On the RoomTemperature QHE in Graphene
Abstract
The unusual quantum Hall effect (QHE) in graphene is described in terms of the composite (c) bosons, which move with a linear dispersion relation. The “electron” (wave packet) moves easier in the direction of the honeycomb lattice than perpendicular to it, while the “hole” moves easier in . Since “electrons” and “holes” move in different channels, the particle densities can be high especially when the Fermi surface has “necks”. The strong QHE arises from the phonon exchange attraction in the neighborhood of the “neck” surfaces. The plateau observed for the Hall conductivity and the accompanied resistivity drop is due to the superconducting energy gap caused by the BoseEinstein condensation of the cbosons, each forming from a pair of oneelectron–twofluxons cfermions by phononexchange attraction. The halfinteger quantization rule for the Hall conductivity: , , is derived.
keywords:
Quantum Hall effect; composite boson (fermion); superconducting energy gap; phonon exchange attraction1 Introduction
In 2005 Novoselov et al. 1 () discovered a quantum Hall effect (QHE) in graphene, a single sheet of graphite. Figure 1 is reproduced after Ref. 1, Fig. 4.
The longitudinal magnetoresistivity and the Hall conductivity in graphene at T and K are plotted as a function of the conduction electron (“electron” or “hole”) density in the scale of cm. The plateau values of the Hall conductivity are quantized in the units of
(1) 
within experimental errors, where is the Planck constant, the electron charge (magnitude). The longitudinal resistivity reaches zero at the middle of the plateaus. These two are the major signatures of the QHE in graphene.
In 2007 Novoselov et al. 2 () reported a discovery of a room temperature QHE in graphene. We reproduceed their data in Figure 2 after Ref. 2, Fig. 1.
The Hall resistivity for “electrons” and “holes” indicate precise quantization within experimental errors at magnetic field 29 T and temperature 300 K. This is an extraordinary jump in the observation temperatures since the QHE in heterojunction GaAs/AlGaAs was reported below 0.5 K. Figure 2 is similar to those in Figure 1 although the abscissas are different, one in gate voltage and the other in carrier density, and hence the physical conditions are different. We give an explanation later. Notice that the quantization in appears in units of , which is a little strange since the most visible quantization for GaAs/AlGaAs appears in units of . We will resolve this mystery in the present work.
From the QHE behaviors in Figures 1 and 2, we observe that the quantization in the Hall conductivity occurs at a set of halfinteger points:
(2) 
The original authors 1 (); 2 () interpreted their data in terms of Dirac fermions. A great number of experimental and theoretical papers followed. The present work deals specifically with the quantization rule in Eq. (2). We shall show this quantization rule based on the cparticles (fermions, bosons) PTEP () model in the present work. We will defer discussion of Dirac fermions and the related matter. The preliminary results were reported in the conference proceedings 3a ().
2 Electron Dynamics in Graphene
The normal carriers in solids are “electrons” (“holes”), which spiral around the applied magnetic field counterclockwise (clockwise) viewed from the tip of the field vector . The “electrons” (“holes”) are excited above (below) the metal’s Fermi energy. These quasiparticles are quotation marked throughout the text. Following Ashcroft and Mermin 3 () we regard the conduction electrons as wave packets.
We consider a graphene, which forms a 2D honeycomb lattice. The WignerSeitz (WS) unit cell 4 (), rhombus (shaded) shown in Figure 3 (a), contains two C’s. We showed in our earlier work 5 () that graphene has “electrons” and “holes” based on the rectangular unit cell (dotted lines) shown in Figure 3 (b). We briefly review our calculations. We must choose the rectangular unit cell to establish the Bloch plane waves 7 () in 2D. For a 1D space, there always exists a 1D space. If one introduces nonorthogonal axes along , then one cannot use Fourier transformation. This difficulty was discussed earlier in our previous work6 (). To establish the electron dynamics we need the orthogonal rectangular unit cell shown in Fig. 3 (b).
