Polytypic nanowhiskers: electronic properties in the vicinity of the band edges

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by  Paulo Eduardo de Faria Junior

Polytypic nanowhiskers: electronic properties in the vicinity of the band edges
Faria Junior, P. E. and Sipahi, G. M.
Instituto de F´ısica de S˜ao Carlos, USP, S˜ao Carlos, SP, Brazil
(ΩDated: January 12, 2011)
The increasing interest of nanowhiskers for technological applications has led to the observation
of the zinc-blend/wurtzite polytypism. Polytypic nanowhiskers could also play, by their character-
istics, an important role on the design of optical and electronic devices. In this work we propose
atheoretical modeltocalculate theelectronic propertiesofpolytypiczinc-blend/wurtzitestructure
1
1 in the vicinity of the band edges. Our formulation is based on the ~k ·p~ method connecting the
0 irreducible representations in the interface of the two different crystalline phases by group theory
2 arguments. Analyzing the composition of the states in theΓ point and the overlap integrals of the
envelope functions we predict energy transitions that agree with experimental photoluminescence
n
spectra.
a
J

1 INTRODUCTION separated: the electrons will be concentrated in the ZB
1
region while the holes will be in the WZ region. A sum-

] Recently an increasing number of papers have re- mary of NWs growth, properties and applications was
ll portedhighqualityandcontrolgrowthofnanowiresand made by Dubrovskii et al. [9].
a
nanowhiskers (NWs) of III-V compounds such as InP, Fromthetheoreticalpointofview,theliteratureabout
h
- InAs, GaAs and GaP [1–7]. Nanoelectronics, nanopho- these III-V NW structures, is focused in growth stabil-
s tonics and solar cells are some of the technological ap- ity,surfaceenergiesandformationofpolytypism[10–16].
e
m plications for these nanostructures like described in the Only recently theoretical calculations of the electronic
references above. properties in the band edges have been done for such
.
at Vapor-liquid-solid(VLS)method[8]wasused,in1964, structures [17]. None has been done in the vicinity of
m byWagnerandEllistogrewasiliconwhisker. Sincethen, the band edges.
other methods like molecular beam epitaxy (MBE) and The polytypism was investigated in a paper of Mu-
-
d metal organic vapor phase epitaxy (MOVPE) have also rayama and Nakayama [18] by ab initio calculations of
n beenappliedtothegrowthofNWs. Thesetechniquesal- the bandoffsetsinWZ/ZBinterfaces. Ausefulschemat-
o
lowthefabricationofhomoandheterostructuresinboth ics of the correspondence between the symmetry of the
c
[ axial and radial directions opening new opportunities to states of the WZ and ZB structures has been presented
engineering novel device applications. in the same article.
2
A heterostructure is composedby assemblingdifferent Theaimofthisstudyistoproposeatheoreticalmodel
v
7 materials side-by-side along a growth direction. The ho- basedongrouptheoryargumentsandthe~k p~methodto
·
2 mostructure is a similar concept, however, it is not the calculate the electronic band structure in the vicinity of
2 kindofmaterialthatisdifferentbutitscrystallinestruc- the band edges andenvelope functions of a WZ/ZB/WZ
0 ture. This effect is also known as polytypism and have homostructureforindiumphosphide(InP).Theeffective
.
2 been observed in NWs structures of several III-V com- mass parameters for the WZ bulk phase of InP were ex-
1 pounds. tractedfromapaperbyPryorandDe[19]thatpredicted
0
Under standard temperature and pressure conditions, thebandstructuresofsomeIII-Vbulkcompoundsinthe
1
: the III-V compounds have as the bulk stable phase the WZphase(phosphides,arsenidesandantimonides). The
v zinc-blend (ZB) structure, exception made for the ni- ZBparametersareextractedfromVurgaftmanetal.[20].
i
X tridesthathavethewurtzite(WZ)structureasthemost The formulation of a theoretical model for this new
r stable phase. However, reducing the dimensions to the kind of nanostructures can lead to band gap engineering
a nanoscale level such as in these NWs, the WZ phase has and novel electronic a and also work as a guide for the
beenobservedfornon-nitrideIII-Vcompounds. Thenat- crystal growth community.
ural explanation to this behavior is the small formation
energy difference between the two crystal structures.
