Theory of Electro-Optical Properties of Graphene Nanoribbons

0.24 MB·English

by  Kondayya Gundra

Theory of Electro-Optical Properties of Graphene Nanoribbons
Kondayya Gundra1,∗ and Alok Shukla1
1Department of Physics, Indian Institute of Technology, Bombay, Mumbai 400076 INDIA
1
1 We present calculations of the optical absorption and electro-absorption spectra of graphene
0 nanoribbons (GNRs) using a π−electron approach, incorporating long-range Coulomb interactions
2 within the Pariser-Parr-Pople (PPP) model Hamiltonian. The approach is carefully bench marked
bycomputingquantitiessuchasthebandstructure,electric-field drivenhalfmetallicity, and linear
n
optical absorption spectra of GNRs of various types, and the results are in good agreement with
a
those obtained using ab initio calculations. Our predictions on the linear absorption spectra for
J
the transversely polarized photons provide a means to characterize GNRs by optical probes. We
6
also compute the electro-absorption spectra of the zigzag GNRs, and argue that it can be used to

determine,whetherornot,theyhaveamagneticgroundstate,therebyallowingtheedgemagnetism
]
i tobe probed through non-magnetic experiments.
c
s
PACSnumbers: 78.20.Bh,78.67.Wj,73.22.Pr,78.40.Ri
-
l
r
t
m I. INTRODUCTION by optical means.
. Most of the theoretical approaches used to study the
t
a electronic structure of GNRs are broadly based upon:
m Discovery of graphene1 has stimulated intense re- (a) tight-binding method,6,7,9 (b) Dirac equation ap-
search in the field from the point-of-view of both fun-
- proach, derived using the linearity of the band struc-
d damental physics, and promising applications2–4. Of ture in the region of interest,18 (c) ab initio DFT
n particular interest are recently synthesized5 quasi-one- and GW based approaches,10,11,13,15 and (d) Hubbard
o dimensional (1D) nanostructures of graphene called model based approaches.19–22 But, it is obvious from
c
graphene nanoribbons (GNRs) which have technologi-
[ the chemical-structure of graphene and GNRs that the
cally promisingelectronicandopticalpropertiesbecause
electrons close to the chemical potential are itinerant π-
2 of the confinement of electrons owing to the reduced di-
electrons which determine their low-energy excitations.
v mensions. As a result, numerous theoretical studies of
Inπ-electronsystemssuchasvariousaromaticmolecules
8
electronic, transport, and optical properties of GNRs of
3 and conjugated polymers, it is well-known that role
varioustypehavebeenperformedovertheyears6–16. The
5 of electron-electron (e-e) interactions cannot be ignored
3 structural anisotropy of GNRs must exhibit itself in an when describing their electronic properties.23Therefore,
8. anisotropic optical response with respect to the photons itisinconceivablethatthelong-rangee-einteractionswill
polarizedalongthelengthofthe ribbons(xpolarized,or
0 be insignificant in graphene and related structures. The
longitudinally polarized) as against those polarized per-
0 effective π-electron approaches such as the Pariser-Parr-
1 pendicular to it (y polarized or transversely polarized), Pople (PPP) model Hamiltonian,24 which incorporate
: with GNRs being in the xy-plane. Despite its obvious
v long-range e-e interaction, have been used with consid-
importance, anisotropy in the optical response of GNRs
i erable success in describing the physics of π-conjugated
X has not been studied in any of the reported optical ab- molecules and polymers.23 Computationally speaking,
r sorptioncalculations,whichconcentrateonlyonthe lon- PPPmodelhastheadvantageofincludingthelong-range
a gitudinal component of the spectra12–16. In this work
Coulombinteractionsofπ electronswithinaminimalba-
we study this anisotropy in detail, and make predictions
sis, thereby allowing calculations on such systems with
which can be tested in optical experiments on oriented
limited computer resources,ascomparedto the abinitio
samples of GNRs, and can serve as a means for their
approaches. Indeed,inourearlierworkswehaveusedthe
optical characterization.
