Phase Transition in Armchair Graphene Nanoribbon Due to Peierls Distortion
CHUONG VAN NGUYEN,1,2NGUYEN VAN HIEU,3HUYNH NGOC TOAN,4 LE CONG NHAN,5NGO THI ANH,6and NGUYEN NGOC HIEU7,8
1.—Institute of Research and Development, Duy Tan University, Da Nang 550000, Viet Nam.
2.—Department of Materials Science and Engineering, Le Quy Don Technical University, Hanoi 100000, Viet Nam. 3.—Physics Department, Da Nang University of Education, Da Nang 550000, Viet Nam. 4.—Department of Natural Sciences, Duy Tan University, Da Nang 550000, Viet Nam.
5.—Department of Environmental Sciences, Saigon University, Ho Chi Minh City 700000, Viet Nam.
6.—Department of Fundamental Sciences, College of Transport No. 2, Da Nang 550000, Viet Nam.
7.—Institute of Research and Development, Duy Tan University, Da Nang 550000, Viet Nam.
8.—e-mail: [email protected]
In this work, the effect of Peierls distortion on the electronic properties of armchair graphene nanoribbons is theoretically studied in a tight-binding approximation. As a consequence of Peierls distortion, when the Kekule-type bond alternation is introduced, the band gap of armchair nanoribbons strongly depends on the difference in bond length between short and long bonds in the honeycomb lattice. We can control the band gap of a nanoribbon by its uniaxial strain and a semiconductor–metal phase transition can occur at certain elongations.
Key words: MoS2, peierls distortion, graphene nanoribbon, tight-binding approximation, phase transition
INTRODUCTION
Graphene nanoribbon (GNR) is a quasi-one-di- mensional (1-D) system which can be obtained from a graphene sheet. The effect of strain on electronic and transport properties of graphene and GNRs has been both theoretically and experimentally studied in the last decade.1–6 It is well known that low- dimensional carbon materials are sensitive to the strain. The dependence of the GNR band gap on the strain is an interesting topic because of its applica- tion in nanodevices based on strained ribbons.
The effect of Peierls distortion on the physical properties of carbon nanotubes has been studied by different methods.7–10In earlier work, the artificial model of Peierls distortion in GNRs has been studied in a tight-binding approximation.11 This work showed that bond alternation plays an impor- tant role in armchair graphene nanoribbons
(AGNRs) while the Peierls instability in zigzag graphene nanoribbons (ZGNRs) can be negligible.11 Recently, evidence for the Peierls instability in deformed AGNRs has been shown using density functional theory (DFT).12In this work, Zhang and Dumitrica˘ showed that an electronic band gap of 40 meV opening in 11-AGNG with H-saturated edges at an elongation of 0.016 is due to Kekule morphology, and the Peierls effect has been recog- nized in 11-AGNR. The Peierls distortion in AGNR can lead to forming a Kekule-type bond alternation structure in the honeycomb lattice.11–14 This phe- nomenon also occurs in other 1-D carbon nanoma- terials that are carbon nanotubes.15
In this work, we consider the semiconductor–
metal phase transition in AGNRs due to Peierls distortion using a tight-binding approximation. An AGNR with Kekule structure (as a result of the Peierls instability) with two different-length C–C bonds in the honeycomb is studied. We focus our calculations on the relationship between uniaxial strain and band gap modulation of the AGNR in the presence of Peierls distortion.
(Received October 31, 2016; accepted February 18, 2017;
published online March 7, 2017)
Ó2017 The Minerals, Metals & Materials Society
3815
MODEL AND METHOD
Due to the periodicity of the honeycomb lattice, the translational period of the AGNR with a Kekule pattern has triple length in comparison with a pristine AGNR. The period along the Oy-axis is Ly 3L0, where L0¼ ffiffiffi
p3
a0 is the undeformed graphene lattice constant with a0 being the C–C bond length (Fig.1). The 2-D primitive cell of a GNR with Kekule structure contains six carbon atoms as shown in Fig. 1. Only 3n-AGNR (nis an integer) has the triple translational period Ly. In principle, we can consider a ribbon with arbitrary dimer number p. In the present work; however, we consider only ribbons withp¼3n. This choice is easy for applying the boundary condition with triple translational period Ly.
In the presence of an alternation bond with Kekule structure, a primitive cell of an AGNR contains six non-equivalent carbon atoms. They are three A and three B carbon atoms (Fig.1). In the two-dimensional sheet as shown in Fig.1, the translational periods along the axes are Lx ¼2aþ2bx and Ly¼4ayþ2by; where a and b are the C–C bond lengths.