We assume that the “electron” (“hole”) wave packet has the charge () and a size of the rectangular unit cell, generated above (below) the Fermi energy . We showed 5 () that (a) the “electron” and the “hole” have different charge distributions and different effective masses, (b) that the “electrons” and “holes” move in different easy channels, (c) that the “electrons” and “holes” are thermally excited with different activation energies, and (d) that the “electron” activation energy is smaller than the “hole” activation energy :
(3) 
The thermally activated electron densities are then given by
(4) 
where and 2 represent the “electron” and “hole”, respectively. In view of Eqs. (3) and (4), . Hence the “electrons” are the majority carriers in graphene. Magnetotransport experiments by Zhang et al. 9 () indicate that the “electrons” are the majority carriers in graphene in agreement with experiments.
3 Fractional Quantum Hall Effect
Fractional QHE were discovered by Tsui, Stormer and Gossard in 1982 Tsui (). In 1983 Laughlin proposed a revolutionary idea 17 () that fractional charges are carried by the elementary excitations for the fractional QHE system. A great number of papers were followed 18 (); 22 (); 16 (); 19 (); 15 (); 20 (). Ezawa wrote books with extensive references for students and researchers 14 (). The prevalent theories 17 (); 18 (); 22 (); 16 (); 19 (); 15 (); 20 () based on the Laughlin wave function 17 () in the Schrödinger picture deal with the QHE at 0 K and immediately above. The system ground state, however, cannot carry a current. To interpret the experimental data it is convenient to introduce composite (c) particles (bosons, fermions). The cboson (cfermion), each containing an electron and an odd (even) number of magnetic flux quanta (fluxons), were introduced by Zhang et al. 22 () and others (Jain 16 ()) for the description of the fractional QHE (Fermi liquid). The cparticles will be regarded as quasiparticles (elementary excitations) existing in the system. A classical electron spirals around the applied static magnetic field. The state has a lower energy relative to the original electron energy because the spiraling current (vortex) is diamagnetic. The fielddressed (attached) electron moves straight. Jain 16 () established a close connection between the integer and the fractional QHE by introducing cfermions. His cfermions are essentially the same as our cfermions. The types of mechanics (classical or quantum) do not change the energy sign. A cfermion is in a negative energy (bound) state. Fujita and Okamura 21 () discussed the formation of a bound cfermion and its connection with Jain’s cfermion. Jain did not include the cbosons in his book Jain (). We view the cbosons as equally important as the cfermions. A cboson is also in a bound state. Besides, cbosons can be BoseEinstein (BE) condensed, which generates a stabilizing (superconducting) energy gap in the excitation spectrum. All QHE states with distinctive Hall plateaus in heterojunction GaAs/AlGaAs are observed below the critical temperature K. The QHE in graphene observed at 300 K is an exception. It is desirable to treat the QHE below and above in a unified manner. The extreme accuracy (precision ) in which each Hall plateau is observed means that the current density must be computed exactly without averaging. In the prevalent theories 17 (); 18 (); 22 (); 16 (); 19 (); 15 (); 20 (), the electronelectron interaction and Pauli’s exclusion principle are regarded as the cause for the QHE. Both are essentially repulsive and cannot account for the fact that the cparticles are bound, that is, they are in negativeenergy states. Besides, the prevalent theories have limitations:

The zero temperature limit is taken at the outset. Then the question why QHE is observed below 0.5 K in GaAs/AlGaAs cannot be answered. We better have a theory for all temperatures.

The highfield limit is taken at the outset. The integer QHE at filling factor (Landau level occupation number) are observed for small integer only. The question why the QHE for high (weak field) is not observed cannot be answered. We better describe the phenomena for all fields.

The Hall resistivity value is obtained in a single stroke. To obtain we need two separate measurements of the Hall field and the current density . We must calculate and take the ratio to obtain .
Fujita and Okamura 21 () developed a quantum statistical theory based on phonon exchange attraction, and used Laughlin’s results to describe the fractional QHE. In the present work we complete the description without using Laughlin’s fractional charge idea with the assumtion that any cfermion has the charge magnitude . See the paper by Fujita, Suzuki and Ho FSH () for more detail. There is a remarkable similarity between the QHE and the HighTemperature Superconductivity (HTSC), both occurring in 2D systems as pointed out by Laughlin 17a (). We regard the phonon exchange attraction as the causes of both QHE and superconductivity. Starting with a reasonable Hamiltonian, we calculate everything using the standard statistical mechanics.