PPP model to extensively to study the electronic struc-
Electro-absorption (EA) spectroscopy, which consists ture and optical properties of finite π-electron systems
ofmeasuringopticalabsorptioninthepresenceofastatic such as conjugated molecules and oligomers at various
external electric (E) field, has been used extensively to levels of theory.25 Therefore, in this work, we have de-
probe the electronic structure and optical properties of cidedto extendourPPP modelbasedapproachtostudy
conjugatedpolymersandothermaterials17. GNRs,being the physics of GNRs in the bulk limit. Because, to the
π-conjugated systems, will also be amenable to similar best of our knowledge, this is the first application of
EA probes, and, therefore, we have calculated the EA the PPP model to the GNR physics, we have carefully
spectrumofzigzagGNRs(ZGNRs)inthiswork. ZGNRs benchmarkeditforquantitiessuchasthebandstructure,
have been predicted to possess a magnetic ground state, electric-fielddrivenhalfmetallicity,andlinearopticalab-
with oppositely oriented spins localized on the opposite sorptionspectraagainstthepublishedab initio workson
zigzagedgesoftheribbons6,8. OurcalculatedEAspectra GNRs, and the results are in very good agreement with
of ZGNRs depends strongly on whether, or not, they each other.
exhibit edge magnetism, thereby, allowing its detection The remainder of this paper is organized as follows.
In the next section, we outline the theoretical aspects
of our work. In section III, we present and discuss our
results. Finally, in section IV we presentour conclusions
and discuss the directions for the future work.
II. THEORETICAL DETAILS
The PPP model Hamiltonian,24 with one π-electron
per carbon atom (half-filled case), is given by
H =−Xtij(c†iσcjσ +c†jσciσ)+
i,j,σ
UXni↑ni↓+XVij(ni−1)(nj −1) (1)
i i<j
Figure 1: The structures of (a) a ZGNR and (b) an AGNR.
above c† creates an electron of spin σ on the p orbital The ribbons are assumed to lie in the xy plane, with the
iσ z
of carbon atom i, n = c† c is the number of elec- periodicity in the x direction.
iσ iσ iσ
trons with spin σ, and n = n is the total num-
i Pσ iσ
ber of electrons on atom i. The parameters U and V
ij
the Coulomb parameters. The tuning of the parameters
aretheon–siteandlong–rangeCoulombinteractions,re-
wasdone for AGNR-12 (AGNR-N , denoting an AGNR
spectively, while t is the one-electron hopping matrix A
ij
with N dimer lines across the width), and with a mod-
element which, if needed, can be restricted to nearest- A
ified set of screened parameters (U = 6.0 eV,κ = 2.0
neighbors(NN). On setting V = 0, the Hamiltonian re- i,j
ij
(i 6= j) and κ = 1), and NN hopping matrix element
ducestotheHubbardmodel. Theparametrizationofthe ii
Coulomb interactions is Ohno like26, t1 =−2.7 eV. As a result, good agreement was obtained
for AGNR-12 between the PPP band gap (1.75 eV) and
V =U/κ (1+0.6117R2 )1/2 , (2) the correspondingGW value ofYang et al.13. Therefore,
i,j i,j i,j
we have decided to use these modified Coulomb param-
where, κ depicts the dielectric constant of the sys- eters throughout these calculations, with the aim that
i,j
tem whichcansimulate the effects ofscreening,and R they will incorporate the GW-level electron-correlation
i,j
is the distance in Å between the i-th and the j-th car- results implicitly in our results.
bon atoms. The Hartree-Fock (HF) theory for periodic
one-dimensional systems, within the linear combination
of atomic orbitals (LCAO) approach is fairly standard, III. RESULTS AND DISCUSSION
and we have implemented both its restricted (RHF) and
unrestricted (UHF) variants. The lattice sums are per-
TheschematicstructuresofAGNRsandZGNRsstud-
formed in the real space by including a large number
iedinthisworkarepresentedinFig. 1. Next,wepresent
of unit cells, and integration along the Brillouin Zone
the results of our PPP model basedcalculations on vari-
(BZ) was performed using the Gauss-Legendre quadra-
ous quantities, for AGNRs and ZGNRs.
ture approach27. The convergence with respect to the
numbers of unit cells included in the lattice sums, as
well as k-points used for BZ integration, was carefully
A. Band Structure
checked.
