Electronic Band Structure of Bulk and Monolayer V2 O5
Gregory Stewart∗
Case Western Reserve University Department of Physics
(Dated: May 2, 2012)
Vanadium oxide (V2 O5 ) is a material that can be fabricated as a thin film of only a few
atomic layers thick during mechanical exfoliation due to the weak interlayer interactions,
much like how graphene is created. During this project we looked at the band structure of
both the bulk and mono-layered V2 O5 . We used computer software called ABINIT, which
is installed on the high performance computer cluster (HPCC), to do all of our calculations.
We used the projector augmented wave (PAW) method in our band structure calculations.
The construction of the PAW potentials for vanadium was studied during our project and we
found that the 3s and 3p bands of vanadium needed to be treated as valence bands in order to
avoid ghost bands from appearing in our calculations. We tested the convergence of the plane
wave energy cut-off parameter and its accuracy first with metallic body centered cubic (BCC)
vanadium and then with rock salt structured vanadium monoxide (VO). The band structure
of V2 O5 can be characterized as oxygen 2s and 2p like bands separated by a gap from the
vanadium 3d like conduction bands. The band gap between them is indirect. There are two
narrow bands that fall just below the conduction band called intermediate bands. Many
interesting properties can arise from these intermediate bands such as transport phenomena
or mediating optical transitions from the valence bands to higher conduction bands.
1.
INTRODUCTION
Vanadium Oxide (V2 O5 ) has many interesting properties, but the most notable (and one that I
will be pursuing) is the ability for it to produce monolayers. Often times, mono-layered materials
∗
Electronic address: ghs9@case.edu
2
FIG. 1: Graphical view of the unit cell and structure of V2 O5 .
(or materials only a few layers thick) have many physical properties that are vastly different than
its’ bulk layered counterpart. V2 O5 has a structure (as shown in Figure 1) that would allow one to
get sheets that are just a few layers thick. One of the most famous of these materials is graphene
which is a single layer of graphite. Graphene has many interesting properties such as its’ strength
and electrical capabilities, to name a few. The strength of graphene can be put in perspective by
noting that ”it would require a force of over 20,000 N to puncture [a layer of graphene 100 µm thick]
with a pencil” Ref. [6] and electrically graphene ”...is the thinnest substance capable of conducting
electricity” Ref. [11]. These properties were not discovered until research was conducted on the
properties of a single layer of graphite and hopefully some interesting properties will arise when
looking at a single layer of V2 O5 .
In Figure 1, you can clearly see the layered structure of V2 O5 . One could almost presume
that in between the layers, the oxygen atoms (red spheres) that are hanging up and down (called
vanadyl oxygen) are repelling each other. The lack of a strong bond between the layers is what
allows someone to pull a couple layers away from the rest of the material. As shown in Figures 2
and 3 we can see some examples of the mechanical exfoliation of V2 O5 as performed by graduate
student Liang Zhao Ref. [15] working under Professor Jie Shan. When looking at the Atomic Force
3
Microscopy (AFM) Ref. 3 of the mechanically exfoliated V2 O5 , we can note a few things. First, the
shape of the thin layered flakes are very symmetrical and rectangular. The second thing to note is
that the thinnest strip is about 4nm thick which corresponds to a sheet about 6 layers thick. With
more experimentation and different exfoliation processes it is evident that the ability to remove a
single layer of V2 O5 is well within reach.
The main way we will be exploring the properties of both bulk and mono-layered V2 O5 is
by using the program called ABINIT. The name ABINIT comes from the Latin term ”ab-initio”
or ”from the beginning”. The beauty of this program is that we only need to know the atomic
number and position of the atoms in the structure we are doing our calculation on and we can
predict everything else from that. A parallel version of ABINIT has been installed on the High
Performance Cluster (HPC) here at Case Western Reserve University to allow my computationally
expensive calculations to be completed in a manageable amount of time. More will be discussed
on how ABINIT works in the Computational Methods section.
FIG. 2: Mechanical exfoliation of V2 O5 as performed
FIG. 3: Atomic force microscopy (AFM) of V2 O5 as
by Liang Zhao Ref. [15] working under Professor Jie
performed by Liang Zhao Ref. [15] working under Pro-
Shan.
fessor Jie Shan. A thickness of about 4nm corresponds
to a sheet of V2 O5 about 6 layers thick.
4
2.
