1. Physics Department, Pittsburg State University Pittsburg, KS
Electronic and transport properties of graphene nanoribbons: influence of edge passivation and uniaxial strain
Benjamin O. Tayo
Physics Department, Pittsburg State University Pittsburg, KS
WSU Physics Seminar
Wichita, KS
November 12, 2014

2. Outline Part I: Graphene Review (Tutorials) Introduction to graphene
Structural properties of graphene
Electronic properties
Graphene’s band structure and the band gap problem
Part II: Graphene nanoribbons
Structural properties, edge passivation
Electronic structure
Effect of quantum confinement
Effect of edges
Effect of external strain
Application: charge transport
Summary and conclusion

3. Part I: Graphene Review
1. Introduction, structural properties
2. Electronic structure
Uniqueness of Graphene’s band structure
Band gap problem
How to solve the band gap problem?

4. Graphene K. S. Novoselov, et al., Science 306, 666 (2004)
A single planar layer of 𝑠𝑝 2 bonded carbon atoms, densely packed in a honey-comb lattice
Carbon to carbon bond length of 𝑎 𝐶−𝐶 =0.142 nm
Thinnest, strongest material known, and has high electrical and thermal conductivities
Room temperature electron mobility of at least 𝑐𝑚 2 /Vs
K. S. Novoselov, et al., Science 306, 666 (2004)

5. Graphene: building block of other carbon materials
Castro Neto et al. Peres, 2006a,
Phys. World 19, 33
Graphene (top left) is a honeycomb lattice of carbon atoms.
Graphite (top right) can be viewed as a stack of graphene layers.
Carbon nanotubes are rolled-up cylinders of graphene (bottom left).
Fullerenes C60 are molecules consisting of wrapped graphene by the introduction of pentagons on the hexagonal lattice.

6. Graphene
Isolation of graphene in 2004 by Manchester group headed by Andre Geim
2010 Nobel prize in physics awarded to Andre Geim and Konstantin Novoselov “for groundbreaking experiments regarding the two-dimensional material graphene”
Country Rankings in Graphene Publications to Date (source: Thomson Reuters ISI Web of Science; search dated 15 June 2012 using “Topic=graphene”; 19,017 records)

7. 2D Graphite (Graphene) Unit Cells
Direct Lattice
Reciprocal Lattice & BZ

8. Energy Pi Bands of Graphene
R. Saito et. al, “Physical Properties of Carbon Nanotubes”

9. Graphene is a zero gap semiconductor
𝐸 𝐹
DOS plot:

10. Graphene’s Low-energy Physics: Dirac Fermions

11. Experimental evidence of massless Dirac fermions in graphene: Cyclotron mass
A = Area in k space enclosed by electron’s orbit
n = carrier concentration
Fitting the theoretical result with experimental data yields:
vF = 106 m/s, t = 3.0 eV
Solid State Physics, Ashcroft and Mermin, 1976
The electronic properties of graphene, Rev. Mod. Phys. Vol. 81, 2009

12. The Band Gap problem in graphene
Graphene’s electrical charge carriers (electrons and holes) move through a solid with effectively zero mass and constant velocity, like photons.
Graphene's intrinsically low scattering rate from defects implies the possibility of almost ballistic transport.
The primary technical difficulty has been controlling the transport of electrical charge carriers through the sheet.

13. How to solve the Band Gap Problem?
Isoelectronic codoping with B and N: 𝐶 (1−2𝑥) 𝐵 𝑥 𝑁 𝑥 .
Graphene substrate interaction, e.g. epitaxial graphene on h-BN or SiC.
N-type doping by Potassium deposition onto the graphene sheet.
Try other materials (2D materials revolution): MoS2, MoSe2, WS2, WSe2, etc.
Lithographic patterning into graphene nanoribbons.
L. Liu & Z. Shen, Appl. Phys. Letts 95, , (2009)
T. Ohta et al., "Controlling the electronic structure of bilayer graphene," Science 313, 951 (2006).
K. S. Novoselov et. al, “Two-dimensional atomic crystals,” PNAS, vol. 102 no. 30, 10451, (2005)

14. Part II: Graphene Nanoribbons
1. Structural properties, edge passivation
2. Electronic structure
Effect of quantum confinement
Effect of edges
Effect of external strain
3. Application: charge transport

15. Graphene Nanoribbons (GNRs)𝑦 x W
GNRs are elongated stripes of single layered graphene with a finite width
Electronic properties depend on edge geometry and width
Structurally very similar to carbon nanotubes

16. AFM image of many graphene nanoribbons parallel to each other
Cançado et al., Phys. Rev. Lett. 93, (2004)

17. Graphene Nanoribbon structural parameters
N = Number of dimer lines
N-AGNR = GNR with armchair edges and N-dimer lines
N-ZGNR = GNR with zig-zag edges and N-dimer lines
N= 3p, 3p+1, 3p+2, where p is a positive integer (family pattern).
Benjamin O. Tayo, Mater. Focus 3, (2014)

18. Effect of edge passivation with Hydrogen
Converged geometry of a H-passivated 7-AGNR
Edge C-C bond lengths are shortened by 3 to 5% compared to those in the middle of the ribbon
*Optimization was performed using DFT with the B3YPL XC potential and the 6-31 G(d) basis set, with the Gaussian 09 code

19. X. Peng, and S, Velasquez, Appl. Phys. Letts., 98, 023112, (2011).
Passivation with other atoms or groups
A. Simbeck et al., Phys. Rev. B 88, (2013)
Different atoms or functional groups provide different levels of perturbations to the nanoribbon.
Electronic properties depend on edge passivation
X. Peng, and S, Velasquez, Appl. Phys. Letts., 98, , (2011).