Ourcalculations,tothebestofourknowledge,arethe
first applications of the PPP model to GNRs in the bulk InFig. 2(a)wepresentthebandstructuresofAGNR-
limit;therefore,itisimportanttoobtainasuitablesetof 11obtainedusing the Hubbardmodelwith U =2.0,and
Coulomb parameters for these systems. In our previous thePPPmodel. Atthetight-bindinglevelalltheAGNRs
calculationsonconjugatedmoleculesandpolymers25,we with N =3p+2 (p a positive integer) are predicted to
A
used two sets of Coulomb parameters namely (a) “stan- be gapless. However, ab initio DFT calculations predict
dard parameters” with U = 11.13 eV and κ = 1.0, all types of AGNRs to be gapped, including those with
i,j
and (b) “screened parameters” with U = 8.0 eV and N = 3p+211,12. Our RHF calculations are in agree-
A
κ = 2.0 (i 6= j) and κ = 1, proposed initially by ment with the DFT results, and also predict all families
i,j i,i
Chandross and Mazumdar to study phenyl-based con- of AGNRs, including N = 3p+2 to be gapped, as is
A
jugated polymers.28 In the absence of extensive exper- obviousfromourPPP results for AGNR-11presentedin
imental data, we adopted the criterion of good agree- Fig. 2 (a). The noteworthy point is that the Hubbard
ment between the ab initio GW band gaps of armchair model, with the currently accepted values of U predicts
GNRs (AGNRs)13 and our PPP band gaps, to choose a negligible gap for N = 11 (cf. Fig. 2), a result in
A
4 (a) Width Total Energy (eV) Band gap (eV)
2 NZ Enm Em ∆E Egnm Egm
V) 4 -23.059 -23.261 -0.025 0.524 2.414
e 0
(k 6 -35.559 -35.825 -0.022 0.336 2.005
ε
-2 8 -48.103 -48.403 -0.019 0.246 1.694
12 -73.237 -73.570 -0.014 0.161 1.287
-4
0 0.2 0.4 0.6 0.8 1 16 -98.417 -98.743 -0.010 0.046 1.037
k(π/a)
3 (b) Table I: Variation of total energy/cell and the band gaps of
ZGNRwiththewidthoftheribbon,computedusingthePPP
1.5
model.
)
V
e 0
(
k
ε
-1.5
-3 energy/cell of the magnetic state (Em) is lower as com-
0 0.2 0.4 0.6 0.8 1 paredtothatofthenon-magnetic(E )one,andtheen-
k(π/a) nm
ergy difference/atom between the non-magnetic and the
3 (c) magnetic states (∆E =(E −E )/N , N ≡number
m nm at at
1.5 of atoms in the unit cell) decreases with the increasing
V) 0.2 ribbon-width, consistent with the non-magnetic ground
(e 0 0 state of graphene. The band gap for the magnetic state
εk -0.2 (Em) is much larger than that of the non-magnetic one
-1.5 0.72 0.76 0.8 g
(Enm). Thenon-zerogapsobtainedforthenon-magnetic
g
-3 statesofZGNRsisanartifactoftheRHFapproach. The
0 0.2 0.4 0.6 0.8 1
k(π/a) bandstructuresofthemagneticandnon-magneticstates
of ZGNR-12 computed using the PPP model are pre-
sented in Fig. 2 (b) , and it is obvious that, for the
Figure 2: (Color online) Band structure near the Fermi en-
ergy (EF =0) of: (a) AGNR-11obtained usingtheHubbard magnetic case, our results are qualitatively very similar
Model (black solid line), with U = 2.0, and the PPP-RHF to the reported ab initio band structures11,15. Quanti-
approach (red broken line), (b) ZGNR-12obtained using the tatively speaking, for ZGNR-8, we obtain Em = 1.70
g
PPP-RHF approach for the non magnetic state (black solid eV, which is higher than the reported GW value of 1.10
line),andthePPP-UHFapproachforthemagneticstate(red eV15. OurbandgapforAGNR-11wasinexcellentagree-
brokenline)inwhichthebandsofupanddownspinsarede- ment with the GW value, but that is not the case with
generate, (c) ZGNR-16, obtained using PPP-UHF model, in
ZGNRs. Webelievethatitcouldpossiblybebecause: (a)
the presence of a lateral electric field of 0.1 V/Å so that the
our Coulomb parametrization was based upon ab initio
up-down degeneracy is lifted (red broken/black solid bands
GW results13 on an AGNR, and (b) electron-correlation
representsup/downspins)with Egm(up) =0.11 eV(magnified effects are stronger in ZGNRs as compared to AGNRs,
in theinset), and Em(down) =0.97 eV.