PREVIOUS WORK
V. Eyert and K. -H. Hock Ref. [5] published a paper on the electronic band structure of V2 O5
in 1998. When going through our work we compared a lot of our findings with what they reported.
Eyert and Hock computed the electronic band structure of bulk V2 O5 using density functional
theory (DFT) and the local-density approximation (LDA) which are both components of ABINIT.
The band gap found is this paper is reported to be 1.74 eV which is smaller than the experimental
value of 2.2 eV. This has to do with the way LDA calculations are carried out and will be discussed
in greater detail later on.
When first checking the validity of our band structure calculation we could easily compare our
results with that of Eyert and Hock. One of the properties we could look for when computing
the band structure were intermediate bands (IB) in the band structure. They lie just below the
conduction bands and are only prevalent in a few materials. Although this paper does not calculate
the band structure of mono-layer V2 O5 , another paper by Chakrabarti and Hermann et al. Ref. [3]
does calculate the mono-layer band structure. They also use DFT and will allow us to compare
our results with theirs. Again, the mono-layer shows IB that lie just below the conduction bands
so again we can look for this pattern during our calculations.
3.
COMPUTATIONAL METHODS
3.1.
ABINIT
ABINIT is a very powerful, free to use, computer program and is currently installed in parallel
on the high performance computing cluster here at Case Western Reserve University. ABINIT’s
main uses are to find the total energy and electronic structure of a system using density functional
theory and a planewave basis Ref. [1]. With ABINIT the only values I needed to know empirically
5
were the type of atoms in my structure and their position within that structure. Both of these are
very easy to find out as we know that V2 O5 has the space group Pmmn (D2h
13 ) with Wyckoff 4f
symmetry for the vanadium, vanadyl and chain oxygen atoms while the bridge oxygen atoms have
Wyckoff 2a symmetry Ref. [7], as stated by Eyert and Hock Ref. [5]. The lattice constants were
also given as a = 11.512Å, b = 3.564Å and c = 4.368Å.
When looking at plane wave basis sets (which ABINIT uses extensively) we first start with a
periodic system with a potential
V (r + na) = V (r)
(1)
where a is the lattice vector and n is some integer. We can use Bloch’s theorem for periodic systems
and get the wavefunction
ψi (r) = expikr fi (r)
(2)
because of the periodicity of f (r) we can write it as a set of plane waves given as
fi (r) =
X
ci,G expiG·r
(3)
G
where G are the reciprocal lattice vectors. If we plug Eqn. 3 into Eqn. 2 we get our electronic
wave functions
ψi (r) =
X
ci,G expi(k+G)·r
(4)
G
These equations can be applied to materials as well. Almost every structure is periodic in nature
if you think about a material as being made up of a bunch of unit cells.
3.2.
Density Functional Theory
ABINIT is based largely on Density Functional Theory (DFT). DFT is a quantum mechanical
modeling method for many body systems and is based largely around the Kohn-Sham equations.
6
DFT has had great popularity in solid state physics and is now used is a wide variety of different
fields.
The Kohn-Sham equations are named after Walter Kohn and Lu Jeu Sham who originally introduced the concept. The Kohn-Sham equations are derived from the application of the Schrdinger
equation to a system of non-interacting particles. In our case, these particles will be electrons. The
Kohn-Sham equations are three separate equations that can be solved in an iterative process which
is why DFT has gained popularity with the rise in the accessibility of high performance computers
that can complete these iterations at a very high rate.
The first Kohn-Sham equation is
1 2
Ei ψi (r) = − ∇ + vef f (r) ψi (r)
2
(5)
which is the Schrdinger equation for electrons in an effective potential, vef f . The second KohnSham equation is
n(r) =
occ
X
|ψi (r)|2
(6)
i
which tell us that the total probability to find an electron somewhere is equal to the modulo squared
of each occupied one electron eigenstate. The final equation is
Z
vef f (r) = vN (r) +
d3 r0
n(r0 )
+ vχc [n(r)]
|r − r0 |
(7)
This relates the effective potential to the density of electrons. The effective potential is equal to
the potential of the nuclei vN (r), plus the Coulomb potential produced by the total density and
the final term is the exchange correlation potential. The last term incorporates the effect that
electrons of the same spin do not interact (due to the Pauli exclusion principle) and should thereby
reduce the total Coulomb potential.
7
3.3.