When the uniaxial strain is applied, the relation between position vectors in the undeformed (R0i) and deformed (Ri) graphene sheets can be written as16
Ri ¼ ð1þeÞR0i; ð1Þ wheree is the strain tensor. In the framework of unique elasticity theory, the strain tensor has the form16
e¼e sin2hrcos2h ð1þrÞcoshsinh ð1þrÞcoshsinh cos2hrsin2h
!
; ð2Þ
whereeis the elongation,ris the Poisson ratio, and h is the angle between the applied strain direction and the x-axis. As shown in Fig.1, the x-axis is parallel to the armchair axis of the graphene sheet.
In this work, we study only the applied strain along they-axis. It means that, in this case, the anglehin the strain tensor (2) is equal top=2.
The change of bond vectors between two nearest neighbor carbon atomsAiBjunder a uniaxial strain can be easily determined via the theory of contin- uum mechanics.17In carbon materials, the hopping integral depends strongly on the C–C bond length.
We have several ways to determine this relation as shown in Refs.16,18, and19. In the present work, we assume that the dependence of the hopping integral on the C–C bond length can be described by Harrison’s expression:20 t¼t0ða0=aCCÞ2; where t0(t) anda0(aCC) are the hopping parameter and C–C bond length of undeformed (deformed) gra- phene, respectively. Harrison’s expression is suit- able for the tight-binding calculations. In this work, we chose a0¼1:42 A˚ and t0¼2:6 eV.21 In general, the bonds between the carbon atoms in a nanorib- bon and graphene are connected with the minimum of the potential energy and the entropy when the carbon structure is being formed. From a physical point of view, in an 1-D carbon material, the Peierls transition can lead to C–C bond alternation.15,22 Also, the scanning tunneling microscope measure- ments of the electron density of states show that the energy gap of the armchair (7,7) carbon nanotube at liquid helium temperature is 0.11 eV.23 We believe that this observed bandgap can be explained by the Peierls transition. Recently, evidence for the Peierls effect (with bond alternation) has been found in strained armchair graphene nanoribbons by DFT calculations12 and in zigzag graphene nanoribbons by experimental work.24 The artificial honeycomb lattice contains two types of C–C bonds with different lengths a and b (a6¼b). We use the assumption of Fujita et al.11 for the difference in length of the C–C bond of the GNR that the short and long bonds differ from the undeformed C–C bond a0 by d (d can be positive or negative). This means that the lengths of the alternation bonds can be represented via the undeformed C–C bonda0as a¼a0þdandb¼a0d.
Fig. 1. (Color online) Schematic of an AGNR with Kekule-type bond alternation. The 1-D unit cell of the AGNR is shown as a dashed rectangle, and six non-equivalent carbon atoms of Ai andBi are shown (withi¼1;2;3). The thin and thick lines refer to the short (a) and long (b) C–C bonds.Lx and Ly are the translational periods along thexandyaxes, respectively.
It is well known that the Peierls transition can be successfully described by the Su-Schrieffer-Heeger model. We assume that all dangling bonds at the GNR edges are terminated by the hydrogen atoms.
The tight-binding approximation is an easy way to consider the electronic properties of materials with periodical structure. In small systems, results by tight-binding calculations are usually close to first- principles calculations. Previously, we have used both the tight-binding approximation and the semiempirical method to consider the phase transi- tion in one-dimensional carbon nanotubes.15,22 We believe that both the tight-binding approximation and first-principles calculations are suitable for calculations of phase transition in GNRs.
In the framework of the nearest-neighbor tight- binding approximation, the Hamiltonian of gra- phene with Kekule structure with primitive cell containing six carbon atoms as shown in Fig.1can be written as
H¼X
RAi
X3
i;j¼1
tAiBjAy
RAiBR
AiþrAiBj þ H.c., ð3Þ whereAy
RAiðBy
RBiÞandAR
AiðBR
BiÞare the creation and annihilation operators of an electron at RAiðRBiÞin sublatticeAiðBiÞ, respectively. These operators sat- isfy anti-commutation relations.
As shown in Fig.1, the primitive contains six carbon atoms. The electronic energy band structure of an AGNR with Kekule structure can be obtained by diagonalization of a (66) matrix with boundary condition. In this study, we calculate numerically for only 6-AGNR under uniaxial strain. The results for other 3n-AGNRs can be obtained by repeating this procedure using the boundary condition.
RESULTS AND DISCUSSION
The electronic energy band structure of a 6-AGNR with Kekule structure under uniaxial strain is shown in Fig.2. Our calculations show that the energy spectrum of a 6-AGNR depends strongly not only on the uniaxial strain but also on the difference in C–C bond lengths. In addition, the sub-bands near the Fermi level are very close to each other.
