International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 3, Issue 5, May 2014
ISSN 2319 - 4847
Density functional theory study of InP quantum
dot and oxidized surface
Mohammed T. Hussein1, Thekra Kasim2, Mudar A. Abdulsattar3 and Askandar.K.Kaka4
1&2
Department of Physics, College of Science, University of Baghdad, Baghdad, Iraq
3
Ministry of Science and Technology, Baghdad, Iraq
Abstract
Density functional theory at the generalized-gradient approximation level coupled with large unit cell method is used to simulate
the electronic structure of zincblende indium phosphide nanocrystals that have dimensions 2-2.8 nm. Nanocrystals structure is
divided into two parts core and surface. The calculated properties include lattice constant, energy gap, valence band width, highest
occupied orbital, lowest unoccupied orbital, density of states etc. of both core and surface. Two kind of cells are investigated,
Bravais and primitive cell multiplets. Results show that shape effects are strong compared with other previously investigated
semiconductor elements and compounds which are attributed to large differences between indium and phosphorus atomic radii
and different atomic electronic properties. Many properties fluctuate indefinitely while other properties converge to definite
values. Quantities such as lattice constant, energy gap and ionicity converge to definite values. However, highest occupied orbital,
lowest unoccupied orbital and Fermi level fluctuates indefinitely depending on the shape and surfaces of the nanocrystal.
Calculations also prove that smaller nanocrystals of InP are more ionic than larger ones. The above shape differences are
reflected on experimental bulk InP based device applications. Oxidized surface shows slightly less lattice constant, smaller gap
and wider both valence and conduction bands with respect to the core region. In addition to the new view and unreported
quantities acquired using present large unit cell method the present method is in good agreement with available literature.
Keywords: Density functional theory , quantum dot , generalized – gradient approximation.
1. INTRODUCTION
Zincblende cubic indium phosphide is a direct gap semiconductor that is used frequently in devices such as optoelectronic
and laser devices. Its 1.4 eV gap [1, 2] is between 1.1 eV of silicon and 1.43 eV of gallium arsenide which are the most
used semiconductors. This property gives InP the opportunity to take similar position in semiconductor industry. InP is
usually used in high power and high frequency devices and as substrate for the growth of other zincblende structures [15]. Nanoscale size reduction that are usually accompanied with blue shift in nanocrystals gap gives InP the opportunity to
maneuver the gap value to be more appropriate in certain applications such as laser devices [3, 5]. The present work will
be involved in calculating electronic properties of InP nanocrystals as the size and shape of these nanocrystals change. We
shall use density functional theory at the generalized-gradient approximation level (Perdew, Burke, and Ernzerhof PBE
approximation) coupled with large unit cell method (LUC- DFT) to simulate the electronic structure of InP which is a
well developed theory that had been applied repeatedly for the nanocrystals electronic structure [6-12].
2. THEORY
The large unit cell method is a supercell method that is restricted to the k=0 point in the wave vector space [6-12]. This
method gets benefit from the fact that after some critical size (usually 1 nm in diameter) of the nanocrystal, a core with
ideal bulk structure is formed that has high symmetry and does not need to go through the geometrical optimization
routine usually done in computing molecular or nanoscale electronic properties [9-12]. On the other hand, the nanocrystal
surface still needs to be geometrically optimized [9, 10, 13]. The k=0 restriction is an essential part of the theory of LUC
since quantum dots are restricted to one point in space and does not need to sum contributions from other points in real or
reciprocal space as is usually done in bulk calculations. An interaction range of four neighbors is imposed to match with
the fact that surface reconstruction is restricted to outer three to four layers [14].
Semiempirical methods were usually the choice in the older calculations of LUC method [6-8] because of computational
time consumed in repeating the central cell in three dimensional space. At the present time most calculations are reported
with ab initio Hartree-Fock method [9-12]. Correlation corrections are rarely added because of computational time needed
using quantum chemistry programs such as the present Gaussian 03 program package [15, 16]. The present LUC-DFT is
an exception from the previous literature. Note that the difference between supercell methods for bulk electronic structure
calculations and the present nanocrystal core electronic structure calculations is the restricted periodicity to the size of the
core part of the nanocrystal [9, 10, 17, 18].