Local Density Approximation
There are two approximations needed when dealing with DFT. The first (and one that was used
in our ABINIT program and all of our results) is the local density approximation (LDA) and the
other is the generalized gradient approximation (GGA).
These approximations are needed because the exchange correlation isn’t known except for a free
electron gas. The LDA is given by
LDA
Eχc
Z
=
ρ(r)χc (ρ)dr
(8)
where χc is the exchange correlation of a uniform electron gas density ρ Ref. [4]. This then gives
us an exchange correlation potential as used in the Kohn-Sham equations of
LDA
vxc
[ρ(r)] =
LDA
δExc
∂xc (ρ)
= χc (ρ) + ρ(r)
δρ(r)
∂ρ
(9)
With this exchange correlation potential we can put this into the Kohn-Sham equations to accurately predict our system.
LDA is an approximation and thus must be treated as such. One of the most common drawbacks
of LDA is that it tends to understate the band gap of a system. As seen in the work by Eyert and
Hock Ref. [5] they reported a band gap of around 1.7 eV with an experimental band gap that is
about 30% higher. Although this may seem like a lot, as long as we are aware of the effect the
approximation has on our calculations we can account for it and if need be, adjust accordingly.
GGA tends to give more accurate band gap values but has other drawbacks and because LDA
tends to have smaller radii for their pseudopotential spheres and our structure is relatively tightly
packed we chose to use LDA in our calculations.
8
3.4.
Pseudopotentials
Pseudopotentials are used with great success when describing the potential directly within or
around the nucleus isn’t necessary. Core electrons are treated as one big system with the nucleus
due to their low interactions with other atoms or electrons and are not a part of the bonds formed
in a material. The valence electrons are the only ones we are interested in because they interact
with other atoms and make up the atomic bonds. Because only valence electrons have an impact
on our calculations, classifying some electrons as core and using a pseudopotential to describe our
system is ideal.
Finding an exact potential for our system would be difficult for two reasons. First, the potential
varies as
1
r
inside the nucleus so we have a divergent function as r goes to zero, with zero being
the center of the nucleus. Secondly, the valence electron wavefunctions must be orthogonal to the
core electron wavefunctions by Pauli’s exclusion principle. Thus, the valence electron wavefunctions must oscillate rapidly as we get closer to the center of the atom. These rapidly oscillating
wavefunctions require large plane waves and thus a large energy cut-off, which both lead to longer
calculations. One can quickly see where the need for pseudopotentials arises.
As stated before, the valence electrons dominate the calculations of most physical properties and
thus the need to have a higher energy cut-off to account for the core electrons can be circumvented
by using a pseudopotential. The pseudopotential does this by replacing the potential within the
core region with a smooth, slowly oscillating potential that lines up with the actual potential
outside of some cut-off radius. This allows us to use a smaller energy cut-off and get results that
are very accurate in a much more time efficient manner.
9
4.
CONVERGENCE STUDIES
4.1.
Core and Valence Tests
When choosing which pseudopotential to use we had a choice of few files to choose from the
ABINIT website which lists different files for each atom. Oxygen is widely used and thus had a file
that was recommended by ABINIT. For oxygen we picked the file with the smallest pseudopotential
radius. Vanadium was different in that no pseudopotentials were recommended by ABINIT. A few
files were listed but it was stressed that we check our results against previously confirmed data
before continuing with our calculations. With this in mind we again chose a pseudopotential with
the smallest radius. When choosing our pseudopotentials for vanadium it is also key to note that all
of the files available had the 3s and 3p orbitals of vanadium treated as valence electrons. Although
this may give a more accurate calculation, the 3s and 3p bands are very tightly bound and it is
feasible for one to think of treating them as core electrons.
Core electrons are called such because they are the electrons that do not participate in bonding.
This means that we can treat the core electrons and nucleus as one system with little interactions
with the other electrons or atoms. The more electrons you can treat as core, especially those of
which are tightly bound, the faster your calculations will go. This is due to how the cut-off energy
is defined which controls the number of wave functions to use in our DFT calculations. The cut-off
energy is
Ecut ≤
(G + k)2
2
(10)
The energy cut-off has a large part to due with the time each calculation takes to complete
because the number of plane waves needed increases as we classify more tightly bound electrons
as valence. When treating the 3s and 3p electrons of vanadium as valence we needed many plane
waves to account for these very tightly bound electrons. As we increased the number of plane
10
waves to use in our calculations we saw completion times for these tests increase at an exponential
rate.
