Quantum Hall Effect in Graphene

The quantum Hall effect in graphene or the unusual quantum Hall effect is the quantization effect of the Hall resistance or conductivity of a two-dimensional electron gas or two-dimensional hole gas in strong magnetic fields in graphene . This effect was predicted theoretically ^[1] ^[2] and confirmed experimentally in 2005 ^[3] ^[4] .

Content

Landau Levels

Landau levels in graphene are described by the Dirac equation for graphene taking into account the magnetic field , which can be written in the form ^[5]

[{\vec {\sigma }}\cdot (i\hbar v_{F}{\vec {\nabla }}+e{\vec {A}}/c)]\psi (x,y)=\varepsilon \psi (x,y)

{\ displaystyle [{\ vec {\ sigma}} \ cdot (i \ hbar v_ {F} {\ vec {\ nabla}} + e {\ vec {A}} / c)] \ psi (x, y) = \ varepsilon \ psi (x, y)}

where the Landau gauge for the vector potential is used ${\vec {A}}=(-By,0)$ ${\ displaystyle {\ vec {A}} = (- By, 0)}$ , the two-dimensional gradient is ${\vec {\nabla }}=({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}})$ ${\ displaystyle {\ vec {\ nabla}} = ({\ frac {\ partial} {\ partial x}}, {\ frac {\ partial} {\ partial y}})}$ , and the vector ${\vec {\sigma }}$ ${\ displaystyle {\ vec {\ sigma}}}$ composed of Pauli matrices $(\sigma _{1},\sigma _{2})$ ${\ displaystyle (\ sigma _ {1}, \ sigma _ {2})}$ . In matrix form, the equation is written as

{\begin{pmatrix}0&-i\hbar v{\frac {\partial }{\partial x}}-\hbar v{\frac {\partial }{\partial y}}+eBy\\-i\hbar v_{F}{\frac {\partial }{\partial x}}+\hbar v{\frac {\partial }{\partial y}}+eBy&0\end{pmatrix}}\psi (x,y)=\varepsilon \psi (x,y).

{\ displaystyle {\ begin {pmatrix} 0 & -i \ hbar v {\ frac {\ partial} {\ partial x}} - \ hbar v {\ frac {\ partial} {\ partial y}} + eBy \\ - i \ hbar v_ {F} {\ frac {\ partial} {\ partial x}} + \ hbar v {\ frac {\ partial} {\ partial y}} + eBy & 0 \ end {pmatrix}} \ psi (x, y) = \ varepsilon \ psi (x, y).}

Here you can easily separate the variables and eventually come to the spectrum for relativistic Landau levels

\varepsilon _{n}=\pm \hbar {\tilde {\omega _{c}}}{\sqrt {n}}=\pm {\sqrt {\hbar v_{F}^{2}eB2n/c}},

{\ displaystyle \ varepsilon _ {n} = \ pm \ hbar {\ tilde {\ omega _ {c}}} {\ sqrt {n}} = \ pm {\ sqrt {\ hbar v_ {F} ^ {2} eB2n / c}},}

Where $n=0,\,1,\,2,...$ ${\ displaystyle n = 0, \, 1, \, 2, ...}$ , " Cyclotron frequency " is ${\tilde {\omega _{c}}}={\sqrt {2}}{\frac {v_{F}}{l_{B}}}$ ${\ displaystyle {\ tilde {\ omega _ {c}}} = {\ sqrt {2}} {\ frac {v_ {F}} {l_ {B}}}}$ magnetic length $l_{B}={\sqrt {\frac {\hbar c}{eB}}}.$ ${\ displaystyle l_ {B} = {\ sqrt {\ frac {\ hbar c} {eB}}}.}$

Quantum Hall Effect

The unusual ( unconventional ) quantum Hall effect was first observed in ^[3] ^[4] , where it was shown that the carriers in graphene really have zero effective mass, since the positions of the plateau on the dependence of the off-diagonal component of the conductivity tensor corresponded to half-integer values of the Hall conductivity $\nu =\pm (|n|+1/2)$ ${\ displaystyle \ nu = \ pm (| n | +1/2)}$ in units $4e^{2}/h$ ${\ displaystyle 4e ^ {2} / h}$ (the factor 4 appears due to four-fold degeneration of energy), i.e.

