Adam Rycerz

[80] Yasumasa Hasegawa, Rikio Konno, Hiroki Nakano, and Mahito Kohmoto. Zero modes of tight‐binding electrons on the honeycomb lattice. *Phys. Rev. B*, 74:033413, 2006.

[81] D. Bercioux and A. De Martino. Spin‐resolved scattering through spin‐orbit nanostruc‐

[82] Kyoko Nakada, Mitsutaka Fujita, Gene Dresselhaus, and Mildred S. Dresselhaus. Edge state in graphene ribbons: Nanometer size effect and edge shape dependence. *Phys.*

[83] M. Oliva‐Leyva and G. G. Naumis. Understanding electron behavior in strained

[84] Y. Lu and J. Guo. Band gap of strained graphene nanoribbons. *Nano Research*, 3(3):189–

[85] D. A. Bahamon and Vitor M. Pereira. Conductance across strain junctions in gra‐

[86] Ginetom S. Diniz, Marcos R. Guassi, and Fanyao Qu. Controllable spin‐charge transport in strained graphene nanoribbon devices. *Journal of Applied Physics*, 116(11),

[87] Markus Morgenstern. Scanning tunneling microscopy and spectroscopy of graphene

[88] Elena Stolyarova, Kwang Taeg Rim, Sunmin Ryu, Janina Maultzsch, Philip Kim, Louis E. Brus, Tony F. Heinz, Mark S. Hybertsen, and George W. Flynn. High‐resolution scanning tunneling microscopy imaging of mesoscopic graphene sheets on an

[89] Guohong Li, Adina Luican‐Mayer, Dmitry Abanin, Leonid Levitov, and Eva Y. Andrei. Evolution of landau levels into edge states in graphene. *Nat. Commun*., 4:1744, 2013.

[90] M. B. Nardelli. Electronic transport in extended systems: Application to carbon

[91] M. P. Lopez Sancho, J. M. Lopez Sancho, and J. Rubio. Quick iterative scheme for the calculation of transfer matrices: application to mo (100). *Journal of Physics F: Metal*

[92] R. Gómez‐Medina, P. San José, A. García‐Martín, M. Lester, M. Nieto‐Vesperinas, and J. J. Sáenz. Resonant radiation pressure on neutral particles in a waveguide. *Phys. Rev.*

[93] S. Datta and B. Das. Electronic analog of the electro‐optic modulator. *Appl. Phys. Lett*.,

[94] J. Wurm, M. Wimmer, and K. Richter. Symmetries and the conductance of graphene

nanoribbons with long‐range disorder. *Phys. Rev. B*, 85:245418, 2012.

on insulating substrates. *physica status solidi (b)*, 248(11):2423–2434, 2011.

graphene as a reciprocal space distortion. *Phys. Rev. B*, 88:085430, 2013.

tures in graphene. *Phys. Rev. B*, 81:165410, 2010.

phene nanoribbons. *Phys. Rev. B*, 88:195416, 2013.

insulating surface. *PNAS*, 104(22):9209–9212, 2007.

nanotubes. *Phys. Rev. B*, 60(11):7828, 1999.

*Physics*, 14(5):1205, 1984.

*Lett*., 86:4275–4277, 2001.

56(7):665–667, 1990.

*Rev. B*, 54:17954, 1996.

199, 2010.

90 Recent Advances in Graphene Research

113705: 1–7, 2014.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/64240

#### **Abstract**

Spectral statistics of weakly disordered triangular graphene flakes with zigzag edges are revisited. Earlier, we have found numerically that such systems may show spectral fluctuations of Gaussian unitary ensemble (GUE), signaling the time‐reversal symme‐ try (TRS) breaking at zero magnetic field, accompanied by approximate twofold valley degeneracy of each energy level. Atomic‐scale disorder induces the scattering of charge carriers between the valleys and restores the spectral fluctuations of Gaussian orthogonal ensemble (GOE). A simplified description of such a nonstandard GUE‒GOE transition, employing the mixed ensemble of 4 × 4 real symmetric matrices was also proposed. Here, we complement our previous study by analyzing numerically the spectral fluctuations of large matrices belonging the same mixed ensemble. Resulting scaling laws relate the ensemble parameter to physical size and the number of atomic‐ scale defects in graphene flake. A phase diagram, indicating the regions in which the signatures of GUE may by observable in the size‐doping parameter plane, is presented.