THEORETICAL MODEL
The phase alternation creates ZB and WZ regions
alongthegrowthdirectionaffectingtheNWsopticaland
electronic properties, generating new forms of band off- The existence two different crystal phases in a single
sets and electronic minibands that can be explored to whisker,canbeexplainedbyasimplejointanalysisofthe
produce new kinds of applications. ZBandWZcrystalstructures. LookingintoZBalongthe
TheWZ/ZBpolytypisminIII-Vcompoundscanmake [111]directionandtoWZ alongthe [0001]direction,one
a type II homostructure characterized by a staggered can found a noticeable similarity. Both can be described
band-edge lineup, i. e., the charge carriers are spatially asstackedhexagonallayers. Infact,theZBstructurehas
2
with SO no SO with SO
(2)
(1)
(2) (1)
Band Gap Band Gap
(2)
(4) (2)
(3) (2)
(2) (1)
(2)
ZB WZ
Figure 1. Γ point symmetry for the two crystal structures.
The numbers in parentheses are the degeneracy of the irre-
ducible representation. This figurewas adapted form [18].
three layers in the stacking sequence (ABCABC) while
the WZ structure has only two (ABAB). So, a stacking
fault in WZ becomes a single segment of ZB and two
sequential twin planes in ZB form a single segment of
WZ [4].
An important difference between the two structures is
the crystalfield energypresentin the WZ structure that
defines the top of the valence band, neglecting spin, as
a composition of two different bands, a non-degenerate
andabidegeneratebands. InZB,thetopofvalenceband Figure 2. (a) ZB and (b) WZ first Brillouin zones. The ar-
rows (red online) represent the growth directions. (c) ZB
is composed by a three-degenerate band. This difference
conventional unit cell with two different coordinate systems.
in the energy bands between the two structures can be
(d) WZ conventional unit cell with its common coordinate
seen schematically for the Γ point in figure 1, with and
system. The [111] growth direction for ZB structure passes
without the spin-orbit interaction. alongthemaindiagonalofthecubeandisrepresentedinthe
To calculate the band structure of a polytypic NW primed coordinate system.
basedinthe~k p~matrixformalism,itis necessarytoun-
·
derstand the differences between the two crystal struc-
tures in the reciprocal lattice or, more precisely, what a smaller set of the states first presented by Murayama
happens with the high-symmetry points in the ZB first andNakamura[18]. Analyzingthis,onemaynoticethat,
Brillouinzonewhenwearedealingwiththe[111]growth in the present method the irreducible representations of
direction and how they can be related to the WZ ones. ZB [111] belong only to the Γ point and no irreducible
Looking through the [111] axis in figure 2a one can representationof the L point will appear in the formula-
see that the L point of ZB structure is projected in the tion.
Γ point. Since the number of atoms in the primitive Anothersubject,asidesthesymmetrycorrespondence,
unit cell for WZ is twice the number in ZB, there are must be pointed out: the ~k ~p matrix for the ZB struc-
·
more states in the Γ point of WZ. This correspondence ture is developed for the cubic main axes, which are the
is done by assigning Γ and L points of the ZB to the [100], [010] or [001] growth direction. This implies that
Γ point of WZ. This compatibility can be seen in the the definition of the x,y,z coordinatesystem defines the
energy diagram presented by Murayama and Nakayama matrix representations of the symmetry operations used
[18],thatrelatestheirreduciblerepresentationsoftheZB to calculate the matrix elements in the~k p~ formulation.
·
Γ andL points to the WZ Γ point. This energy diagram Changingthe coordinatesystemtox′,y′,z′,whichis the
providestheessentialinsightforourmethodsinceweare coordinatesystemoftheWZHamiltonian,willenableus
going to relate ~k ~p matrices developed with irreducible to relate the ZB and WZ matrices.