In order to simulate zincblende nanocrystals electronic structure using LUC method two kinds of cells are available:
primitive and Bravais cell multiplets (Fig. 1 and 2 respectively). The main difference between these two kinds of cells is
the shape and its associated surfaces. Primitive cells are parallelograms while Bravais cells are cubic in shape [14]. This
difference in shape results in many fluctuations in the electronic structure of diamond or zincblende structured materials
Volume 3, Issue 5, May 2014
Page 1
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 3, Issue 5, May 2014
ISSN 2319 - 4847
[9-12]. However, the present fluctuations are the strongest between the investigated materials as we shall see in the next
sections. Three primitive cell multiplet cores are investigated namely: 16, 54 and 128 atoms. Two Bravais cell multiplet
cores are investigated namely 8 and 64 atoms. The next Bravais large unit cell (216 atoms) is out of the Gaussian 03
program memory capabilities [16]. For details on the properties of these cells we can refer particularly to reference [17] in
addition to the previously mentioned LUC references. The periodic boundary condition (PBC) utility is used to calculate
(001)-(1×1) InP nanocrystal oxidized surface using slab geometry of the stoichiometry In 8P8 O4. Computational time and
resources was the main obstacle in investigating larger lattices or surfaces.
Fig. 1: (color online) InP 54 atoms core LUC (parallelepiped shape primitive cell multiplet).
Fig. 2: (color online) InP 64 atoms core LUC (cubic Bravais cell multiplet).
3. CALCULATIONS AND RESULTS
The three dimensional (3D) periodic boundary condition (PBC) utility is used for nanocrystal core calculations and the
2D-PBC for the surface calculations. To optimize core structure only the lattice constant is needed to be optimized. At the
surface part we need to optimize distances between layers including oxygen layer in addition to optimizing lattice
constant. In the optimization procedure we pick the minimum energy structure for every investigated LUC. From the set
of minimum structures we can draw the following figures for the core part: Figs. 3 and 4 show the variation of the lattice
constant and the energy gap with the variation of the number of core atoms respectively. Fig. 5 shows the variation of
both valence and conduction band widths with the variation of the number of core atoms. Highest occupied molecular
orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) variation with the variation of the number of core
atoms are shown in Fig. 6. Fig. 7 shows the variation of ionicity with the variation of the number of core atoms. The
properties of the InP nanocrystal (001)-(1×1) oxidized surface is also investigated using slab geometry of the
stoichiometry In8 P8O4 LUC. This stoichiometry is repeated periodically (in a 2D surface) and restricted to the actual
surface size of the nanocrystal. The oxidized surface has an optimized lattice constant of 0.53 nm. The energy gap of this
surface is 0.0198 eV while the surface valence and conduction bands have the values of 23.8 and 13.16 eV respectively.
Surface calculations are more demandable in computer time than core because of lack of symmetry which is the reason for
restricting our calculations for the upper mentioned surface only. Fig. 8 shows the surface slab ionic charges in a path that
is perpendicular to the investigated surface. Finally Fig. 9 shows the degeneracy of states of the core part of 54 and 64
atom LUCs.
Volume 3, Issue 5, May 2014
Page 2
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 3, Issue 5, May 2014
ISSN 2319 - 4847
Lattice constant (nm)
0.572
0.57
0.568
0.566
0.564
0.562
0.56
0.558
0
20
40
60
80
100
120
140
Number of core atoms
Fig. 3: Lattice constant variation with the number of core LUC atoms of InP nanocrystals.
1.35
Energy gap (eV)
1.3
1.25
1.2
1.15
1.1
1.05
1
0
20
40
60
80
100
120
140
Number of core atoms
Fig. 4: Energy gap variation with the number of core LUC atoms of InP nanocrystals.
16
14
Band width (eV)
12
10
Valence
Conduction
8
6
4
2
0
0
20
40
60
80
100
120
140
Number of core atoms
Fig. 5: Valence and conduction bands variation with the number of core LUC atoms of InP nanocrystals.