**Keywords:** graphene, quantum chaos, random matrix, time-reversal symmetry, gaus‐ sian ensemble

## **1. Introduction**

The notion of emergent phenomena was coined out by Anderson in his milestone science paper of 1979 [1]. In brief, emergence occurs when a complex system shows qualitatively different properties then its building blocks. Numerous examples of emergent systems studied in contemporary condensed matter physics, including high‐temperature superconductors and heavy‐fermion compounds [2], are regarded as systems with spontaneous symmetry break‐ ing [3]. A link between emergence and spontaneous symmetry breaking, however, does not

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

seem to have a permanent character. In a wide class of electronic systems, such as semicon‐ ducting heterostructures containing a two‐dimensional electron gas (2DEG), physical proper‐ tiesofitinerant electrons are substantiallydifferentthanpropertiesoffree electrons (or electrons in atoms composing the system), and are also highly‐tunable upon variation of external electromagnetic fields [4]. To give some illustration of this tunability, we only mention that electrons inGaAsheterostructures canbeusuallydescribedbyastandardSchrödinger equation of quantum mechanics with the effective mass *m*eff=0.067*m*e (where *m*e is the free electron mass), whereas in extreme cases of quantum states formed in quantum Hall systems, effective quasiparticles may not even show the Fermi‒Dirac statistics [5, 6].

It is rather rarely noticed that graphene, a two‐dimensional form of carbon just one atom tick [7], also belongs to the second class of emergent systems (i.e., without an apparent spontaneous symmetry breaking) described briefly above. In a monolayer graphene, effective Hamiltonian for low‐energy excitations has a Dirac‒Weyl form, namely

$$\mathcal{H}\_{\text{eff}} = \upsilon\_F \left[ \mathbf{p} + e \mathbf{A}(\mathbf{r}, t) \right] \cdot \boldsymbol{\sigma} + \mathcal{U}(\mathbf{r}, t), \tag{1}$$

where *v*<sup>F</sup> <sup>=</sup>10<sup>6</sup> m/s is the energy‐independent Fermi velocity, *<sup>σ</sup>* <sup>=</sup>(*σx*, *<sup>σ</sup>y*) with the Pauli matrices *σx* and *σy*, *p* = −*iħ*(∂*<sup>x</sup>* , ∂ *<sup>y</sup>* ) is the in‐plane momentum operator, the electron charge is −*e*, and the external electromagnetic field is defined via scalar and vector potentials, *U* (*r*, *t*) and *A*(*r*, *t*), with the in‐plane position *r* =(*x*, *y*) and the time *t*. 1 In other words, the system build of nonrelativistic elements (carbon atoms at normal conditions) turns out to host ultrarelativ‐ istic quasiparticles, providing a beautiful example of an emergent phenomenon, which binds together two rather distant areas of relativistic quantum mechanics and condensed matter physics [8]. This observation applies generically to bilayer or multilayer graphenes [9], as well as to HgTe/CdTe quantum wells [10], although microscopic models describing such *other Dirac systems* are slightly different. It is also worth to mention so‐called artificial graphenes, in which waves (of different kinds) obey their effective Dirac equations [11‒13].

A peculiar nature of Dirac fermions in graphene originates from the chiral structure of the Hamiltonian ℋeff, accompanied by the fact that coupling to the external electromagnetic field is described by additive terms, which are linear in both scalar and vector potentials. A remarkable consequence of these facts is the quantization of the visible light absorption [14], an unexpected macroscopic quantum effect recently found to have analogs in other Dirac systems [15, 16], and even in a familiar graphite [17]. Another intriguing effect of this kind appears for dc conductivity of ballistic graphene [18]. In the so‐called pseudodiffusive transport regime, the conductance of a rectangular sample (with the width *W* and the length *<sup>L</sup>* ) scales as *<sup>G</sup>* <sup>=</sup>*σ*<sup>0</sup> <sup>×</sup>*<sup>W</sup>* / *<sup>L</sup>* for *<sup>W</sup>* <sup>≫</sup>*<sup>L</sup>* , where *σ*<sup>0</sup> <sup>=</sup>(<sup>4</sup> / *<sup>π</sup>*)*<sup>e</sup>* <sup>2</sup> / *<sup>h</sup>* is the universal quantum value of the conductivity [19, 20], whereas the shot‐noise power and all the other charge‐transfer characteristics are indistinguishable from those of a classical diffusive conductor regardless

<sup>1</sup> Strictly speaking, ℋeff Eq. (1) applies to quasiparticles near the *K* valley in the dispersion relation. To obtain the effective Hamiltonian for other valley (K') it is sufficient to substitute *σy*→−*σy*.

the sample shape [21]. At high magnetic fields, the pseudo diffusive charge transport is predicted theoretically to reappear for resonances with Landau levels in both monolayer [22] and bilayer graphene [23]. In the presence of disorder, a fundamental property of the Hamil‐ tonian—the time reversal symmetry (TRS)—starts to play a decisive role. In particular, effective TRS in a single valley may be broken even in the absence of magnetic fields, leading to observable (and having the universal character) consequences for the conductance and spectral fluctuations [24, 25], as well as for the peculiar scaling behavior predicted for the conductivity [26, 27].