·
representations belonging to different crystal structures. Wecaneitherrecalculatethematrixrepresentationsof
Theusual~k ~pmatrixformulationisconstructedinthe thesymmetryoperationsanddeveloptheZB~k ~pmatrix
· ·
vicinity ofthe Γ pointforZBandWZ structures[21,22] or use the rotation matrices presented in the paper of
considering the first three valence bands and the first Park and Chuang [23] to transform the ZB ~k ~p Hamil-
·
conduction band, represented by a 4 4 matrix. When tonian in the x,y,z to the x′,y′,z′ coordinate system.
×
the spin-orbitinteractionis includedthis numberis mul- The basis, c , in the prime coordinate system is
l
{| i}
tiplied by 2, resulting in a 8 8 matrix. These bands definedas(fromthispointon,theprimewillbedropped
×
are exactly the ones presented in figure 1, and represent out of the notation):
3
300 a) HH
LH
1 200 SO
|c1i=−√2|(X +iY)↑i 100
1 0
c2 = (X iY)
| i √2| − ↑i -100
c3 = Z -200
| i | ↑i
1 -300
c4 = (X iY)
| i √2| − ↓i -400
1 )
|c5i=−√2|(X +iY)↓i meV --650000
|c6i=|Z ↓i gy ( 300
|c7i=i|S ↑i ner 200 b)
|c8i=i|S ↓i (1) E 100 HLHH
CH
which is convenient to describe the WZ Hamiltonian. 0
Since wearenottreatingtheinterbandinteractionex- -100
plicitinthematrix,theconductionbandhaveaparabolic -200
formforbothcrystalsandisnotaffectedbytherotation,
-300
reading as
-400
-500
¯h2 ¯h2
E (~k)=E +E + k2+ k2+k2 (2) -600
c g VBM 2mke z 2m⊥e x y 9 6 3 0 3 6 9 12 15
(cid:0) (cid:1)
whereE isthegapenergy,E istheenergyofthe
g VBM
top valence band and for the ZB structure the electron
Figure3. (a) BulkZBvalencebandstructurealongthe[111]
effective masses are the same mke =m⊥e. direction. (b) Bulk WZ valence band structure reproducing
Applying the rotation matrices to the valence band resultsin [19]alongthe[0001]growthdirection. Bothvalence
ZB Hamiltonianwe obtainacommonmatrixfor the two band structures were plotted for 20% of the first Brillouin
structures: Zoneinkx andkz directions. Withoutspin-orbitthe(x,y,z)
coordinates belong tothesame irreducible representation for
ZB structure and then we can see that the band structure is
F K∗ H∗ 0 0 0 isotropic in kx and kz directions. In WZ structure (x,y) and
K −G −H 0 0 ∆ (z)coordinatesbelongtodifferentirreduciblerepresentations
 −H H∗ λ 0 ∆ 0  makingthebandstructureanisotropicinkxandkzdirections.
HV = −0 0 0 F K H  (3)
 − 
 0 0 ∆ K∗ G H∗ 
 − −  The A (i = 1,...,6,z), and ∆ (i = 1,2,3) WZ param-
 0 ∆ 0 H∗ H λ  i i
 −  eters can be related to the γ (i = 1,2,3) and ∆ ZB
  i SO
The matrix terms are defined as: parameterscomparingthe ZB rotatedmatrix to the WZ
matrix.
InthisworkwewillassumetheWZasidealandthere-
F =∆1+∆2+λ+θ fore the lattice parameters will considered the same as
G=∆1 ∆2+λ+θ ZB. As a consequence, strain effects will not be taken
−
λ=A1kz2+A2 kx2+ky2 into account.
θ =A3kz2+A4(cid:0)kx2+ky2(cid:1)
K =A5k+2 +2√(cid:0)2Azk−kz(cid:1) RESULTS AND DISCUSSION
H =A6k+kz +Azk−2
∆=√2∆3 (4) We first apply our model for bulk ZB and WZ InP to
check the reliability of the matrix for both crystal struc-
where k =k ik . tures. This initial investigation is useful to fit the ef-
± x y
±
The only difference from the usual WZ formulation is fective mass parameters and to understand the valence
the inclusion of a new parameter, A , that takes into band states in the band edge since now we are dealing
z
account the reorientationof the ZB growth axis. with different basis states for the ZB matrix.