FIG. 4: Total energy as a function of cutoff energy. By having a smaller energy cutoff our calculations will
inherently be faster due to there being less plane waves at each k-point.
As shown in Figure 4 we see that as we increase the cut-off energy we see a convergence
in the total energy of our system. To put some of these calculations into perspective, the selfconsistent calculation for an Ecut of 40 Hartrees (Ha) took around 20 hours to compute on the HPC.
Thankfully, ABINIT allows for this calculation to be performed only once and then subsequent
calculations can use that data. We chose to use an Ecut of 40 Ha which we considered a good
compromise between speed and accuracy.
This length of time in calculation was due largely in part to the use of the vanadium 3s and
3p electrons as valence electrons. In order to bypass this we looked into creating our own psuedopotential with the vanadium 3s and 3p electrons treated as core electrons. When creating
pseudopotentials for vanadium we used a program called AtomPAW which is also installed on the
HPC. We were able to come up with a pseudopotential that treated the tightly bound and highly
problematic 3s and 3p electrons as part of the core electrons and thus we were only left with 3d
and 4s electrons as valence. The next thing we did to test our new pseudopotential was to calculate
11
the band structure of rock-salt structure VO. The band structure from this calculation is shown
on the left in Figure 5.
FIG. 5: Band structure of rock-salt structure VO us-
FIG. 6: Band structure of rock-salt structure VO us-
ing pseudopotential with vanadium 3s and 3p elec- ing pseudopotential with vanadium 3s and 3p electrons treated as core.
trons treated as valence.
On the right, in Figure 6, we have the (correct) band structure of VO using the pseudopotentials
we found online where the 3s and 3p electrons of vanadium were treated as valence. One difference
that is noticeably clear is the number of bands in Figure 5 is greater than the number of bands
in Figure 6. These bands are called ”ghost” or ”phantom” bands and have no physical meaning.
These bands come from the inability of DFT and LDA to correctly approximate our system when
we treat the 3s and 3p electrons as core.
Going back to our original pseudopotential that treated these electrons as core we wanted
to check that this pseudopotential would not give us false data as well. To do this, we used our
pseudopotential to check many of the physical qualities of metallic vanadium and rock-salt structure
VO. These quantities include the minimum volume of the unit cell and from that we could find the
minimum lattice constant, the bulk modulus and the derivative of the bulk modulus. All of these
quantities can be easily found for a lot of materials and we can use these to check the accuracy of
our pseudopotential.
12
FIG. 7: Plot of total energy as a function of unit cell volume. This gave us the data plotted in the circles.
The red line is the plot of the Vinet equation of state Eqn. 11
To find these values is relatively straight forward and easy to do in ABINIT for a simple
structure. We used ABINIT to calculate the total energy as we slowly varied the volume of the
unit cell. From that we get a curve as shown in Figure 7. To this data we fit the Vinet equation
of state (EOS) which is given as
E(V ) = E0 +
o
h
i
2B0 V0 n
0
1/3
1/3
− 32 (B00 −1)[(V /V0 )1/3 −1]
2
−
5
+
3B
((V
/V
)
−
1)
−
3(V
/V
)
exp
0
0
0
(B00 − 1)2
(11)
The Vinet EOS allows us to iteratively fit a curve to our data and from that curve find the minimum
volume, V0 , bulk modulus, B0 , and the derivative of the bulk modulus, B00 . After fitting the Vinet
EOS of state were to our data for both metallic vanadium and rock-salt structure vanadium oxide
we obtained the values in question. These values are listed in Table I. We can clearly see that for
both VO and vanadium metal the pseudopotential we are using gives the values we would expect.
As seen above for both vanadium metal and VO our pseudopotential produced results within
a few percent of the value we would expect and also gives us a the correct band structure for VO
(as we saw) and vanadium metal (which is not shown in this paper). With this, we felt confident
that the pseudopotential was giving the correct results and moved on to working with V2 O5 with a
pseudopotential that treated the 3s and 3p electrons of vanadium as valence instead of core as we
13
TABLE I: Comparison of the values of the minimum lattice constant, bulk modulus and derivative of bulk
modulus for both metallic vanadium and rock-salt structure vanadium oxide with previously calculated
experimental values to test our pseudopotential’s accuracy.
would have liked. The first thing to do with V2 O5 is to optimize the unit cell and atomic positions
of each atom in the unit cell.
