inﬂuence of bilayer ratio for the r value of Fe/V superlattices, including
the data analyzed from previous research. Hydrogen barely stay in the interface layers in a bi-layer superlattice, and the obtained concentrations, either
from the optic transmission or analysis of electrical resistivity, are the mean
values. Therefore these mean concentrations need to be converted into interior
concentration. Thus we suppose the number of deplete layers are 0.0, 0.5, 1.0
and 1.5 and plot r as a function of nFe /nV . In all of these Fe/V superlattices,
r does not exceed 0.4 and it seems to decrease with the ratio of nFe /nV . This
is consistent with the increasing of Fe layers are likely to increase the electron
density in V and affect the electronic band structure of the interface, in order to
block the available interstitial sites until the phase transition occurs. However
it is still not clear whether the site occupancy has changed or the uncertainties
from the analysis of concentration by different methods should be taken into
account.
48
5.4 Dependency of critical temperature
9ILOP
&U9
)H9Q)H Q 9 7 &>.@
)H9Q)H Q 9 !
0R9
Q
Figure 5.5. Critical temperature as a function of reciprocal of V thickness.
The critical temperature of hydrogen in Fe/V superlattice is found to be
thickness dependent [27]. Tc for hydrogen in many different V superlattices
are shown as the function of V thickness in Fig. 5.5. This plot includes the
Tc analyzed from reported isotherms. Tc is deﬁned as the maximum point of
spinodal temperature Ts in this work. Ts is a function of hydrogen concentration c: Ts = uc(r − c)/kB r. Thus one approach to determine Ts is to analyze the
derivative of enthalpy ∂ h/∂ = u = f (c) and the constant blocking parameter
r. Another method is to determine the temperature at which ∂ μ/∂ c = 0 for
each concentration. The latter way is applied for this analysis due to the relative smaller uncertainties in the procedure. In Fig. 5.5, Tc of 50 nm, 10 nm V
ﬁlms and bulk V are plotted as the thin ﬁlm group. Hydrogen occupies T site
at lower concentrations in these ﬁlms, as they are thick enough to neglect the
effect of compressive strain in V due to MgO substrates. Comparing the two V
ﬁlms, the energy of elastic interaction in these V ﬁlms is similar to that in bulk
V, thus effects of substrate and interface contribute to the change in thermodynamics. The ﬁnite size effect is more notable when the thickness reduces to
10 nm comparing with the 50 nm ﬁlm. Samples with nFe /nV < 0.2 are shown
as red circles. It is observed that Tc reduces gradually with the reciprocal of
V thickness and all of the Fe/V sample with nFe /nV = 1/7 fall on this ﬁtted curve. This ratio is found to be ideal for Fe/V superlattice growth since
its structural characterization shows ﬁne quality compared with other mismatched samples. This also indicates that under the identical interior strain
state, thinner superlattices are more affected by the interface. Samples with
nFe /nV > 0.2 are shown as green triangles in Fig. 5.5. Tc of these samples are
below the ones with less Fe layers, this denotes either the site occupancies are
different in these two Fe/V groups, or there are dislocations in the bilayers if
Fe/V= 1/7. Finite size effect again observed in the two Cr/V (001) samples,
49
the curve of Tc is above that of Fe/V. For Mo/V samples there is no systematic
relation between Tc and V thickness found. Tc of Mo/V 1/4 is 458 K, which
is almost the same as for bulk V. This indicates one Mo monolayer in bilayers
does not inﬂuence thermodynamics. While for 3/4 Mo/V the interaction energy is below that of Mo/V 1/4, this occurs when more Mo layers contribute
to the band structure of the interfaces or apply stronger tensile strain in V.
It seems contributions from Mo layers reduces Tc of hydrogen in these thin
Mo/V (001) superlattices. However Tc is observed to be lower in Mo/V 8/14
than that of Mo/V 14/14. Mo layers contribute reversely in the thicker Mo/V
superlattices. It seems that Tc is reduced under a weaker tensile strain ﬁeld
when Mo layers are thick enough to exhibit identical effects to the interfaces.
However results of more samples are needed to verify this hypothesis.