They just separate out when the elongation is large enough (see Fig.2e). In the framework of the tight- binding approximation, the hopping parameter plays an important role in the electronic properties of carbon materials. The original symmetry is broken by the appearance of bond alternation, and a new symmetry has been found. The distance between carbon atoms has changed due to Peierls distortion. Consequently, this leads to fluctuation in the hopping parameter of the system via Harrison’s above-mentioned formula. In comparison to a pris- tine AGNR, when the alternation bond was intro- duced, the band gap of a 6-AGNR with Kekule structure has dropped dramatically. The band gap
(a) (b) (c) (d) (e)
Fig. 2. (Color online) Electronic energy band structure of 6-AGNR with Kekule structure under uniaxial strain near the Fermi level: (a) pristine ribbon (d¼0 nm,e¼0); (b)d¼0:01 nm ande¼0; (c)d¼0:02 nm ande¼1%; (d)d¼0:03 nm ande¼0}; (e)d¼0:03 nm ande¼7%. The p-bands are symmetrical to thep-bands at the Fermi levelEF ¼0.
Fig. 3. (Color online) Dependence of the band gap of a 6-AGNR with Kekule-type bond alternation on uniaxial strain. The inset is the condition for the semiconductor–metal phase transition:dversuseat Eg¼0.
of pristine 6-AGNR is about 1.4 eV. In the presence of a Kekule-type bond alternation of d¼0:01 nm, the band gap of 6-AGNR is just 0.15 eV at e¼0 as shown in Fig. 2b. This value is about nine times smaller than that of pristine 6-AGNR. These prove that the electronic properties of an AGNR are very sensitive to changes in its geometry and topological characteristics. In Fig.3, we show our calculated results for the dependence of the ribbon band gap on the elongationewith differentd. At e¼0, the band gap is directly proportional to the difference in C–C bond lengthd. We can see that with the limitation of small uniaxial strain, the dependence of the band gap of a 6-AGNR with Kekule structure on uniaxial strain can be represented as a W-shaped curve. The amplitude of the gap variation of an AGNR with Kekule structure under uniaxial strain is not too large, about 0.5 eV. The semiconductor–metal phase transition can occur at certain elongations.
The critical value of the uniaxial strain e corre- sponding to the semiconductor–metal transition point depends strongly on the difference in C–C bond length d. As mentioned above, the electronic properties of an AGNR depend strongly on the hopping parameters, i.e., the C–C bond lengths. The dependence of the band gaps on uniaxial strain is described by similarly shaped curves because the calculated model for the strain problem is unique elasticity theory. The inset of Fig. 3 shows the condition for a semiconductor–metal phase transi- tion in a 6-AGNR with Kekule structure under uniaxial strain. We figured out the condition for Eg¼0 (phase transition point) in a 6-AGNR with Kekule structure. We can see that, at the phase transition point (Eg¼0), the difference in bond length ddepends linearly on the elongatione.
In this work, we also investigate the dependence of the electronic band gap of a 6-AGNR on the difference in C–C bond length d. The semiconductor–metal phase transition can be found at certain d. For
instance, in Fig.4, we show the dependence of the band gapEgon the difference in C–C bond lengthdat e¼5%. It seems that the role of uniaxial strain and the difference in bond length in the band gap modulation of a 6-AGNR is the same. The dependence of the band gap individually on uniaxial strain or difference in bond length can be described by W- shaped curves. The semiconductor–metal phase transition can be found in a 6-AGNR in both cases.
CONCLUSION
In conclusion, the effect of Peierls distortion on the electronic properties of 6-AGNR has been the- oretically studied using the tight-binding approxi- mation. The semiconductor–metal phase transition under uniaxial strain of a 6-AGNR with Kekule structure has been found. Our calculations show that, in the presence of Kekule-type bond alterna- tion, the band gap of 6-AGNR suddenly decreases in comparison to the pristine AGNR case. The phase transition can occur at different elongations, and it also depends on the difference in C–C bond length of the honeycomb lattice. The phase transition is an important characteristic of material for its applica- tions in nanoelectro-mechanical devices, and our results open a new approach for applications of strained AGNRs in nanodevices.
ACKNOWLEDGEMENTS
This research was funded by the Vietnam Na- tional Foundation for Science and Technology Development (NAFOSTED) under Grant No.
103.01-2014.04.
CONFLICT OF INTEREST
The authors declare that they have no conflict of interest.
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