4.2.
Atom Position Optimization
Due to V2 O5 ’s complex structure we can’t optimize the structure by just changing the size of
the unit cell and finding the minimum lattice constant because this would need to be done for
each atom and would be a near impossible process. Luckily, ABINIT has built in parameters to
automatically optimize the structure. This was done for both bulk and mono-layer V2 O5 .
The tables above show the change of the atomic positions from the original positions in Å
for both the mono-layer and bulk structure of V2 O5 . The lattice constants for bulk V2 O5 after
optimization are also given. Lattice constants for the mono-layer are not given because the monolayer was defined the same as the bulk, except the lattice constant c was made quite large (around
14
20 Å) to make inter-layer interactions as close to zero as possible. Because of this definition of the
mono-layer, optimizing the unit cell of the mono-layer would be counter-productive and thus was
not done.
The only notable changes in the structure are the changes in the z-direction of both the vanadyl
oxygen and lattice constant of bulk V2 O5 . Both of these values suggest that the vanadyl oxygens
were originally defined to be a little farther away than they actually should be but it is not a
change that should be too worrisome as it is only about a 10% change from the original value with
all other changes being much smaller.
5.
5.1.
RESULTS
Bulk Structure
FIG. 8: Graphical view of the Brillouin zone of orthorhombic V2 O5 with the high symmetry points used in
the band structure calculation given.
With a working pseudopotential and appropriately defined value for the structure of V2 O5 and
the corresponding energy cut-off the only thing left to do was to calculate the band structure. We
will first present the band structure of bulk structure V2 O5 and then will look at the mono-layer.
The Brillouin zone for V2 O5 is shown in Figure 8. It is a simple orthorhombic structure with the
15
high symmetry points we use in our band structure calculations given.
FIG. 9: Band structure of bulk V2 O5 .The indirect
FIG. 10: Density of states (DOS) for bulk structure
band gap lies between the points R and Γ and has a
V2 O 5
value of 1.67 eV.
The band structure of bulk V2 O5 is shown in Figure 9 with the DOS corresponding to this band
structure given in Figure 10. The dotted red line is the Fermi energy and tells us what the highest
occupied band is in our band structure. This band structure and all subsequent band structures
were shifted down by the Fermi energy to set the Fermi energy to lie at zero. The high symmetry
points that my calculation passes through in reciprocal space are shown at the bottom of the band
structure. There are different spaces between each line because some paths correspond to longer
distances for our calculation to travel and thus required more steps between each point.
Another thing to note is that the full band structure is not shown. The band structure extended
all the way down to nearly -60 eV. These bands are the vanadium 3s like bands. Similar bands
appeared at other very low energies for both the vanadium 3p and oxygen 2s like bands, although
both of these were around -30 eV and -15 eV respectively. The reason we cut these off is because
they aren’t very interesting outside of telling us that the vanadium 3s and 3p and the oxygen 2s
electrons are all very tightly bound, which was already known. Also keeping all of these bands
16
meant that we had very low definition of the higher bands.
The first thing to note is the intermediate bands lying in between the valence (lower) bands
and conduction (upper) bands. This is the first thing we looked for in our calculation as this was a
very distinct property when Eyert and Hock Ref. [5] carried out their band structure calculation.
The split off bands are very clearly separated from the other bands as seen by the DOS around 2
eV. The band gap is indirect and lies between the points Γ and R and has a value of 1.6223 eV.
Eyert and Hock reported an indirect band gap of around 1.74 eV so our calculation of the band
structure is indeed accurate when compared to the literature but our band gap is even smaller than
the experimental value of 2.2 eV.
Finally, the upper valence bands and lower conduction bands can be characterized by the orbitals
they most represent. The lower conduction bands are oxygen 2p like. They lie at this high energy
because they are the bonding electrons of oxygen and whereas the lowest valence bands account
for the anti-bonding from the vanadium 3d electrons. If we were to show the higher energies we
would see vanadium 4s like bands lying at a much higher energy but much like the vanadium 3s
and 3p bands, these aren’t interesting outside of knowing their existence.
5.2.