seem to have a permanent character. In a wide class of electronic systems, such as semicon‐ ducting heterostructures containing a two‐dimensional electron gas (2DEG), physical proper‐ tiesofitinerant electrons are substantiallydifferentthanpropertiesoffree electrons (or electrons in atoms composing the system), and are also highly‐tunable upon variation of external electromagnetic fields [4]. To give some illustration of this tunability, we only mention that electrons inGaAsheterostructures canbeusuallydescribedbyastandardSchrödinger equation of quantum mechanics with the effective mass *m*eff=0.067*m*e (where *m*e is the free electron mass), whereas in extreme cases of quantum states formed in quantum Hall systems, effective

It is rather rarely noticed that graphene, a two‐dimensional form of carbon just one atom tick [7], also belongs to the second class of emergent systems (i.e., without an apparent spontaneous symmetry breaking) described briefly above. In a monolayer graphene, effective Hamiltonian

where *v*<sup>F</sup> <sup>=</sup>10<sup>6</sup> m/s is the energy‐independent Fermi velocity, *<sup>σ</sup>* <sup>=</sup>(*σx*, *<sup>σ</sup>y*) with the Pauli matrices *σx* and *σy*, *p* = −*iħ*(∂*<sup>x</sup>* , ∂ *<sup>y</sup>* ) is the in‐plane momentum operator, the electron charge is −*e*, and the external electromagnetic field is defined via scalar and vector potentials, *U* (*r*, *t*) and

of nonrelativistic elements (carbon atoms at normal conditions) turns out to host ultrarelativ‐ istic quasiparticles, providing a beautiful example of an emergent phenomenon, which binds together two rather distant areas of relativistic quantum mechanics and condensed matter physics [8]. This observation applies generically to bilayer or multilayer graphenes [9], as well as to HgTe/CdTe quantum wells [10], although microscopic models describing such *other Dirac systems* are slightly different. It is also worth to mention so‐called artificial graphenes, in which

A peculiar nature of Dirac fermions in graphene originates from the chiral structure of the Hamiltonian ℋeff, accompanied by the fact that coupling to the external electromagnetic field is described by additive terms, which are linear in both scalar and vector potentials. A remarkable consequence of these facts is the quantization of the visible light absorption [14], an unexpected macroscopic quantum effect recently found to have analogs in other Dirac systems [15, 16], and even in a familiar graphite [17]. Another intriguing effect of this kind appears for dc conductivity of ballistic graphene [18]. In the so‐called pseudodiffusive transport regime, the conductance of a rectangular sample (with the width *W* and the length *<sup>L</sup>* ) scales as *<sup>G</sup>* <sup>=</sup>*σ*<sup>0</sup> <sup>×</sup>*<sup>W</sup>* / *<sup>L</sup>* for *<sup>W</sup>* <sup>≫</sup>*<sup>L</sup>* , where *σ*<sup>0</sup> <sup>=</sup>(<sup>4</sup> / *<sup>π</sup>*)*<sup>e</sup>* <sup>2</sup> / *<sup>h</sup>* is the universal quantum value of the conductivity [19, 20], whereas the shot‐noise power and all the other charge‐transfer characteristics are indistinguishable from those of a classical diffusive conductor regardless

Strictly speaking, ℋeff Eq. (1) applies to quasiparticles near the *K* valley in the dispersion relation. To obtain the effective

1

In other words, the system build

(1)

quasiparticles may not even show the Fermi‒Dirac statistics [5, 6].

for low‐energy excitations has a Dirac‒Weyl form, namely

922 Recent Advances in Graphene Research

*A*(*r*, *t*), with the in‐plane position *r* =(*x*, *y*) and the time *t*.

Hamiltonian for other valley (K') it is sufficient to substitute *σy*→−*σy*.

1

waves (of different kinds) obey their effective Dirac equations [11‒13].

Although the interest in graphene and other Dirac systems primarily focus on their potential applications [28, 29], quite often linked to the nonstandard quantum description [8], we believe that the fundamental perspective sketched in the above also deserves some attention. In the remaining part of this article, we first overview basic experimental, theoretical and numerical findings concerning signatures of quantum chaos in graphene and its nanostructures (Section 2). Next, we present our new numerical results concerning the additive random matrix model originally proposed in Ref. [25] to describe a nonstandard GUE‒GOE transition, accompanied by lifting out the valley degeneracy (Section 3). The consequences of these findings for prospective experiments on graphene nanoflakes, together with the phase diagram depicting the relevant matrix ensembles in the system size‐doping plane, are described in Section 4. The concluding remarks are given in Section 5.
