DiracDirac’’s Equation and its s Equation and its Implications in PhysicsImplications in Physics

RiazuddinRiazuddinCentre for Advanced Mathematics and PhysicsCentre for Advanced Mathematics and Physics

NUST, RawalpindiNUST, Rawalpindiandand

National Centre for PhysicsNational Centre for PhysicsQauidQauid--ii--AzamAzam University, Islamabad, PakistanUniversity, Islamabad, Pakistan

•• There have been two kinds of scientific revolutions, There have been two kinds of scientific revolutions, which F. J. Dyson has classified as those driven bywhich F. J. Dyson has classified as those driven by new new conceptsconcepts and those driven by and those driven by new toolsnew tools. According to . According to him the former are the ones him the former are the ones ““that attract the greatest that attract the greatest impact on public awareness of science while the latter impact on public awareness of science while the latter are not so impressive to the general public but of equal are not so impressive to the general public but of equal importance for the progress of science. The effect of a importance for the progress of science. The effect of a conceptconcept--driven revolution is to explain new things that driven revolution is to explain new things that have to be explainedhave to be explained”” or to discover things predicted by or to discover things predicted by a concepta concept--driven theory so as to verify or discord that driven theory so as to verify or discord that theory. The concepttheory. The concept--driven scientific revolutions have driven scientific revolutions have been rare.been rare.

•• Taking Quantum Mechanics as a prime example of a Taking Quantum Mechanics as a prime example of a conceptconcept--driven revolution, Thomas Kuhn in his book, driven revolution, Thomas Kuhn in his book, ‘‘The structure of Scientific RevolutionThe structure of Scientific Revolution’’, has listed, in , has listed, in addition to quantum mechanics, only six major addition to quantum mechanics, only six major conceptconcept--driven revolutions in the last 500 years, driven revolutions in the last 500 years, associated with the names of Copernicus, Newton, associated with the names of Copernicus, Newton, Darwin, Maxwell, Freud and Einstein. During the same Darwin, Maxwell, Freud and Einstein. During the same period there have been about twenty toolsperiod there have been about twenty tools--driven driven revolutions, mostly in Biology and Astronomy, using revolutions, mostly in Biology and Astronomy, using tools created by Physics.tools created by Physics.

•• The first half of the last century witnessed the creation The first half of the last century witnessed the creation of Relativity and Quantum Mechanics, the two out of of Relativity and Quantum Mechanics, the two out of seven conceptseven concept--driven revolutions mentioned above and driven revolutions mentioned above and these will serve forever as a testimonial to the highest these will serve forever as a testimonial to the highest intellectual achievements of humanity.intellectual achievements of humanity.

•• Dirac combined Special Theory of Relativity with Dirac combined Special Theory of Relativity with Quantum Mechanics.Quantum Mechanics.

•• Dirac Equation:Dirac Equation:•• After the discovery of quantum mechanics in 1926After the discovery of quantum mechanics in 1926--

1927, the big problem was that the Schrodinger 1927, the big problem was that the Schrodinger equation equation

was not consistent with special theory of relativity. Its was not consistent with special theory of relativity. Its natural extension to Kleinnatural extension to Klein--Gordon equationGordon equation

( )ψψ 22 mt

i +∇−=∂∂

0222

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛+∇−

∂∂ φμt

had some difficulties in interpretation at that time. There did had some difficulties in interpretation at that time. There did not not seem to be any other possibility until Paul Dirac made a break seem to be any other possibility until Paul Dirac made a break through. He did it from pure logic and by introducing what are through. He did it from pure logic and by introducing what are called spinor fields in addition to more familiar scalar, vectorcalled spinor fields in addition to more familiar scalar, vector and and tensor fields.tensor fields.

•• Fred Hoyle: Write down 2 equations, just to show how simple Fred Hoyle: Write down 2 equations, just to show how simple science is, provided you know what it means: The first defines ascience is, provided you know what it means: The first defines aspinor field spinor field from a given spinor fieldfrom a given spinor field ::

where where ‘‘ee’’, , ‘‘mm’’ are charge and mass of the electron.are charge and mass of the electron.spinor form of spacespinor form of space--time derivativetime derivativespinor form of electromagnetic field.spinor form of electromagnetic field.

.βη

αβαβαβ ξη ⎟⎠⎞

⎜⎝⎛ +∂=

...

eAim

.βα∂.βαA

αξ

•• Now comes the big statement:Now comes the big statement:

•• Repeat the same operation and you get back to where Repeat the same operation and you get back to where you started. you started.