50
6. Conclusion
Hydrogen and vanadium are chosen to study thermodynamic and kinetic properties of lattice gas system MH (metal hydride): hydrogen has the lowest embedding energy than any other elements within the limit of density of homogeneous electron gas in metals; hydrogen is weakly bonded in V than other metals and the isotherms of vanadium hydride are experimentally observable and
measurable at relatively low pressure. Substantial efforts have been devoted
on hydrogen in bulk V, however the formation and properties of VH or MH
become different and not fully understand when the lattices are strained, conﬁned etc.. Therefore we use V based superlattices (Fe/V and Cr/V), combined
previous results on Mo/V, W/Nb and V ﬁlms, to explore both effects of strain
and ﬁnite size on thermodynamics properties. Light transmittance is used to
determine and watch hydrogen content and diffusion in the experiments. We
found by reducing the thickness of the Fe/V superlattices the energy of H-H
interaction and critical temperatures of V layers are decreased. This is further conﬁrmed by the investigation of ﬁnite size effects of magnetic ordering
and hydrogen uptake in other superlattices such as Cr/V. The isotope effects
of thermodynamics are reduced in Fe/V superlattices as compared with that
in bulk V, this is understood as the T site occupancy is replaced by Oz type
in Fe/V. The X layers (Fe, Cr or Mo) also affect the thermodynamics and
electrical resistivity. Hydrogen diffusion decreases at higher concentrations
in thinner V ﬁlms comparing with bulk V. The activation energy is increased
especially in the Fe/V samples as H occupies Oz sites rather than T sites. By
combining with literature on hydrogen in V based superlattices and ﬁlms, we
again veriﬁed the enthalpy of hydrogen solution depends on the electron density in V. Besides the substrates, the thickness ratio of boundary layers to V
layers affects contributes to the density in V lattice. We therefore conclude that
ﬁnite size effects can have substantial effect on the thermodynamic properties of metal hydride systems. However, these effects only become prominent
when the extension of the absorbing layer is in the nm range. To utilize ﬁnite
size effects for tailoring the thermodynamic properties of hydrogen storage
devices will therefore be challenging.
51
Summary in Swedish
Väte är en kandidat för framtida energilagring eftersom det ﬁnns i överﬂöd och
inte ger några oönskvärda restprodukter. Industriellt framställs väte oftast från
petroleum (kolväten), vatten eller andra föreningar. Väte skulle även kunna
framställas via spjälkning av vatten i bränsleceller med energi från förnybara
källor. Den kemiska energin som frigörs vid förbränning av väte är många
gånger större än energin från andra typer av kemiska bränslen och vid förbränning av väte är den enda restprodukten vattenånga. Ett möjligt tillämpningsområde för väte är i transportsektorn. Idag används väte som bränsle i både
vanliga förbränningsmotorer men även i elektriska hybridbilar. Den viktigaste
aspekten hos bränslet för fordon är att det har en hög gravimetriska densiteten,
vilket betyder mycket energi i liten massa. Tyvärr har väte lagrat i metallhydrider en väldigt låg gravimetrisk densitet. För många andra tillämpningar
däremot är inte den gravimetriska densiteten något problem, t.ex. gaffeltruckar
eller stationära reservkraftverk etc.
Vätelagring kan baseras på fysisk lagring (t.ex. komprimerad gas eller
ﬂytande väte) eller kemisk lagring. Vid fysisk lagring fortsätter väte vara i
molekylform medans vid kemisklagring binder vätet med jon-bindingar till ett
värdmaterial. Grundämnen från d-blocket eller f-blocket kan bilda metalliska
(interstitiella) hydrider när de utsätts för väte. Inte bara rena metaller utan
även legeringar med metallbindningar kan bilda hydrider. Metall-väte bindningen kan i princip separera elektronen från vätets atomkärna och kvar blir
en proton eller en väteradikal. Därför är hydrider elektriska ledare och icke
stökiometriska vilket betyder att vätet kan röra sig relativt fritt inuti dessa material. Metallhydrider (MH) kan reversibelt och snabbt både absorbera och
släppa ifrån sig atomisk eller diatomiskt väte eftersom vätet positionerar sig
interstitiellt i metallhydriden. De kan därför användas för att lagra väte eller
som en del i bränsleceller. Men en nackdel är att i dessa material kan de interstitiella väte atomerna orsaka försprödning.
Den mest intressanta frågan för dessa vätelagringsmaterial är hur man kan
öka mängden lagrat väte. Bildningsentalpin för interstitiella hydrider används
ofta för att bestämma vätets löslighetsgrad och diffusiviteten är ett mått på
tiden som krävs för väte att absorberas. Bildningsentalpin har visats vara linjärt proportionell mot elektrondensiteten hos metallen. Bildningsentalpin för
väte i vanadin är ungefär -0.3 eV (vid 1 bar referenstryck), detta är det minst
negativa värdet för exotermisk absorption i interstitiella metallhydrider. Detta
betyder att en första ordningens fasövergång sker på den mittersta av isotermerna under 1 bar. Å andra sidan är väte den enda gasen som har negativ
52
bildningsentalpi för de elektrondensiteter som ﬁnns hos metaller. Därför har
vi valt väte och vanadin som modell system för att studera gittergas systemet.
Väte absorption i V supergitter kan klassiﬁceras i många olika steg, fysisorption och kemisorption båda vid ytan samt även absorption i V gittret.
Skillnaden i energinivåer mellan initiala och slutgiltiga tillstånd bestämmer
den frigjorda värmeenergin vid absorptionen. Energibarriärerna som vätet
måste överstiga bestämmer diffusionen. Teorier om elektronbandstruktur och
effektiva medium teorin används ofta för att beskriva hydridbildningens inverkan på entalpin. Vätets interstitiella positioner är viktigt för både termodynamiken och kinetiken, därför är metallhydridens fasdiagram baserat på typen
av interstitiell position och ordningsgrad. Ordningen påverkar den elektriska
resistiviteten och vätehalten kan bestämas från ljustransmission.