4
Energy Set of states 1 Set of states 2 Energies α β
E1 |c1i |c4i ZB E1=E2>E3 0.57735 0.81650
E2 α|c2i+β|c6i β|c3i+α|c5i WZ E1>E2>E3 0.98345 0.18116
E3 β|c2i−α|c6i −α|c3i+β|c5i
Table II. The energy ordering and the α and β coefficients.
Table I. The valence band energies and the respective eigen- The knowledge of these coefficients enable the identification
states at the Γ point. of the states.
ZB identification WZ identification
The ZB parameters were obtained from Ref. [20] and
HH HH
the WZ parameters were derived using the effective
LH CH
masses presented in Ref. [19]. Since in our model there
SO LH
isnoexplicitinterbandinteractiononlythevalenceband
is presented in figure 3 for the bulk form of both crystal
Table III.The WZ identification for theZB states.
structures.
The usualidentificationofthe bands wereusedfor ZB
while in WZ we had to analyze the composition of the
states in the Γ point. Table I summarizes the energies
Since in our model we are using the WZ basis to de-
and states compositions in the Γ point that helps us to
scribe the ZB structure it is useful to think that ZB is
identify the WZ states.
a WZ structure with zero crystal-field energy. Noting
this feature we can identify the ZB valence band states
in the Γ point using the WZ identification. This helps
The energies and the α and β coefficients are given by
us to understand the band edge in the WZ/ZB interface
[22]:
in a quantum well structure. Table III relates the two
different identifications.
E1 =∆1+∆2 (5)
AsthedimensionsofNWsarebigenoughtoavoidlat-
2 eral confinement, this structure can be considered as a
E2 = ∆1−2 ∆2 +s ∆1−2 ∆2 +2∆23 (6) 2-dimensionalsystem, like a quantumwell or a superlat-
(cid:18) (cid:19) tice. In this context, we simulated a single WZ/ZB/WZ
polytypic quantum well with 160˚A well width and 320˚A
barrier width so that the whole system has 480˚A.
2
E3 = ∆1−2 ∆2 −s(cid:18)∆1−2 ∆2(cid:19) +2∆23 (7) poUsssiibnlge tthoecboannstdrumctismthaetcqhuavnaltuuemgiwveenll ipnroRfielfe..[1T9o] iutnis-
derstandtheconfinementprofileweplotted, infigure4a,
E the bandedge energyin the WZ/ZB interface. Since the
2
α= (8) compositionofthe states LHandCHis differentin both
E2+2∆2
2 3 crystal structures we did not think it was convenient to
p draw a line connecting the energies. We believe that
√2∆3 there’s a mix of these values to form the effective poten-
β = (9)
E2+2∆2 tial. However,wecannotethattheconductionbandand
2 3
the first valence band form a type-II homostructure.
The states identificatipon were made as described in It is also convenient to plot the diagonal terms of the
Ref. [24]: HH (heavy hole) states are composed only by total Hamiltonian to check the variation of the param-
c1 or c4 , LH (light hole) states are composed mainly eters along the growth direction, which can be found in
| i | i
of c2 or c5 and CH (crystal-field split-off hole) are figure4b. Althoughthe nondiagonaltermsofthe matrix
| i | i
composedmainlyof c3 or c6 . Accordingto theidenti- willmixthethreeprofileswecanexpectthattheconfine-
| i | i
ficationoftheenergiespresentedintableI, E1 isthe HH ment of the electrons will be in the ZB region while the
state and, if α>β, E2 is the LH state and E3 is the CH holeswillbe inthe WZregionbelongingintheirmostto
state or E2 is the CH state and E3 is the LH state oth- the HH state.
erwise. The valence band ordering in the Γ point, which The band structure of the quantum well plotted up to
dependsonthevaluesof∆1,∆2 and∆3,andthevalueof 10% of the kx direction and 100% of the kz direction is
the coefficients for both crystal structures can be found shownin figure 5. We canobservethat the bound states
in table II. in the valence band are mainly HH states, which means
5
1850
a)
1825
1800
1775
1750
1725
)
V
e
m
(
y
Figure 4. (a) The band edge energy in the WZ/ZB interface g b)
r
relating the WZ and ZB states. (b) The diagonal potential e
n
profile of the total Hamiltonian along the growth direction. E
325
they are composed entirely by the c1 and c4 states of
| i | i
the basis that have only X and Y components. This
| i | i
feature directly reflectsin the luminescence propertiesof 300
the system: the light will be almost fully polarized in
the xy plane, i. e., transversal to the growth axis of the
polytypic quantum well. The LH states have less than
275
5% of Z component.