The countability concept of the fluxons, known as the flux quantization:
(5) 
where sample area, fluxon number (integer), flux quantum, is originally due to Onsager 23 (). The magnetic (electric) field is an axial (polar) vector and the associated fluxon (photon) is a halfspin fermion (fullspin boson). The magnetic (electric) flux line cannot (can) terminate at a sink, which supports the fermionic (bosonic) nature of the associated fluxon (photon). No halfspin fermion can annihilate itself because of angular momentum conservation. The electron spin originates in the relativistic electron equation (Dirac’s theory of electron) 24 (). The discrete (two) quantum numbers cannot change in the continuous limit, and hence the spin must be conserved. The countability and statistics of the fluxon is the fundamental particle properties. We postulate that the fluxon is a halfspin fermion with zero mass and zero charge.
We assume that the magnetic field is applied perpendicular to the 2D plane. The 2D Landau level energy,
(6) 
with the states have a great degeneracy; the is the effective mass of an “electron” and the the cyclotron frequency. The CenterofMass (CM) of any cparticle moves as a fermion (boson). The eigenvalues of the CM momentum are limited to 0 or 1 (unlimited) if it contains an odd (even) number of elementary fermions. This rule is known as the EhrenfestOppenheimerBethe’s (EOB’s) rule 24 (); 25 (); 26 (). Hence the CM motion of the composite containing an electron and fluxons is bosonic (fermionic) if is odd (even). The system of the cbosons condenses below the critical temperature and exhibits a superconducting state while the system of cfermions shows a Fermi liquid behavior.
A longitudinal phonon, acoustic or optical, generates a density wave, which affects the electron (fluxon) motion through the charge displacement (current). The exchange of a phonon between electron and fluxon generate an attractive transition.
Bardeen, Cooper and Schrieffer (BCS) 28 () assumed the existence of Cooper pairs 29 () in a superconductor, and wrote down a Hamiltonian containing the “electron” and “hole” kinetic energies and the pairing interaction Hamiltonian with the phonon variables eliminated. We start with a BCSlike Hamiltonian for the QHE: 21 ()
(7) 
where is the number operator for the “electron” (1) [“hole” (2), fluxon (3)] at momentum and spin with the energy with annihilation (creation) operators satisfying the Fermi anticommutation rules:
(8) 
The fluxon number operator is represented by with satisfying the anticommutation rules:
(9) 
The phonon exchange can create electronfluxon composites, bosonic or fermionic, depending on the number of fluxons. We call the conductionelectron composite with an odd (even) number of fluxons cboson (cfermion). The electron (hole)type cparticles carry negative (positive) charge. Electron (hole)type Cooperpairlike cbosons are generated by the phononexchange attraction from a pair of electron (hole)type cfermions. The pair operators are defined by
(10) 
The prime on the summation in Eq. (7) means the restriction: , Debye frequency. The pairing interaction terms in Eq. (7) conserve the charge. The term , where , sample area, is the pairing strength, generates a transition in the electrontype cfermion states. Similarly, the exchange of a phonon generates a transition between the holetype cfermion states, represented by . The phonon exchange can also paircreate (pairannihilate) electron (hole)type cboson pairs, and the effects of these processes are represented by .
The Cooper pair is formed from two “electrons” (or “holes”). Likewise the cbosons may be formed by the phononexchange attaraction from two likecharge cfermions. If the density of the cbosons is high enough, then the cbosons will be BEcondensed and exhibit a superconductivity.
The pairing interaction terms in Eq. (7) are formally identical with those in the generalized BCS Hamiltonian 30 (). Only we deal here with cfermions instead of conduction electrons.