\sigma _{xy}=\pm {\frac {4e^{2}}{h}}\left(|n|+{\frac {1}{2}}\right)

{\ displaystyle \ sigma _ {xy} = \ pm {\ frac {4e ^ {2}} {h}} \ left (| n | + {\ frac {1} {2}} \ right)}

.

This quantization is consistent with the theory of the quantum Hall effect for Dirac massless fermions ^[1] . For a comparison of the integer quantum Hall effect in an ordinary two-dimensional system and graphene, see Figure 1. Here are shown the broadened Landau levels for electrons (highlighted in red) and for holes (blue). If the Fermi level is between the Landau levels, then the dependence of the Hall conductivity $\sigma _{xy}$ ${\ displaystyle \ sigma _ {xy}}$ a number of plateaus are observed. This dependence differs from ordinary two-dimensional systems (a two-dimensional electron gas in silicon, which is a two-valley semiconductor in planes equivalent to {100}, can be used as an analog, that is, it also has four-fold degeneracy of the Landau levels and Hall plateaus are observed at $\nu =4|n|$ ${\ displaystyle \ nu = 4 | n |}$ ).

The quantum Hall effect (QHE) can be used as a resistance standard, because the numerical value of the plateau observed in graphene is $h/2e^{2}$ ${\ displaystyle h / 2e ^ {2}}$ performed with good accuracy, although the quality of the samples is inferior to the highly mobile 2DEG in GaAs , and, accordingly, the quantization accuracy. The advantage of QHE in graphene is that it is observed at room temperature ^[6] (in magnetic fields above 20 T ). The main limitation on the observation of QHEs at room temperature is imposed not by the smearing of the Fermi – Dirac distribution itself, but by scattering of carriers by impurities, which leads to a broadening of the Landau levels.

Fig. 1. a) The quantum Hall effect in an ordinary two-dimensional system. b) Quantum Hall effect in graphene.

g=g_{s}g_{v}=4

{\ displaystyle g = g_ {s} g_ {v} = 4}

- spectrum degeneracy

pn transition

Due to the absence of a forbidden gap in graphene in structures with an upper gate, a continuous pn junction can be formed when the voltage at the upper gate allows us to invert the carrier sign specified by the back gate in graphene, where the carrier concentration never vanishes (except for the point of electroneutrality) and no areas devoid of carriers as in ordinary pn junctions . In such structures, the quantum Hall effect can also be observed, but due to the inhomogeneity of the sign of the carriers, the values of the Hall plateaus differ from those given above. For a structure with one pn junction, the quantization values of the Hall conductivity are described by the formula ^[7]

G={\frac {2e^{2}}{h}}{\frac {|\nu ^{'}||\nu |}{|\nu ^{'}|+|\nu |}},

{\ displaystyle G = {\ frac {2e ^ {2}} {h}} {\ frac {| \ nu ^ {'} || \ nu |} {| \ nu ^ {'} | + | \ nu | }},}

Where $\nu$ ${\ displaystyle \ nu}$ and $\nu ^{'}$ ${\ displaystyle \ nu ^ {'}}$ - filling factors in the n- and p-regions, respectively (the p-region is under the upper shutter), which can take values $\pm 2,\pm 6,\pm 10$ ${\ displaystyle \ pm 2, \ pm 6, \ pm 10}$ etc. Then, plateaus in structures with one pn junction are observed at 1, 3/2, 3, 5/3, etc. Such plateau values were observed experimentally. ^[eight]

pnp transition

For a structure with two pn junctions ^{[9], the} corresponding values of the Hall conductivity are equal

G={\frac {e^{2}}{h}}{\frac {|\nu ^{'}||\nu |}{2|\nu ^{'}|+|\nu |}}={\frac {2}{3}},{\frac {6}{5}},{\frac {6}{7}},...(\nu \nu ^{'}<0).