g and the HF approachadopted here ignores those effects.
Ina pioneeringworkSonet al.10, basedupon ab initio
complete disagreement with the DFT, and our PPP re- DFTcalculations,predictedthatinthepresenceofalat-
sults. Thus,fromthiscaseitis obviousthatforAGNRs, eral electric field, ZGNRs exhibit half-metallic behavior
long-rangeCoulombinteractions as included in the PPP leadingtotheirpossibleuseinspintronics. Theydemon-
model play a very important role of opening up the gap strated that for the field strength 0.1 V/Å, the gap for
for the N = 3p+2 case. Our PPP value of the band oneofthe spins ofZGNR-16willclose,leading to metal-
A
gap1.06eVofthisAGNRisagaininexcellentagreement lic behavior for that spin orientation. In Fig. 2(c) we
with the ab initio GW result reported by Yang et al.13. present the band structure of the same ZGNR exposed
The case of the ground state of ZGNRs is an in- to the identical field strength, calculated using the PPP
teresting one with several authors reporting the exis- model, and the tendency towards half-metallicity is ob-
tence of a magnetic ground state, with oppositely ori- vious. While the band gap in the absence of the field
ented spins localized on the opposite zigzag edges of the was 1.037 eV, in the presence of the field up-spin band
ribbons6,8, a result verifiedalso in severalfirst principles gap is reduced to 0.11 eV, while the down-spin gap de-
DFT calculations11,15. We investigated this in our PPP creases to 0.97 eV. Therefore, considering the fact that
modelcalculationsbyusingtheRHFmethodforthenon- our PPP model based approach does not incorporate
magnetic state and the UHF method for the magnetic electron-correlation effects, its quantitative predictions
one, and the results are summarized in Table I. We find are in very good agreement with the ab initio ones10,
that for a ZGNR of width N (N ≡ number of zigzag and thus it is able to capture the essentialphysics of the
Z Z
lines across the width), ZGNR-N in short, the total electric-field driven half-metallicity in ZGNRs.
Z
5
B. Optical Absorption Σ (a)
11
)4
s
t
Nextwepresentthelinearopticalabsorptionspectraof ni
u3
GNRs,computedwithinthePPPmodel. Theopticalab- b.
sorption spectrum for the x-polarized (y-polarized) pho- ar2 Σ Σ
( 12 22
tons is computed in the form of the corresponding com- 2) Σ
(ε 1 33 Σ Σ
ponents of the imaginary part of the dielectric constant 21 44
tensor, i.e., ǫ(x2x)(ǫ(y2y)(ω)), using the standard formula 00 2 ω(eV)4 6
π/a |hc(k)|p |v(k)i|2 6
ǫ(ii2)(ω)=CXv,c ˆ−π/a {(Ecv(k)−~ωi)2+γ2}Ec2v(k)dk, nits)45 Σ11 (b)
(3) u
where a is the lattice constant, p denotes the momen- b.3 Σ12+ Σ21
i r
a
tsuenmtsotpheeraatonrguinlarthfereqi-utehncCyarotfestiahne dinirceidcteinotn,raωdiraetpiorne-, (2)ε(21 Σ22
E (k)=ǫ (k)−ǫ (k), with ǫ (k)(ǫ (k)) being the con-
cv c v c v 0
ductionband(valenceband)eigenvaluesoftheFockma- 0 2 ω(eV) 4 6
trix,γ isthe line width,while C includesrestofthe con-
stants. We have set C =1 in all the cases to obtain the
Figure3: (Coloronline)Imaginarypartsofthedielectriccon-
absorptionspectrainarbitraryunits. Thecomponentsof
the momentum matrix elements hc(k)|p|v(k)i needed to stant (ǫ(x2x)(ω) in black solid, and ǫ(y2y)(ω) in red broken lines)
computed using the PPP model, and modified screened pa-
compute ǫ(2)(ω), for a generalthree-dimensionalsystem, rameters, for: (a) AGNR-11, (b) ZGNR-8, with a magnetic
ii
can be calculated using the formula,29 ground state. Labels of the peaks denote the bands involved
inthetransition(seetextforanexplanation),andalinewidth
of 0.05 eV was assumed throughout.