The cbosons, having the linear dispersion relation, can move in all directions in the plane with the constant speed 21 (); 30 (). The supercurrent is generated by cbosons monochromatically condensed, running along the sample length. The supercurrent density (magnitude) , calculated by the rule: , is given by
(11) 
where is the effective charge of carriers. The Hall field (magnitude) equals . The magnetic flux is quantized as in Eq. (5). Hence we obtain
(12) 
Here, we assumed that the cfermion containing an electron and an even number of fluxons has a charge magnitude . For the integer QHE, , , then we obtain , explaining the plateau value observed for the integer QHE.
The supercurrent generated by equal numbers of cbosons condensed monochromatically is neutral. This is reflected in the calculations in Eq. (11). The supercondensate whose motion generates the supercurrent must be neutral. If it has a charge, it would be accelerated indefinitely by the external field because the impurities and phonons cannot stop the supercurrent to grow. That is, the circuit containing a superconducting sample and a battery must be burnt out if the supercondensate is not neutral. In the calculation of in Eq. (12), we used the unaveraged drift velocity , which is significant. Only the unaveraged drift velocity cancels out exactly from numerator/denominator, leading to an exceedingly accurate plateau value.
We now extend our theory to include elementary fermions (electron, fluxon) as members of the cfermion set. We can then treat the superconductivity and the QHE in a unified manner. The cboson containing one electron and one fluxon can be used to describe the integer QHE.
Important pairings and the effects are listed below.

a pair of conduction electrons, superconductivity

a fluxon and cfermions, QHE

a pair of likecharge conduction electrons, each with two fluxons, QHE in graphene.
4 The Room Temperature QHE
The QHE behavior observed for graphene is remarkably similar to that for GaAs/AlGaAs. The physical conditions are different however since the gate voltage and the applied magnetic field are varied in the experiments. The present authors regard the QHE in GaAs/AlGaAs as a manifestation of superconductivity generated by the magnetic field. Briefly, the magnetoresistivity for a QH system reaches zero (superconducting) and the accompanied Hall resistivity generates a plateau by the Meissner effect. The QHE state is not easy to destroy because of the superconducting energy gap in the cboson excitation spectrum. If an extra magnetic field is applied to the system at optimum QHE state (the center of the plateau), then the system remains in the same superconducting state by expelling the extra field. If the field is reduced, then the system stays in the same state by sucking in extra field fluxes, thus generating a Hall conductivity plateau. In the graphene experiments, the gate voltage is varied. A little extra gate voltage relative to the optimum voltage (the center of the plateau) polarizes the system without changing the superconducting state, thus generating a Hall conductivity plateau. This state has an extra electric field energy:
(13) 
where is the sample area, the dielectric constant, and is the extra electric field, positive or negative, depending on the field direction. If the gate voltage is further increased (or decreased), then it will eventually destroy the superconducting state, and the resistivity will rise from zero. A strong current generates high magnetic field around it, which eventually destroys the supercurrent. This explains the flat plateau and the rise in resistivity from zero.
We now examine the data shown in Figure 2. We first observe that the rightleft symmetry is broken. “Electrons” and “holes” move in different channels with different masses, breaking symmetry. The applied gate voltage induce the surface conduction electrons and hence changes the Fermi surface. A relatively high voltage 20 V may bring the system to the van Hove singularity points in the neighborhood of which the conduction electron densities are high. This is where the prominent QHE is observed. We note that such discussions are possible only with the rectangular unit cell model, and not with the WS unit cell model, which predicts a gapless semiconductor with the electronhole symmetry: , .
We wish to derive the quantization rule in Eq. (2). Let us first consider the case of . The QHE requires a BEC of cbosons. Its favorable environment is near the van Hove singularities, where the Fermi surface changes its curvature sign. For graphene, this happens when the 2D Fermi surface just touches the Brillouin zone boundary and “electrons” or “holes” are abundantly generated. The quantization rule given by Eq. (2) is realized if the cbosons are formed from a pair of likecharge cfermions, each containing a conduction electron and two (2) fluxons. By assumption, each cfermion has the effective charge :
(14) 
After studying the lowfield QH states of cfermions we obtain
(15) 
for the density of the cfermions with fluxons, where is the electron density. All fermionic QH states (points) lie on the classicalHall straight line passing the origin with a constant slope when is plotted as a function of the inverse magnetic field. For higher fields the LL spacing is greater, and hence the fermion formation is more difficult if is greater. The cboson contains two (2) cfermions. Using Eq. (12), we obtain
(16) 
Here, the field at is used, where the cboson density is equal to the flux density . We note that the value obtained here is in agreement with the experiments shown in Fig. 1.