{\ displaystyle G = {\ frac {e ^ {2}} {h}} {\ frac {| \ nu ^ {'} || \ nu |} {2 | \ nu ^ {'} | + | \ nu |}} = {\ frac {2} {3}}, {\ frac {6} {5}}, {\ frac {6} {7}}, ... (\ nu \ nu ^ {'} < 0).}

Landau Basic Level Cleavage

In ^[10] , spin splitting of the relativistic Landau levels and the removal of fourfold degeneracy for the lowest Landau level near the electroneutrality point are observed. Several theories have been proposed to explain this effect ^[11] .

Links

↑ ¹ ² Gusynin VP et al. Unconventional Integer Quantum Hall Effect in Graphene Phys. Rev. Lett. 95 , 146801 (2005) DOI : 10.1103 / PhysRevLett. 95.146801
↑ Peres NMR, et. al. Electronic properties of disordered two-dimensional carbon Phys. Rev. B 73 , 125411 (2006) DOI : 10.1103 / PhysRevB.73.125411
↑ ¹ ² Novoselov KS et al. “Two-dimensional gas of massless Dirac fermions in graphene”, Nature 438 , 197 (2005) DOI : 10.1038 / nature04233
↑ ¹ ² Zhang Y. et. al. "Experimental observation of the quantum Hall effect and Berry's phase in graphene" Nature 438 , 201 (2005) DOI : 10.1038 / nature04235
↑ Peres NMR et. al. “Algebraic solution of a graphene layer in transverse electric and perpendicular magnetic fields” J. Phys .: Condens. Matter 19 , 406231 (2007) DOI : 10.1088 / 0953-8984 / 19/40/406231
↑ Novoselov KS et. al. Room-Temperature Quantum Hall Effect in Graphene Science 315 , 1379 (2007) DOI : 10.1126 / science.1137201
↑ Abanin DA, Levitov LS Quantized Transport in Graphene pn Junctions in a Magnetic Field Science 3 , 641 (2007) DOI : 10.1126 / science.1144672
↑ Williams JR et. al. Quantum Hall Effect in a Gate-Controlled pn Junction of Graphene Science 317 , 638 (2007) DOI : 10.1126 / science.1144657
↑ Özyilmaz B. et. al. Electronic Transport and Quantum Hall Effect in Bipolar Graphene pnp Junctions Phys. Rev. Lett. 99 , 166804 (2007) DOI : 10.1103 / PhysRevLett. 99.166804
↑ Zhang Y., et al. , “Landau-Level Splitting in Graphene in High Magnetic Fields” Phys. Rev. Lett. 96 , 136806 (2006) DOI : 10.1103 / PhysRevLett. 96.136806
↑ Fuchs J. et al . Spontaneous Parity Breaking of Graphene in the Quantum Hall Regime Phys. Rev. Lett. 98 , 016803 (2007) DOI : 10.1103 / PhysRevLett . 98.016803 ; Nomura K. et al ., Quantum Hall Ferromagnetism in Graphene Phys. Rev. Lett. 96 , 256602 (2006) DOI : 10.1103 / PhysRevLett . 96.256602 ; Abanin DA et al ., Spin-Filtered Edge States and Quantum Hall Effect in Graphene Phys. Rev. Lett. 96 , 176803 (2006) DOI : 10.1103 / PhysRevLett . 96.176803 ; Fertig HA et al ., Luttinger Liquid at the Edge of Undoped Graphene in a Strong Magnetic Field Phys. Rev. Lett. 97 , 116805 (2006) DOI : 10.1103 / PhysRevLett . 97.116805 ; Goerbig MO et al ., Electron interactions in graphene in a strong magnetic field Phys. Rev. B 74 , 161407 (2006) DOI : 10.1103 / PhysRevB.74.161407 ; Alicea J. et al ., Graphene integer quantum Hall effect in the ferromagnetic and paramagnetic regimes Phys. Rev. B 74 , 075422 (2006) DOI : 10.1103 / PhysRevB.74.075422 ; Gusynin VP et al ., Excitonic gap, phase transition, and quantum Hall effect in graphene Phys. Rev. B 74 , 195429 (2006) DOI : 10.1103 / PhysRevB.74.195429