Mono-layer
For the mono-layer of V2 O5 we defined this by making the unit cell lattice constant c very
large. This pushes the next layer very far away and makes inter-layer interactions minimal. When
talking about the Brillouin zone what this does is make the z-direction very small. When defining
the Brilloin zone we have that
kx =
π
,
a
ky =
π
,
b
kz =
π
c
(12)
From this we can see that as c becomes larger, kz becomes smaller. Although we didn’t choose an
outrageous value of a couple centimeters, we chose a c value of nearly five times the original value.
17
This means that the Brillouin zone for the mono-layer has very little movement in the z-direction
and thus we should (and did) see very flat bands along paths like Γ to Z.
FIG. 11: Plot of band structure of mono-layered
FIG. 12: Density of states (DOS) for mono-layer
V2 O5 . The indirect band gap lies between the points V2 O5
S and Γ and its value is 2.07 eV.
The band structure for the mono-layer of V2 O5 is shown in Figure 11 and the DOS for that
band structure is next to it in Figure 12. These two figures look very much like the bulk structure
band structure and density of states shown previously. Again, we have an indirect band gap of
about 2.07 eV which compares well with the literature values we found of 2.1 eV [3].
In the mono-layer band structure we again see these intermediate bands, but as for any qualities
that are prevalent in the mono-layer band structure that might make the mono-layer interesting
for fabrication nothing jumps out. The mono-layer acts much like the bulk structure does, as is the
case for most materials. Although this may sound like a let down the presence of the intermediate
bands could possibly make both bulk and mono-layer V2 O5 a very sought after material with the
interesting possibilities that come from those. More information on the intermediate bands is in
the discussion section.
18
TABLE II: Change of position for each atom in Å with respect to the original positions used in our calculations.
5.3.
Vanadyl Oxygen Vacancy
V2 O5 is very commonly used in catalysis because of its’ ability to ”give” and ”take” oxygen
very easily. From this it is a logical step that many different materials could arise from this
exchange of oxygen. One such material is V4 O9 . We obtain this new material by taking one of the
vanadyl oxygen and removing it from V2 O5 , as would be what happens in catalysis. Because we
are removing an oxygen we know that the Fermi level should move up because we are essentially
adding two electrons back into the structure but beyond that there is no previous work of this for
which we can compare our work.
The first thing we needed to do was optimize the structure of V4 O9 . This was again done using
ABINIT and the same scheme that we used to optimize V2 O5 . When optimizing V4 O9 there is a
strong possibility that some of the atoms will end up moving a great deal, especially the atoms in
the immediate vicinity to the vandyl oxygen we removed.
The results from our optimization of V4 O9 is shown in Table II. At a quick glance the one thing
that stands out is how much the vanadium that was directly below the vanadyl oxygen changes.
The vanadium moves up over an angstrom which is not something that can be ignored. We also
see that one of the bridge oxygen atoms near the vanadium moves down about three quarters of
19
FIG. 13: Caption
FIG. 14: Caption
an angstrom. This too, is a sizeable amount. An initial band structure calculation was taken with
just one of the vanadyl oxygen atoms removed and no optimization put into affect. This can be
seen in Figure 13 with its’ density of states shown in Figure 14.
As we predicted, the Fermi level has indeed moved up from the top of the valence bands to the
bottom of the conduction bands. Also to note is that the intermediate bands are less pronounced
but if one looks closely at the DOS you can see that there is some slight separation between the
lowest conduction bands and those that lie above them. The band gap is again indirect and has
a value of 1.66 eV which is almost exactly the same as the band gap found for the bulk structure
V2 O5 .
The intermediate bands have also changed. Like I mentioned earlier they have moved closer
to the conduction bands but there is now three broad intermediate bands instead of two narrow
intermediate bands like we saw in previous calculations. With broader split off bands and possibly
the shifting down of the main conduction bands could account for the lessened gap between the
intermediate bands and the main conduction bands. The intermediate bands may be broader due
to the high concentration of vacancies with a tenth of our total oxygen atoms removed in our
20
FIG. 15: Caption
FIG. 16: Caption
structure and a quarter of our vanadyl oxygen atoms removed.
The band structure for V4 O9 with the newly optimized positions for the atoms is shown in Figure
15 with the DOS for that band structure given in Figure 16. From a quick glance comparing the
last figures for the band structure and DOS Figures 13 and 14 we see a few things. First we can
see that the intermediate bands which were already not as detached from the conduction bands
are almost non-existent. If we look at the DOS we see that there is no longer a clearly defined
gap between the intermediate bands and the high conduction bands. One reason for this is that
the conduction bands may have shifted down. The band gap is again indirect but only has a value
of about 0.748 eV which is less than half the value of the band structure with the non-optimized
atom positions.