Volume 3, Issue 5, May 2014
Page 3
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 3, Issue 5, May 2014
ISSN 2319 - 4847
16
Molecular orbital energy (eV)
14
12
10
LUMO
8
HOMO
6
4
2
0
0
20
40
60
80
100
120
140
Number of core atoms
Fig. 6: LUMO and HOMO levels variation with the number of core LUC atoms of InP nanocrystals.
0.25
Ionicity (a.u.)
0.2
0.15
0.1
0.05
0
0
20
40
60
80
100
120
140
Number of core atoms
Fig. 7: Ionicity variation with the number of core LUC atoms of InP nanocrystals.
0.6
P
0.3
Atomic charge (a.u)
In
In
In
In
0
0
2
4
6
8
10
12
P
P
P
O
-0.3
O
-0.6
Layer number
Fig. 8: Atomic charges as a function of layer depth in a slab geometry calculation of InP.
A path that is perpendicular to the investigated surface is used to draw the figure.
Volume 3, Issue 5, May 2014
Page 4
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 3, Issue 5, May 2014
ISSN 2319 - 4847
35
a
30
Degeneracy of States
25
20
15
10
Eg
5
0
-5
0
5
10
15
20
25
Energy (eV)
64InP
50
45
Density of states
40
35
30
25
20
Eg
15
10
5
0
-15
-10
-5
0
5
10
15
20
Energy (eV)
Fig. 9: Degeneracy of states of 54 and 64 atom LUCs (a and b respectively) as a function of energy levels are shown
in this figure. Valence band is shown with bold lines while conduction band is shown with ordinary lines. The energy
gap is indicated between the two bands.
4. DISCUSSION AND CONCLUSIONS
Fig. 3 shows the variation of the optimized lattice constants as the core of InP nanocrystal grows up in size. The
differences between points decrease as the number of core atoms increases which shows that some limit for the variation
of lattice constant exist. On the other hand, the 8 atom LUC core lattice constant is nearer to the value of 64 atom core
lattice constant which is also the case between 16 atom, 54 atom and 128 atom core lattices. Both 8 and 64 atom core
cells are Bravais lattice multiplets while 16, 54 and 128 atom cores are primitive cell multiplets. Although this shape
effect was found in previous literature [9, 10, 17], the present one is the largest between these effects. Normally, IV
elements [9, 17] or IV-IV compounds [18] does not show the present strong shape effects. Even III-V compounds [10]
between elements in the same period of the periodic table do not show this effect. However, all previous nanocrystal
calculations using LUC show variation in lattice constant length [9, 10, 12, 17, 18]. As a result of the above mentioned
calculations we attribute this shape effect to the large differences between indium and phosphorus atomic electronic
properties and atomic sizes. The two atoms (indium and phosphorus) are two rows and two columns apart in the periodic
table that invokes different electronic properties such as electron affinity and ionization potential in addition to the d
orbital electrons that exist in indium atom and not in phosphorus atom. A surface that is terminated by one of these two
atoms is completely different in electronic properties than that which is terminated by the other atom as we shall see later
in Fig. 8. The present largest cell (128 atoms) value of lattice constant is 0.56 nm which is comparable to the
experimental bulk value of 0.58 nm [19]. Although there is a long way between the present 128 atom core nanocrystal
and bulk size, the present value of the lattice constant is acceptable taking into consideration the usual systematic error
inhibited in molecular orbital calculations [6-12] that underestimates this property especially for high atomic number
elements such as germanium [9]. Theoretical variation of lattice constant was first pointed out in reference [12]; however,
precise experimental measurement that confirms this variation was accomplished for diamond nanocrystals only [20].
Fig. 4 shows the variation of the energy gap of InP nanocrystals with the number of core atoms. This figure also show the
same trend mentioned above for the lattice unit length. Although shape effect was also recognized for the energy gap in
the previous LUC studies [9, 10, 17, 18], this effect is stronger in InP nanocrystals.