[gusynin_PRL_2005-1] ¹ ² Gusynin VP et al. Unconventional Integer Quantum Hall Effect in Graphene Phys. Rev. Lett. 95 , 146801 (2005) DOI : 10.1103 / PhysRevLett. 95.146801

[Peres_PRB_2006-2] Peres NMR, et. al. Electronic properties of disordered two-dimensional carbon Phys. Rev. B 73 , 125411 (2006) DOI : 10.1103 / PhysRevB.73.125411

[novoselov_Nature_2005-3] ¹ ² Novoselov KS et al. “Two-dimensional gas of massless Dirac fermions in graphene”, Nature 438 , 197 (2005) DOI : 10.1038 / nature04233

[zhang_Nature_2005-4] ¹ ² Zhang Y. et. al. "Experimental observation of the quantum Hall effect and Berry's phase in graphene" Nature 438 , 201 (2005) DOI : 10.1038 / nature04235

[5] Peres NMR et. al. “Algebraic solution of a graphene layer in transverse electric and perpendicular magnetic fields” J. Phys .: Condens. Matter 19 , 406231 (2007) DOI : 10.1088 / 0953-8984 / 19/40/406231

[Novoselov_science_2007-6] Novoselov KS et. al. Room-Temperature Quantum Hall Effect in Graphene Science 315 , 1379 (2007) DOI : 10.1126 / science.1137201

[abanin_science_2007-7] Abanin DA, Levitov LS Quantized Transport in Graphene pn Junctions in a Magnetic Field Science 3 , 641 (2007) DOI : 10.1126 / science.1144672

[williams_science_2007-8] Williams JR et. al. Quantum Hall Effect in a Gate-Controlled pn Junction of Graphene Science 317 , 638 (2007) DOI : 10.1126 / science.1144657

[ozyilmaz_PRL_2007-9] Özyilmaz B. et. al. Electronic Transport and Quantum Hall Effect in Bipolar Graphene pnp Junctions Phys. Rev. Lett. 99 , 166804 (2007) DOI : 10.1103 / PhysRevLett. 99.166804

[10] Zhang Y., et al. , “Landau-Level Splitting in Graphene in High Magnetic Fields” Phys. Rev. Lett. 96 , 136806 (2006) DOI : 10.1103 / PhysRevLett. 96.136806

[fuchs_PRL_2007-11] Fuchs J. et al . Spontaneous Parity Breaking of Graphene in the Quantum Hall Regime Phys. Rev. Lett. 98 , 016803 (2007) DOI : 10.1103 / PhysRevLett . 98.016803 ; Nomura K. et al ., Quantum Hall Ferromagnetism in Graphene Phys. Rev. Lett. 96 , 256602 (2006) DOI : 10.1103 / PhysRevLett . 96.256602 ; Abanin DA et al ., Spin-Filtered Edge States and Quantum Hall Effect in Graphene Phys. Rev. Lett. 96 , 176803 (2006) DOI : 10.1103 / PhysRevLett . 96.176803 ; Fertig HA et al ., Luttinger Liquid at the Edge of Undoped Graphene in a Strong Magnetic Field Phys. Rev. Lett. 97 , 116805 (2006) DOI : 10.1103 / PhysRevLett . 97.116805 ; Goerbig MO et al ., Electron interactions in graphene in a strong magnetic field Phys. Rev. B 74 , 161407 (2006) DOI : 10.1103 / PhysRevB.74.161407 ; Alicea J. et al ., Graphene integer quantum Hall effect in the ferromagnetic and paramagnetic regimes Phys. Rev. B 74 , 075422 (2006) DOI : 10.1103 / PhysRevB.74.075422 ; Gusynin VP et al ., Excitonic gap, phase transition, and quantum Hall effect in graphene Phys. Rev. B 74 , 195429 (2006) DOI : 10.1103 / PhysRevB.74.195429