6.
6.1.
DISCUSSION
Intermediate Bands
Intermediate bands have made some great headway when looking at solar cells. In a recent paper
in Nature Photonics by Luque, Marti and Stanley Ref. [10] the impact these intermediate bands
21
could have on solar cell efficiency is laid out. The main way these solar cells work is by taking an
intermediate band material and sticking it between an n- type and p- type semiconductors. These
semiconductors would act as contacts between the conduction band for the n- type and the valence
band for the p- type.
For a single gap solar cell, the maximum efficiency has been estimated to be around 41% with
the solar cell at room temperature Ref. [9]. On the other hand, intermediate band solar cells could
reach efficiencies over 60% at room temperature Ref. [10]. The ideal intermediate band would
have a total band gap of around 1.95 eV with two sub-band gaps between the valence band and
intermediate bands and then between the intermediate band and conduction bands of 0.71 eV and
1.24 eV Ref. [10].
Although V2 O5 and V4 O9 do not quite have these values, there are a few tricks people can use
when fabricating materials to acquire the desired band gap. Bulk V2 O5 has a total band gap of
around 2.2 eV in experiments but this could be reduced. When constructing intermediate band
materials they are usually doped with specific amounts of impurities to artificially make these
intermediate bands appear; whereas V2 O5 (and V4 O9 to some extent) have these intermediate
bands with no doping required.
One problem with these intermediate bands is making sure that they do not act like a trap
for electrons. This would be the case when electrons are both excited from the valence bands to
the intermediate bands and electrons falling from the upper conduction bands back down to the
intermediate bands. This would greatly reduce the efficiency of the solar cells and is something
that needs to be avoided when constructing these intermediate band solar cells.
22
6.2.
Vanadyl Oxygen Vacancy
The removal of a vandyl oxygen from V2 O5 showed us a couple interesting properties. Firstly,
this removal showed us how greatly the structure would change. No longer did the optimization
yield a change of a few hundredths angstroms and sometimes tenths of angstroms but we saw
changes greater than one angstrom in the case of the vanadium atom sitting below the oxygen
atom we removed. We also saw a broadening in and addition of a band to the intermediate bands.
This most likely came from the the high concentration of vacancies in our new structure.
With this optimization we would expect to see some changes in the band structure and DOS,
but what we ended up seeing were vast differences between the band structure of the original
positions and those of the optimized position, with a band gap decrease of over 50% and even less
pronounced split off bands. These major changes to our calculations further reinforced the need
for us to optimize our structure and use those new positions in our calculations.
7.
CONCLUSIONS
When looking at whether or not to classify the vanadium 3s and 3p electrons as valence or core
in our pseudopotential we found that they need to be included as valence electrons. Even though
they are very tightly bound, evidenced by the fact that their bands lie about 30 eV or lower than
the upper valence bands. We also saw that the psuedopotentials we created with the vanadium
3s and 3p electrons treated as core electrons gave us ”phantom” bands in our band structure of
rock-salt structure VO. With a pseudopotential that had the vanadium 3s and 3p electrons treated
as valence electrons we saw the correct band structure for rock-salt structure VO, the correct values
for the minimum lattice constant, bulk modulus and derivative of the bulk modulus for both VO
and vanadium metal. This allowed us to be confident in our pseudopotential and we moved on to
our calculations of optimizing the structure of both bulk and mono-layer V2 O5 and eventually the
23
band structure of both of these.
When optimizing the structures of bulk and mono-layer V2 O5 we used a scheme within ABINIT
to minimize the force below a specified tolerance, at which point the calculation was converged.
This optimization showed a maximum difference of only about 10% in the z-direction with the
difference for the x and y-directions being smaller than this. This comes from the difficulty of
DFT to treat van der Waals interactions. With our optimized structure we moved on to our band
structure calculations.
The bulk structure and mono-layer band structures for V2 O5 had been previously calculated in
literature and we were able to get results that agreed quite nicely with the literature. We had a
band gap of 1.67 eV for the bulk structure and a band gap of 2.07 eV for the mono-layer. This
difference in band gap comes from the fact that the weak interactions between the up and down
pointing vanadyl oxygen atoms has been greatly reduced and thus we have a less broad valence
band, effectively widening the band gap.