The QHE states with integers are generated on the weaker field side. Their strengths decrease with increasing as shown below. The magnetic field magnitude becomes smaller with increasing . The LL degeneracy is proportional to , and hence LL’s must be considered. First consider the case . Without the phononexchange attraction the electrons occupy the lowest two LL’s with spin. The electrons at each level form fundamental (f) cbosons. In the superconducting state the cbosons occupy the monochromatically condensed state, which is separated by the superconducting gap from the continuum states (band) as shown in the righthand figure in Fig. 4.
The cboson density at each LL is onehalf the density at , which is equal to the electron density fixed for the sample. Extending the theory to a general integer , we have
(17) 
This means that both the critical temperature and the energy gap are smaller, making the plateau width (a measure of ) smaller in agreement with experiments. The cbosons have lower energies than the conduction electrons. Hence at the extreme low temperatures the supercurrent due to the condensed cbosons dominates the normal currents due to the conduction electrons and noncondensed cbosons, giving rise to the dip in . The superconducting energy gap is obtained and discussed earlier. For completeness the derivation of is given in Appendix. Thus, we have obtained Eq. (2) within the framework of our fractional QHE theory in terms of cparticles.
In summary, we established that

The halfinteger FQHE arises from the BEC of cbosons, each containing a pair of cfermions with two fluxons.

The Hall conductivity is quantized at , .

The strengths of the plateaus become smaller with increasing .
Appendix: Temperature Dependent Energy Gap
The cbosons can be bound by the interaction Hamiltonian . The fundamental cbosons (fcbosons) can undergo a BoseEinstein condensation (BEC) below the critical temperature . The fcbosons are condensed at a momentum along the sample length. Above , they can move in all directions in the plane with the Fermi speed . The ground state energy can be calculated by solving the Cooperlike equation:
(A1) 
where is the reduced wave function for the stationary fcbosons; the prime on the integral sign means that the restriction: , =Debye frequency. We obtain after simple calculations
(A2) 
where is the density of states per spin at . Note that the binding energy does not depend on the “electron” mass. Hence, the fcbosons have the same energy .
At 0 K only stationary fcbosons are generated. The ground state energy of the system of fcbosons is
(A3) 
where is the (or ) fcboson number.
At a finite there are moving (noncondensed) fcbosons, whose energies are obtained from31 ()
(A4) 
For small , we obtain
(A5) 
where is the Fermi speed. The energy depends linearly on the momentum magnitude .
The system of free massless bosons undergoes a BEC in 2D at the critical temperature :
(A6) 
where is the boson speed, and the density. Briefly the BEC occurs when the chemical potential vanishes at a finite . The critical temperature can be determined from
(A7) 
After expanding the integrand in powers of and using , we obtain
(A8) 
from which we obtain formula (A6). Substituting in Eq. (A6), we obtain
(A9) 
The interboson distance calculated from this equation is . The boson size calculated from Eq. (A9), using the uncertainty relation and , is , which is a few times smaller than . Thus the bosons do not overlap in space, and the free boson model is justified.
In the presence of the BEcondensate below , the unfluxed electron carries the energy , where the quasielectron energy gap is the solution of
(A10) 
Note that the gap depends on . At there is no condensate, and hence vanishes.
The moving fcboson below with the condensate background has the energy , obtained from
(A11) 
where replaced in Eq. (A4). We obtain
(A12) 
where is determined from
(A13) 
The energy difference
(A14) 
represents the dependent energy gap between the moving and stationary fcbosons. The energy is negative. Otherwise, the fcboson should break up. This limits to be less than . The energy gap is at 0 K. It declines to zero as the temperature approaches .