Volume 3, Issue 5, May 2014
Page 5
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 3, Issue 5, May 2014
ISSN 2319 - 4847
Valence and conduction bands in Fig. 5 show the same trend of shape effects. Cubic cells (8, 64 atoms) have wider bands
than parallelepiped cells. This effect was also recognized but with less obvious shape effects in previous studies [9, 10, 17,
18]. The same is true in Fig. 6 for the HOMO and LUMO orbital energies. HOMO and LUMO orbitals do not only show
shape effects, but also continues to fluctuated indefinitely due to the formation of different surfaces in larger nanocrystals
[10]. Other quantities that can be derived from HOMO and LUMO orbital energies include affinity, ionization energy and
Fermi level. Affinity and ionization potential is connected approximately by a negative sign to LUMO and HOMO
energies respectively [13]. Fermi level is the average of LUMO and HOMO energies [21]. As a result of these relations,
these three quantities also have the same shape effect that affect HOMO and LUMO energies.
The only property that does not show shape effects (at least in our calculations range) is the atomic ionicity as depicted in
Fig. 7. This figure shows that smaller nanocrystals are more ionic than larger ones. This property was also noted
previously in LUC calculations for other nanocrystals [18].
Surface electronic structure calculations show that energy gap is very small (0.0198 eV) at the oxidized surface which is
the case of previously investigated oxidized surfaces [9, 17, 18]. The unit lattice length is also slightly smaller (0.53 nm)
which reflects good adherence of the oxide layer. Surface reconstruction calculations are nearly similar to full geometrical
optimization with the difference that we can perform calculations for a single surface and not all surfaces as in full
geometrical optimization of a nanocrystal. Surface valence and conduction bands widths (23.8 and 13.16 eV respectively)
are very much wider than their core counter part which is a consequence of symmetry breaking at the surface. The highly
degenerate states at the core are split at the surface to full the energy gap region and widen both valence and conduction
bands. Fig. 8 shows atomic charges as a function of layer depth in a slab calculation. A path that is perpendicular to the
investigated surface is used to draw the figure. As we can note from this figure, the phosphorus terminated surface is
more disturbed (heavily reconstructed) than that of indium. The reason is that phosphorus layer is usually negatively
charged. As oxygen atoms are attached to the surface they absorb this negative charge from phosphorus atoms. This
change induces disturbances in layers charges that continue three to four layers inside. This process does not occur at the
indium terminated surface since indium atoms are already positively charged. Combining Fig. 8 with Fig. 7 shows that
InP nanocrystals are ionic nanocrystals with decreasing ionicity as the nanocrystal size increases.
Figure 9 summarizes the physics of many of the previous figures. Degeneracy of states of 54 and 64 atom LUCs show the
above valence and conduction band widths variation between primitive and Bravais cells. The variation of HOMO and
LUMO movements are also obvious in this figure. The highest degeneracy of primitive cells is higher than that of Bravais
cells. This shows that symmetry of primitive cells is more definite than Bravais cells. The primitive cell multiplets have
{111} bounding surfaces while Bravais cell multiplets have {100} surfaces [14]. The present theoretical differences
between these surfaces are also found experimentally at the bulk scale in n-channel metal-oxide-semiconductor fieldeffect transistors [22]. As a result the maximum drain current is greater by a factor of 3.5 on the (111) surface compared
to devices fabricated on the (100) surface at the same bias conductions. We also believe that the present differences are
responsible for electrical interface instabilities observed in InP-based devices [23] and different surfaces [24].
Comparison of present LUC-DFT method with other theoretical investigations strengthens our choice of this theory. The
approximation (k=0) can not be applied for example to k.p method since the wave vector (k) expansion is an essential
part of the theory. Instead, the k.p theory is more appropriate to nanowires or nanoflowers …etc or at least quantum dots
that are larger than a critical size [25, 26] so that empirical parameters can be used to manage the rest of the calculations.
Using k.p theory for quantum dots that are less than these critical sizes results some times with an errors or the use of
empirical parameters that are far from there expected usual values [27, 28]. It is difficult to find one model that satisfies
all the requirements of all situations encountered in nanoscience calculations. As an example, the present LUC that does
not use any empirical parameters can not reach large number of atoms or large sizes of nanoparticles. Empirical
parameters relax these restrictions by avoiding some part of the computational procedure and replacing it by experimental
findings. Most of present nanoscale models have their origin from bulk solid state physics and chemistry calculations.