20. Electronic structure: effect of quantum confinement𝑦 𝑊 x

21. Effect of strain and H-passivation: model Hamiltonian

23. Model Hamiltonian At k = 0, Hamiltonian is:
Y.W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev.
Lett. 97, (2006).
Benjamin O. Tayo, Mater. Focus 3, (2014).

24. Tight-Binding Parameters
Hopping integrals are calculated using analytic expressions for TB matrix elements between C atoms
For edge carbon atoms, additional strain due to H passivation has to be taken into account
D. Porezag, et al., Phys. Rev. B 51,
12947 (1995).

25. (a) Band Gap of Unstrained H-passivated GNR
M. Han et al.,“Energy Band-Gap
Engineering of Graphene Nanoribbons,”
PRL 98, (2007).
𝐸 𝑔𝑎𝑝 = 1.5 𝑒𝑉 𝑛𝑚 𝑊[𝑛𝑚]
Benjamin O. Tayo, Mater. Focus 3, (2014).

26. (b) Effective mass of Unstrained H-passivated GNR
TB approx.

27. (c) Band Gap and Effective mass of strained H-passivated GNR
X. Peng & S. Velasquez, “Strain
modulated band gap of edge
passivated armchair GNRs,”
APL 98, (2011).

28. (a) Asymmetry of Band Gap variation with strain
N = number of dimer lines
Ne = number of pi electrons in the GNR
Ω = 2 𝜋 𝑁 𝑐 𝑇 is the length of the 1D BZ per allowed state
Line width = 0.01 eV
Energy range -2 eV < E < 2 eV
EF = 0 (Fermi Energy)
B. Tayo, Mater. Focus 3, (2014).

29. Application: Charge transport in AGNR
Carrier scattering by longitudinal acoustic phonons plays a significant role in charge transport in intrinsic semiconductors.
Within the deformation potential theory (dp), the carrier relaxation time ( 𝜏 𝑑𝑝 ) and mobility ( 𝜇 𝑑𝑝 ) are given by:
C = stretching modulus; E1 = Deformation potential constant
J. Bardeen and W. Shockley, Phys. Rev. 80, 72 (1950).
F. B. Beleznay, F. Bogr, and J. Ladik, J. Chem. Phys. 119, 5690 (2003).

30. Application: Charge transport in AGNR
The advantage of gapless graphene is its high carrier mobility.
When a non-zero gap is engineered by patterning graphene into nanoribbons, the mobility has been shown to decrease dramatically
The hardness to achieve high mobility and large on/off ratio simultaneously limits the development of graphene electronics.
Suitable choice of strain and edge passivation could be used to open the band gap while maintaining a low effective mass.
X. R. Wang, Y. J. Ouyang, X. L. Li, H. L. Wang, J. Guo, and H. J. Dai, Phys. Rev. Lett. 100, (2008).
J. Wang, R. Zhao, M. Yang, Z. Liu, and Z. Liu, Chem. Phys. 138, (2013).

31. Future of Graphene Electronics
Walt A. de Heer: “Researchers should stop trying to use graphene like silicon, and instead use its unique electron transport properties to design new types of electronic devices that could allow ultra-fast computing”
“Exceptional ballistic transport in epitaxial graphene Nanoribbons,” J. Baringhaus, M. Ruan, F. Edler, A. Tejeda, M. Sicot, A. Taleb-Ibrahimi, A. Li, Z. Jiang, E. H. Conrad, C.
Berger, C. Tegenkamp, and Walt A. de Heer, Nature Physics 10, 182, (2014).

32. Summary
Edge passivation and strain can both be described within the TB approx. by simply renormalizing the C-C hopping integral.
Studied relationship between carrier mass and band gap energy for strained H-passivated AGNRs belonging to different families: N = 3p, 3p+1, 3p+2
For unstrained H-passivated AGNRs, the effective mass exhibits a linear dependence on band gap energy for small energy gaps or large ribbon width.
However for ribbons with small width or larger band gaps, the effective mass dependence on energy gap is parabolic.
In the presence of strain, both band gap and effective mass displays a nearly zigzag periodic pattern, indicating that the effective mass remains proportionate to the band gap even in the presence of applied strain.
Finally, we discussed the implications of non-zero band gap on carrier mobility

33. Acknowledgement
Pittsburg State University summer faculty fellowship
Supercomputer core time from LBNL
Use of Gaussian 09 software for DFT calculations

34. THANK YOU FOR YOUR ATTENTION