One of the more interesting properties to arise from the band structure calculations were intermediate bands that were just below the main conduction bands. These intermediate bands have
been getting a lot of recent attention for their theoretical use in solar cells. With a maximum
efficiency for single band gap solar cells estimated to be around 41%, two stage band gap solar cells
could achieve an efficiency above 60%. This increase could make solar cells a much more viable
option with nearly a 50% increase in yield coming from each solar cell.
Finally we looked at what would happen if we doped our structure with more electrons. This
was done by removing a vanadyl oxygen atom which effectively adds two electrons back into our
structure, as evident in the Fermi energy moving from the top of the valence band to the bottom of
the conduction band. Initial band structure calculations were carried out with this new structure
and we found that three split off bands existed instead of two as we saw previously. The band gap
was almost identical to the bulk structure band gap with a value of 1.66 eV.
24
We then optimized the structure of V4 O9 and saw massive changes to the nearest neighbours
of the vanadyl oxygen we removed, with changes on the order of angstroms instead of tenths of
hundredths of angstroms. This optimized position was then used in our band structure calculations
with surprising results. We saw that the band gap was now 0.74 eV which is less than half of the
originally calculated band gap. Also, the intermediate bands were far less prevalent which had
to do with the fact that the bottom conduction bands moved down which both intermixed the
intermediate bands with the other conduction bands and also lessened the band gap.
8.
ACKNOWLEDGEMENTS
I would like to thank my adviser Professor Walter Lambrecht in the Physics Department for
his constant guidance and help throughout this project. When first starting I was unsure how to
even navigate a Unix server, so his ever lasting patience was much appreciated. I would also like
to thank Professors Singer and Petschek for their guidance throughout the senior project seminar
course.
25
9.
BIBLIOGRAPHY
[1] ”ABINIT.” Home ABINIT. Web. 24 Apr. 2012. <http://www.abinit.org/>.
[2] ”Calculation of Electronic Band Structures GPAW.” Campos-wiki Pages. CAMd Et Al, 09 Dec. 2011.
Web. 09 Dec. 2011.
[3] Chakrabarti, A., K. Hermann, R. Druzinic, M. Witko, F. Wagner, and M. Petersen. ”Geometric and
Electronic Structure of Vanadium Pentoxide:A Density Functional Bulk and Surface Study.” Physical
Review B 59.16 (1999): 10583-0590. Print.
[4] Clark, Dr S. J. ”Plane Wave Basis Sets.” Condensed Matter Theory. 30 Jan. 2003. Web. 27 Apr. 2012.
[5] Eyert, V., and K. H. Hck. ”Electronic Structure of V2 O5 : Role of Octahedral Deformations.” Physical
Review B 57.20 (1998).
[6] ”Graphene Has Record-breaking Strength.” Physicsworld.com Homepage. Web. 28 Apr. 2012.
<http://physicsworld.com/cws/article/news/2008/jul/17/graphene-has-record-breaking-strength>.
[7] International Tables for Crystallography (2006). Vol. A, ch. 7.1, Space group 59.
[8] Londero, and Schrder. Role of van der Waals bonding in the layered oxide V2 O5 : First-principles
density-functional calculations. Physical Review B 82, 054116 (2010).
[9] Luque, A. & Marti, A. ”Increasing the efficiency of ideal solar cells by photon induced transitions at
intermediate levels.” Phys. Rev. Lett. 78, 5014-5017. (1997).
[10] Luque, Antonio, Antonio Marti, and Colin Stanley. ”Understanding Intermediate-band Solar Cells.”
Nature Photonics 6 (2012): 146-52. Print.
[11] ”New Graphene-based Material Shows Promise to ’revolutionize Electronics Industry’” Zeenews Technology Web. 28 Apr. 2012.
[12] van Zeghbroeck, B. ”Energy Bands.” Department of Electrical, Computer, and Energy Engineering
(ECEE) — University of Colorado at Boulder. 2011. Web. 09 Dec. 2011.
[13] X. Gonze, et al. Computer Phys. Commun. 180, 2582-2615 (2009). ”ABINIT : First-principles approach
of materials and nanosystem properties.”
26
[14] X. Gonze, G.-M. Rignanese, M. Verstraete, et al. Zeit. Kristallogr. 220, 558-562 (2005).
[15] Zhao, Liang. Layered Vanadium Pentoxide (V2 O5 ) November 17th, 2010.