| i
The probability densities calculatedfromthe envelope
functions for the Γ point of the EL1, EL2, EL3, HH1,
HH2andHH3bandsareshowninfigure6. Theseresults 250
6 0 6 12 18 24 30 36 42 48 54 60 66 72 78
agrees with the experimental data suggesting a spatial
separation of the carriers, due to the fact that is more
likely to find holesin the WZ regionandelectronsin the Figure 5. Conduction (a) and valence (b) band structure of
ZB region. thesingleWZ/ZB/WZpolytypicquantumwell. Frombottom
Inaquantumwelltheselectionrulesfordipole optical totop theconduction bandstates are: EL1 - EL5. From top
transition are given by the matrix element M [25]. to bottom the valence band states are: HH1 - HH15, LH1,
LH2, HH16, LH3,HH17.
M =hE|~r·εˆ|Hi=McMg (10) Mg (%) Energy difference (meV)
This matrix element can be divided in two parts: M EL1-HH1 ∼2.6 1370.76
c
and M . M is the matrix element between the electron EL1-HH3 ∼6.4 1372.89
g c
and hole states in the basis c that depends on the EL2-HH2 ∼12 1405.06
l
{| i}
polarization of the light and M is the overlap integral EL3-HH1 ∼23 1453.86
g
between the electron and hole envelope functions. EL3-HH3 ∼33.5 1455.99
TableIV.Finiteoverlapintegrals (Mg)andenergydifference
M = c ~r εˆ c (11) between some conduction and valence band states at the Γ
c e h
h | · | i point. Wecan note that overlap is bigger for excited states.
∞
M = g∗(z)g (z)dz (12) parities, they are either even or odd and we can expect
g e h
Z−∞ to have null overlap integrals between conduction and
valence states. The non-zero overlap integrals and the
The M matrix elements are known from symmetry
c
difference energy in the Γ point are presented in table
propertiesofthe Γpointandthenon-vanishingelements
IV.
are S x X , S y Y and S z Z . We just have to
h | | i h | | i h | | i
calculate the overlap integral M . Since the potential
g
profile is even, the envelope functions have well defined
6
5
ACKNOWLEDGEMENTS
a)
EL1
4 EL2
EL3
We like to thank the Brazilian funding agencies
3 CAPES and CNPq for the financial support.
2
y
sit
n 1
e
d [1] J. Bao, D. C. Bell, F. Capasso, N. Erdman, D. Wei,
y
bilit b) HH1 LM.aFtreorbiaelrsg,,2T1.,M36a5r4te(n2s0s0o9n),. andL.Samuelson,Advanced
a 4 HH2
b HH3 [2] J. Johansson, L. S. Karlsson, C. P. T. Svensson,
o
r T. M˚artensson, B. A. Wacaser, K. Deppert, L. Samuel-
P 3 son, and W. Seifert, Journal of Crystal Growth, 298,
635 (2007).
2
[3] J. Johansson, L. S. Karlsson, C. P. T. Svensson,
T. Martensson, B. A. Wacaser, K. Deppert, L. Samuel-
1 son, and W.Seifert, NatureMaterials, 5, 574 (2005).
[4] P. Caroff, K. A. Dick, J. Johansson, M. E. Messing,
0 K.Deppert, andL.Samuelson,NatureNanotechnology,
-240 -160 -80 0 80 160 240 4, 50 (2009).
z
[5] M.Tchernycheva,G.E.Cirlin,G.Patriarche,L.Travers,
V. Zwiller, U.Perinetti, and J.-C. Harmand,Nano Let-
Figure 6. The probability densities of the electrons (a) and ters, 7, 1500 (2007).
holes (b). We can note thespatial separation of the carriers. [6] Y. Kitauchi, Y. Kobayashi, K. Tomioka, S. Hara,
K. Hiruma, T. Fukui, and J. Motohisa, Nano Letters,
10, 1699 (2010).