m
hc(k)|p|v(k)i = ~0hc(k)|∇kH(k)|v(k)i
im (ǫ (k)−ǫ (k))
+ 0 c v hc(k)|d|v(k)i(,4) top)tothen-thconductionband(countedfrombottom),
~
thepeakofǫ(2)(ω)at1.1eVisΣ11,at3.1eVisΣ22,at3.8
xx
where m0 is the free-electron mass, ∇kH(k) repre- eVisΣ33,andat5.8eVisΣ44. Thepeaksofǫ(y2y)(ω)at2.1
sents the gradient of the Hamiltonian (Fock matrix, in eV and 5.6 eV both correspond to Σ12 and Σ21. The re-
the present case) in the k space, hc(k)|d|v(k)i denotes markablefeatureofthepresentedspectrumisthatowing
the matrix elements of the position operator d defined to the symmetry of the AGNRs, the peaks correspond-
with respect to the reference unit cell, and accounts ing to x- and y-polarized photons are well separated in
for the so-called intra-atomic contribution.29Note that energy,and their relative intensities can be measuredby
Eq. 4 can also be used to compute the matrix element performing experiments on oriented samples. On com-
hc(k)|py|v(k)i needed to compute the absorption spec- paring our PPP spectrum (ǫ(2)(ω)) with the ab initio
xx
trum for the y-polarizedlight for GNRs (which are peri- GWspectrumofYangetal.13,wenotethatthelocations
odic only inthe x direction), by setting the firstterm on
of the first peaks close to 1.1 eV are in excellent agree-
itsrighthandsidetozero,becauseforaone-dimensional
ment with each other. However,our calculations predict
system periodic along the x direction, the Hamiltonian
several higher energy peaks with significant intensities
has no ky dependence. hc(k)|∇kH(k)|v(k)i for the case located around 3.0 eV absent in the GW work. Further-
of GNRs is obtained easily by calculating the numerical
more, we also predict the intensities of the y-polarized
derivative of the Fock matrix at various k-points of the peaks, which was absent in the work of Yang et al.13.
one-dimensional Brillouin zone. For the d operator, the
In Fig. 3(b) we present our calculated optical ab-
usual diagonal representation was employed. The cal-
(2) (2)
culation of the absorption spectra of the GNRs for the sorptionspectrum (ǫxx(ω) and ǫyy(ω)) for the ZGNR-8.