Den kemiska potentialen hos MH kan härledas från väte trycket vid kemisk
jämvikt. Enligt statistisk mekanik så ges den kemiskapotentialen hos MH
som en funktion av väte-väte (H-H) interaktioner, gitter expansion, vätetryck,
vätekoncentration och vätets vibrationsenergi. Därför är differensen i kemiskpotential mellan väteisotoper endast beroende av skillnaden i vibrationsfrekvenserna hos väte och deuterium. Den kritiska temperaturen beror på väte-väte
interaktioner som beror på den elastiska dipol tensorn, det lokala töjningsfältet, koncentrationen, isotermiska kompressionsförmågan och gitter parametern. Vätets diffusionsförmåga kan beskrivas som en funktion av vibrations
energi och kemiskpotential.
Mycket arbete har gjorts på väte-metallsystem i bulk. Men för metallhydrider i supergitter form påverkas material egenskaperna av volymen, töjningen och det intermediära materialet. För att undersöka detta har tunnﬁlmer
med enbart vanadin och tunnﬁlmer med supergitter bestående av antingen
krom-vanadin eller järn-vanadin bilager tillväxts på MgO substrat. Substratet
modiﬁerar då det lokala töjningsfältet, den interstitiella besättningsgraden och
volym expansionen vid väte absorption hos tunnﬁlmen. Dessa supergitter har
också annorlunda elektron densitet jämfört med samma material i bulkform
vilket deﬁnitivt påverkar lösningsentalpin. I det här arbetet har vi speciellt
fokuserat på Fe/V och Cr/V supergitter tunnﬁlmer och kombinerat detta med
tidigare publicerade resultat om Mo/V och V tunnﬁlmer.
Resultaten visar att en tjockleksreduktion av Fe/V bilageren i supergittret
minskar både energin hos H-H interaktionen och även den kritiska temperaturen. Detta är i överrensstämmelse med andra supergitter t.ex Cr/V och även
med effekten av begränsad utsträckning på den magnetiska ordningen i magnetiska tunnﬁlmer. Isotopberoende termodynamiska effekter försvinner i Fe/V
supergitter jämfört med V i bulk, detta kan förklaras med att besättningen av
tetragonala-positioner (T) byts mot oktahedrala z positioner (Oz ) i Fe/V. Vidare påverkar gränslagren t.ex. Fe, Cr eller Mo även termodynamiken och den
elektriska resistiviteten.
Väte diffusionen minskar vid högre koncentrationer i tunnare V ﬁlmer jämfört med i bulk V. Aktiveringsenergin förbättras speciellt i Fe/V prover då
53
H besätter oktahedrala interstitial positioner Oz istället för tetragonala positioner. I kombination med tidigare litteratur om väte i V baserade supergitter
och tunnﬁlmer kan vi veriﬁera att lösningsentalpin hos väte beror på elektrondensiteten hos V. Denna elektron densiteten i V gittret påverkas förutom av
substraten även av tjocklekskvoten mellan gränslagren och V lagren.
54
Acknowledgment
I take this opportunity to thank my supervisor Björgvin Hjörvarsson, who
brings the joy of discovery in fundamental science and opens the gate of western world to me. I have learned great philosophy, methodology and communication skills from him. I appreciate his patience and looking after during the
past years.
I would like to thank my second supervisor Max Wolff for his help from
logical analysis to scientiﬁc writing. I am impressed by his distinctive angle
(view) to every issue, spirit of questioning and good temper.
I must acknowledge my greatest home country, China Scholarship Council, and Björgvin Hjörvarsson for the ﬁnancial supporting during my research
career. Hydrogen team, material physics group, Uppsala University, Swedish
nation and University of Science and Technology Beijing also have my acknowledgment, I would not accomplish without the support from all of these
units.
I would like to express my gratitude to my former supervisor Wan Farong at
USTB for supporting me for further development, Ola Hartman for the knowledge of diffusion, Gunnar Pálsson for the wonderful ideas and solutions (thesis
revision), Jan Prinz for the guidance to the experiments, Andreas Bliersbach
for thesis revision and ﬁerce criticizing, Sotirios Droulias for providing samples, Lennard Mooij, Anders Olsson, Emil Melander, Bengt lindgren, Vassilios Kapaklis, Gabriella Andersson, Huang Wen, Atieh Zamani and other
millions of names. Thank you all who gave me help but I do not even know
your names, please forgive my bad memory. I especially thank my kindest
ofﬁce mate Andreas Frisk for the summary translation and accompanying.
Thank you my family and my friends who were or are in Uppsala, you made
me beat all the difﬁculties and shared the happiness with me. Thank you all
the tough people who gave me hard time, you made me grow and ﬁnd out how
strong and capable I can be.
55
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