[7] S.-G. Ihn, J.-I. Song, Y.-H. Kim, J. Y. Lee, and I.-
Although we have achieved reliable results with the H. Ahn, Nanotechnology, IEEE Transactions on, 6, 384
model considered here for InP, we believe that improve- (2007).
ments should be made to treat other III-V compounds. [8] R. S. Wagner and W. C. Ellis, Applied Physics Letters,
Insystemswherethelowerconductionbandhassymme- 4, 89 (1964).
try Γ7 in the WZ phase and the spin-orbit splitting is [9] V. Dubrovskii, G. Cirlin, and V. Ustinov, Semiconduc-
tors, 43, 1539 (2009).
bigger or equivalent to the gap energy (InAs and InSb
[10] F.Glas,JournalofAppliedPhysics,104,093520(2008).
[19])thefull8 8~k p~methodshouldbeconsideredwith
× · [11] K.Sano,T.Akiyama,K.Nakamura, andT.Ito,Journal
the inclusion of the Kane interband momentum matrix of Crystal Growth, 301-302, 862 (2007).
elements(P)andalsoKanequadraticterms(B). Insys- [12] T. Akiyama, K. Sano, K. N. Akamura, and T. Ito,
tems with Γ8 symmetry in the lower conduction band Japanese Journal of Applied Physics, 45, L275 (2006).
for WZ phase,like AlAs orGaAs [19], the full symmetry [13] M. D. Moreira, P. Venezuela, and R. H. Miwa, Nan-
otechnology, 21, 285204 (2010).
analysis should be redone.
[14] N.Sibirev,M.Timofeeva, A.Bol′shakov,M.Nazarenko,
and V. Dubrovskii, Physics of the Solid State, 52, 1531
(2010).
[15] A.Nastovjak,I.Neizvestny,I.Shwartz, andZ.Yanovit-
CONCLUSIONS skaya, Semiconductors, 44, 127 (2010).
[16] M. Lubov, D. Kulikov, and Y. Trushin, Technical
Physics Letters, 35, 99 (2009).
In summary, we have presented a theoretical method
[17] E. G. Gadret, G. O. Dias, L. C. O. Dacal, M. M.
based in group theory arguments and the ~k p~ method de Lima Jr., C. V. R. S. Ruffo, F. Iikawa, M. J. S. P.
·
to calculate the electronic properties of InP polytypic Brasil, T. Chiaramonte, M. A. Cotta, L. H. G. Tizei,
structures in the vicinity of the band edges. D. Ugarte, and A. Cantarero, Physical Review B, 82,
125327 (2010).
Understanding the behavior of the band structure of [18] M.MurayamaandT.Nakayama,Phys.Rev.B,49,4710
the calculated system we could predict the trends in the
(1994).
luminescenceenergytransitions. Thepreliminaryresults [19] A. De and C. E. Pryor, Physical Review B, 81, 155210
are in good agreementwith the experimental data avail- (2010).
able so far such as carriers’ spatial separation and light [20] I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan,
polarization in luminescence spectra. Journal of Applied Physics, 89, 5815 (2001).
[21] E.O.Kane,PhysicsofIII-Vcompoundsvol.1,editedby
Withtheproposedmodel,itwillbepossibletopredict
R. K. Willardson and A. C. Beer, Semiconductors and
theopticalpropertiesofselectedstructuresandtodesign Semimetals, Vol. 1 (World Scientific, 1966) Chap. 3, pp.
systems that best fit new device applications. 75–100.
7
[22] S. L. Chuang and C. S. Chang, Physical Review B, 54, [24] S.L. Chuangand C. S.Chang, AppliedPhysicsLetters,
2491 (1996). 68, 1657 (1996).
[23] S.H.ParkandS.L.Chuang,JournalofAppliedPhysics, [25] M. Fox, Optical Properties of Solids, Oxford Master Se-
87, 353 (2000). ries in Condensed Matter Physics (Oxford University
Press, c2001) Chap. 6, pp.115–142.