y−polarized photons (ǫ(y2y)(ω)), to the best of our knowl- The peaks in ǫ(x2x)(ω) are located at 1.7 eV (Σ11), 2.9 eV
edge,has notbeen done earlier. Becausesuchtransverse (Σ12+Σ21),at4.0eV(Σ22),whiletheprominentpeaksof
excitations do not couple to the photons polarized along ǫ(2)(ω) are at 1.7 eV (Σ11) and 2.9 eV (Σ12+Σ21). The
yy
thexdirection,theyhavealsobeencalled“darkexcitons” noteworthy point is that most of the prominent peaks
in the literature13,15. have mixed polarization characteristics, unlike the case
TheopticalabsorptioninAGNRshasbeenstudiedex- of AGNRs. This is because of the fact that for mag-
tensively by ab initio approaches in recent works12,13,16. netic ground states, the reflection about the xz-plane is
In Fig. 3(a) we present the optical absorption spectrum broken,leadingtomixedpolarizations. Thisisanimpor-
ofthe AGNR-11. IfΣmn denotesapeakinthespectrum tant result which can also be tested in oriented samples
duetoatransitionfromm-thvalenceband(countedfrom ofZGNRs. OurPPPopticalabsorptionspectrumofthis
ZGNR compares qualitatively well to the GW spectrum It will also be of interest to perform similar studies on
computed by Yang et al.15, although our peaks are con- bilayer and other multilayer GNRs, to investigate how
sistentlyblue-shifted comparedto the GWresult, due to various properties of the ribbons evolve, as the number
the corresponding disagreement in the band structure. of layers are increased. Of particular interest is the case
Moreover,Yang et al.15 did not compute the peak inten- of multilayer ZGNRs, to probe as to what is the nature
sities for the y-polarized photons. ofedgemagnetisminthosesystems. Furthermore,itwill
Σ
11 (a)
20
C. Electro-Absorption
10
In Figs. 4 we present the EA spectrum of ZGNR-8
computedas the difference ofthe linear absorptionspec-
trawithandwithoutanexternalstaticE-fieldofstrength
0
0.1 V/Å along the y-axis. In Fig. 4(a) we present the
1 2 3 4 5 6
EA spectrum for the non-magnetic ground of ZGNR-8, ω(eV)
computed using the PPP-RHF approach. Without the 6
external E-field, the Σ transition is disallowed for the (b)
11
non-magnetic state of such a ZGNR for the x-polarized 3 Σ11 Σ12+Σ21
light due to symmetry selection rules14. However,in the
presence of the field, due to the broken symmetry, this 0
Σ
transition becomes strongly allowed leading to a very 11
strong peak in the EA spectrum. Fig. 4(b) portrays the -3
EAspectrumofthesameZGNRforthemagneticground
state, and, here the physics of half-metallicity manifests -6
1 2 3 4 5 6
itself in that one observes two energetically split peaks ω(eV)
correspondingto two different Σ transitions among up-
11
and down-spin electrons. Thus, our calculations predict
Figure 4: (Color online) Linear absorption spectrum (black
that the EA signal is different for the ZGNRs depend-
solid)andelectroabsorption(redbroken)ofZGNR-8forpho-
ing on whether they have a magnetic or a non-magnetic tonspolarized along thex axis for: (a) non magnetic ground
ground state, a result which can be used to determine state, and (b) magnetic ground state. A line width of 0.05
the nature ofthe groundstate ofZGNRs using EA spec- eV was assumed throughout, and the bands involved in the
troscopy. electro-absorption peaksare indicated.
IV. SUMMARY AND OUTLOOK also be of interest to include excitonic effects in the op-
tical absorption spectrum of ZGNRs so as to perform a
In summary, we have used a PPP model based π- complete comparison with the future experimental work
electronapproach,incorporatinglong-rangeCoulombin- on these systems. For the purpose, it is important to go
teractions, to study the electronic structure and optical beyond the Hartree-Fockapproachand include electron-
properties of GNRs in the bulk limit. In particular, we correlation effects. Work along all these directions is in
computed the optical absorption spectra of GNRs for progress in our group, and the results will be communi-
transversely polarized photons, in addition to the lon- cated in the future publications.
gitudinal ones, thereby allowing us to investigate the
anisotropic optical response of these materials. Our pre-
dictions that forAGNRs longitudinalandtransversepo-
Acknowledgments
larized components will be well separated energetically,
while ZGNRs will exhibit absorption with mixed polar-
ization,canbetestedinexperimentsonorientedsamples. We thank the Department of Science and Technol-
Furthermore, we also presented first calculations of the ogy (DST), Government of India, for providing financial
EAspectraofZGNRs,andourresultssuggestapossibil- support for this work under Grant No. SR/S2/CMP-
ity of an optical determination of whether, or not, they 13/2006. K.GisgratefultoDr. S.V.G.Menon(BARC)
possess a ground state with edge magnetism. for his continued support of this work.
∗
PermanentAddress: TheoreticalPhysicsDivision,Bhabha tronic address: [email protected], [email protected]
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