Modifications on these models are necessary to be applied successfully to the nanoscale. In this sense, Pseudopotential
theory seems more successful than k.p theory in modeling nanowires in available literature [5, 29, 30]. However,
pseudopotential parameters have to be modified to fit the implications of quantum dot or nanowires sizes. On the
contrary, LUC method does not use any parameters and it is suitable (theoretically) to be applied for all sizes starting
from nanoscale particles and ending by bulk solids with the exception of computational efforts that can not be afforded by
present computational facilities for large sizes of nanoparticles or bulk. Comparison with available experimental and
theoretical results of other methods reveals the correctness and good results of the present LUC method at least in the
present range of sizes. As an example quantum dots and nanowires of comparable diameter to the present quantum dots
have energy gaps very near to the present range (1.0 to 1.6 eV) and are in good agreement with our results [26, 29, 30,
31]. This includes the bulk experimental value 1.4 eV which is within the above mentioned interval [1, 2]. Notably,
nanowires calculations proved the effect of different crystallographic axis using pseudopotential theory [30] in similar
results to our quantum dots. Many of the InP present calculated quantities such as the variation of HOMO, LUMO,
ionicity, and lattice constant as a function of particle shape are not reported previously.
Volume 3, Issue 5, May 2014
Page 6
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 3, Issue 5, May 2014
ISSN 2319 - 4847
4. Conclusions
We conclude from the above results that many properties of InP nanocrystals change abruptly at the nanoscale.
Calculations show that shape and bounding surfaces of the nanocrystal is an important factor in its electronic properties.
Lattice unit length is larger for cubic (Bravais) lattices than in parallelepiped (primitive) lattices in InP nanocrystals.
Energy gap, valence band, conduction band, HOMO and LUMO orbitals all show shape effects. However some properties
such as ionicity do not show this effect but decreases smoothly as nanocrystal core size increases. Nanocrystal oxidized
surfaces show smaller energy gaps and lattice unit lengths. Symmetry breaking surfaces widen both valence and
conduction bands At the expense of reducing the energy gap. Degeneracy of states as a function of energy show a
summary of the above mentioned shape properties. Experimental evidence of the present shape differences are found and
utilized in devices fabricated on different InP bulk surfaces. In addition to the new view and unreported quantities
acquired using present large unit cell method the results of present method is in good agreement with available literature.
References
[1] R. A. Lewis, Y. J. Wang, Phys. Rev. B 71, 115211 (2005).
[2] J. Peretti, H. Drouhin, D. Paget and A. Mircea, Phys. Rev. B 44, 7999 (1991).
[3] M. M. G. Alemany, Luis Tortajada, Xiangyang Huang, Murilo L. Tiago, L. J. Gallego, and James R. Chelikowsky,
Phys. Rev. B 78, 233101 (2008).
[4] G. Chen, S. F. Cheng, D. J. Tobin, L. Li, K. Raghavachari, and R. F. Hicks, Phys. Rev. B 68, 121303(R) (2003).
[5] Randy J. Ellingson, Jeff L. Blackburn, Jovan Nedeljkovic, Garry Rumbles, Marcus Jones, Huaxiang Fu, and Arthur J.
Nozik, Phys. Rev. B 67, 075308 (2003).
[6] R. Evarestov, M. Petrashen, E. Lodovskaya, Phys. Status Solidi b 68, 453-461 (1975).
[7] A. Harker, F. Larkins, J. Phys. C 12, 2487-2495 (1979).
[8] Ibtesam O. Radi, Mudar A. Abdulsattar, Ahmed M. Abdul-Lettif, Phys. Status Solidi (b) 244, 1304-1317 (2007).
[9] N. H. Aysa, M. A. Abdulsattar, A.M. Abdul-Lettif, Micro & Nano Letters 6, 137-140 (2011).
[10] M. A. Abdulsattar, Electron. Mater. Lett. 6, 97-101 (2010).