The experimental electron density is cm and the Fermi velocity ms. The critical temperature is expected to be much above 300 K. The temperature 50 K can be regarded as a very low temperature relative to . Hence the QH state has an Arheniusdecay type exponential stability factor:
(A15) 
where is the zerotemperature energy gap.
References
 (1) K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, A. A. Firsov, Nature 438, 97 (2005).
 (2) K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. I. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim A. K. Gaim, Science 315, 1379 (2007).
 (3) S. Fujita, A. Suzuki, H. C. Ho, the paper submitted to PTEP.
 (4) S. Fujita and A. Suzuki, Journal of Physics, Conference Serie 490 012064 (2014).
 (5) N. W. Ashcroft, N. D. Mermin, Solid State Physics, (Saunders, Philadelphia,1976), p. 214.
 (6) E. Wigner, F. Seitz, Phys. Rev. 43, 804 (1933).
 (7) S. Fujita, A. Suzuki, J. Appl. Phys. 107, 013711 (2010).
 (8) S. Fujita, Y. Takato, A. Suzuki, Mod. Phys. Lett. B 25, 223 (2011).
 (9) S. Fujita, A. Jovaini, S. Godoy, A. Suzuki, Phys. Lett. A 376, 2808 (2012).
 (10) F. Bloch, Zeits. Phys. 52, 555 (1928).
 (11) Y. Zhang, Y.W. Tan, H.L. Stormer, P. Kim, Nature 438, 201 (2005).
 (12) D. C. Tsui, H. L. Stormer, A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982).
 (13) R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
 (14) S. M. Girvin, A. H. MacDonald, Phys. Rev. Lett. 58, 1252 (1987).
 (15) S. C. Zhang, T. H. Hansson, S. Kivelson, Phys. Rev. Lett. 62, 82 (1989).
 (16) J. K. Jain, Phys. Rev. Lett. 63 (1989) 199; Phys. Rev. B 40, 8079 (1989); ibid. B 41, 7653 (1990); Surf. Sci. 263, 65 (1992).
 (17) N. Read, Phys. Rev. Lett. 62, 86 (1989).
 (18) B. I. Halperin, P. A. Lee, H. Read, Phys. Rev. B 47 (1993) 7312.
 (19) R. Shankar, G. Murthy, Phys. Rev. Lett. 79, 4437 (1997).
 (20) Z. F. Ezawa, Quantum Hall Effects, 2nd ed., (World Scientific, Singapore, 2008).
 (21) S. Fujita, Y. Okamura, Phys. Rev. 369, 155313 (2004).
 (22) J. K. Jain, Composite Fermions, (Cambridge University Press, Cambridge, UK, 2007).
 (23) S. Fujita, A. Suzuki and H. C. Ho, Arxiv. 1304.7631v1 [condmat.mes–hall].
 (24) R. B. Laughlin, Science 242, 525 (1988).
 (25) L. Onsager, Phil. Mag. 43, 1006 (1952).
 (26) P. A. M. Dirac, Principles of Quantum Mechanics, 4th ed., (Oxford Univ. Press, Oxford, 1958). pp. 248–252, pp. 253–263, p. 267.
 (27) P. Ehrenfest, J. R. Oppenheimer, Phys. Rev. 37, 333 (1931).
 (28) H. A. Bethe, R. Jackiw, Intermediate Quantum Mechanics, 2nd ed.,(Benjamine, New York, 1968). p. 23.
 (29) S. Fujita, SP Gau, A. Suzuki, J. Korean Phys. Soc. 38, 456 (2001).
 (30) J. Bardeen, L. N. Cooper, J. R. Schriefler, Phys. Rev. 108, 1175 (1957).
 (31) L.N. Cooper, Phys. Rev. 104, 1189 (1956).
 (32) S. Fujita, K. Ito, S. Godoy, Quantum Theory of Conducting Matter, Superconductivity, (Springer, New York, 2009). pp. 7375, pp. 77–79.
 (33) S. Fujita and A. Suzuki, Electrical Conduction in Graphene and Nanotubes, (WileyVCH, Weinheim, Germany, 2013). pp. 212–215.