[11] M. A. Abdulsattar, Khalil H. Al-Bayati, Phys. Rev. B 75, 245201 (2007).
[12] M. A. Abdulsattar, Physica E 41, 1679-1688 (2009).
[13] S.J. Sque, R. Jones, and P.R. Briddon, Phys. Rev. B 73, 085313 (2006).
[14] C. Kittel, "Introduction to Solid State Physics", 7th edition, Wiley, (1996).
[15] J. B. Foresman and AEleen Frisch, "Exploring Chemistry with Electronic Structure Methods: A Guide to Using
Gaussian", 2nd edition, Gaussian Inc. (1996).
[16] M. J. Frisch, G. W. Trucks, H. B. Schlegel, et al.: Gaussian 03, Revision B.01, Gaussian, Inc., Pittsburgh, PA,
(2003).
[17] M. A. Abdulsattar, Solid State Sci. 13, 843 (2011).
[18] H.M. Abduljalil, M.A. Abdulsattar, S.R. Al-Mansoury, Micro & Nano Lett. 6, 386 (2011).
[19] J. A. McCaulley, V. M. Donnelly, M. Vernon, and I. Taha, Phys. Rev. B 49, 7408 (1994).
[20] C. Wen, Z.H. Jin, X.X. Liu, X. Li, J.Q. Guan, D.Y. Sun, Y.R. Lin, S.Y. Tang, G. Zhou, J.D. Lin, Spectrosc. Spetral
Anal. 5, 681 (2005).
[21] S. M. Sze and K. K. Ng, "Physics of semiconductor devices", 3rd edition, Wiley (2007).
[22] C. Wang, M. Xu, R. Colby, D. W. Zhang, and P. D. Ye, Electrochem. Solid State Lett. 15, H27 (2012).
[23] P. Tomkiewicz, B. Adamowicz, M. Miczek, H. Hasegawa, and J. Szuber Appl. Surf. Sci., 254, 8046 (2008).
[24] O. Pulcia, K. Lüdgeb, W. Schmidta and F. Bechstedt, Surf. Sci., 464, 272 (2000).
[25] O. Stier, M. Grundmann, and D. Bimberg, Phys. Rev. B 59, 5688 (1999).
[26] P. E. Faria Junior and G. M. Sipahi, J. Appl. Phys. 112, 103716 (2012).
[27] S. Tomic and N. Vukmirovic, Phys. Rev. B 86, 159902(E) (2012).
[28] G. Bester and A. Zunger, Phys. Rev. B 71, 045318 (2005).
[29] T. Akiyama, K. Nakamura, and T. Ito, Phys. Rev. B 73, 235308 (2006).
[30] D. Karanth and H. Fu, Phys. Rev. B 74, 155312 (2006)
[31] C. L. dos Santos and P. Piquini, Phys. Rev. B 81, 075408 (2010).
AUTHOR
Dr.Mohammed T.Hussein completed his Ph.D. at the physics department in laser spectroscopy from
Complutense University – Madrid-Spain in 1995. His research interests lie in the field of organic
semiconductor and molecular spectroscopy. He is currently a member of the Nanotechnology &
Optoelectronics Research Group at the Physics department of Baghdad University.
Volume 3, Issue 5, May 2014
Page 7
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 3, Issue 5, May 2014
ISSN 2319 - 4847
Dr. Thekra Kasim completed her Ph.D. at the physics department , College of Science , Baghdad
University , specialization in Solid state Physics . She is currently a member of the Materials research group at
the physics department of Baghdad University .
Dr. Mudar Ahmed Abdulsattar completed his Ph.D in Solid State Physics: Baghdad University, his worked
in ministry of science and Technology , Director of materials science , Building Materials Center in Ministry
of science and Technology ,2007 . his research and lectures in industrial applications, mostly related to
material’s science and computer simulation methods, theoretical research covers areas of electronic structure of
covalent and zinc-blend .
Mr. Askandar K.Kaka has MS.C in physics , he is a member at physics department – Koya university , Iraq.
Now he is Ph.D student in physics department , College of Science , University of Baghdad .
Volume 3, Issue 5, May 2014
Page 8