•• This is the Dirac equation This is the Dirac equation -- the equation which the equation which controls the structure of the world, even down to the controls the structure of the world, even down to the behavior of the electronic gadgets used in our daily behavior of the electronic gadgets used in our daily lives. lives.

•• Dirac equation is also a good example to illustrate a Dirac equation is also a good example to illustrate a difference between a mathematician and a theoretical difference between a mathematician and a theoretical physicist.physicist.

.

..β

βαβαα ηξ ⎟

⎠⎞⎜

⎝⎛ +∂= eAim

•• A certain operation acting on a spinor field led to a A certain operation acting on a spinor field led to a second spinor field:second spinor field:

•• A similar operation to the second fields led to a third, and A similar operation to the second fields led to a third, and so on, adso on, ad--infinite.infinite.

...

00β

αβαβα ηξ meAi =⎟⎠⎞

⎜⎝⎛ +∂

αβ

βαβα

βα

βαβα

αβ

βαβα

ξη

ηξ

ξη

21

11

10

.

..

...

.

..

meAi

meAi

meAi

=⎟⎠⎞⎜

⎝⎛ +∂

=⎟⎠⎞

⎜⎝⎛ +∂

=⎟⎠⎞⎜

⎝⎛ +∂

•• There is nothing much one could do further to this There is nothing much one could do further to this process, as far as mathematics is concerned. But what process, as far as mathematics is concerned. But what Dirac did was indeed something further, namely the Dirac did was indeed something further, namely the odd spinors in sequence are all the same, as are the odd spinors in sequence are all the same, as are the eveneven--numbered spinors, numbered spinors,

•• Mathematically this need not to be true. What Physics Mathematically this need not to be true. What Physics does is to make it true, thereby does is to make it true, thereby ““definingdefining”” the the Universe. Thus the Universe is a set of restrictions on Universe. Thus the Universe is a set of restrictions on mathematical quantities of the kind discovered by mathematical quantities of the kind discovered by Dirac.Dirac.

ξξξηηη

======

........

10

10

It can be written in more familiar form by introducing It can be written in more familiar form by introducing 44--component spinorcomponent spinor ;;

wherewhere•• ““Dirac equation is written in one lineDirac equation is written in one line……! ! •• From the solution of this little equation come details From the solution of this little equation come details

about the Hydrogen atom, the spin of the electron, about the Hydrogen atom, the spin of the electron, and the existence of antiand the existence of anti--matter. Poets bring us fresh matter. Poets bring us fresh insight with right sequence of words, Dirac brought insight with right sequence of words, Dirac brought us fresh insights with right sequence of symbolsus fresh insights with right sequence of symbols””. . John S. John S. TigdenTigden, , Hydrogen: The Essential ElementHydrogen: The Essential Element

( ) 0=+ ψγ μμ mDi

μμμ eAD +∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

β

α

ηξ

ψ&

Vacuum (Nothing):Vacuum (Nothing):

•• Dirac also gave a new concept of vacuum, an infinite sea Dirac also gave a new concept of vacuum, an infinite sea of electrons occupying negative energy states, with far of electrons occupying negative energy states, with far reaching consequences.reaching consequences.

•• Dirac picture of vacuum at Dirac picture of vacuum at A 0, t →

•• Pair creationPair creation:: a quantum of high energy radiation a quantum of high energy radiation disappears in the process, giving rise to electron plus a disappears in the process, giving rise to electron plus a positron. What the radiation quantum does is to lift one positron. What the radiation quantum does is to lift one of the electrons from its negative energy state, into a of the electrons from its negative energy state, into a positive energy state, where it behaves observable as an positive energy state, where it behaves observable as an ordinary electron. And the gap in the infinite sea is ordinary electron. And the gap in the infinite sea is positron. positron.

•• Nothing is something:Nothing is something:In fact, nothing is a lot. On subIn fact, nothing is a lot. On sub--microscopic scales the microscopic scales the vacuum is a foam of particlesvacuum is a foam of particles--antiparticle pairs popping antiparticle pairs popping up in and out of a virtual existence, on energy up in and out of a virtual existence, on energy borrowed from the energy fields.borrowed from the energy fields.

•• Nothing has energy:Nothing has energy:–– We have learned to seek symmetry in the laws of Nature, not We have learned to seek symmetry in the laws of Nature, not

necessarily in the consequences of these laws. If the laws of necessarily in the consequences of these laws. If the laws of nature have a deep and profound symmetry, the symmetry is nature have a deep and profound symmetry, the symmetry is not manifest in the Universe today since the observed not manifest in the Universe today since the observed character of the four forces of nature is very different. character of the four forces of nature is very different.

–– One of the most beautiful ideas of modern Physics is that the One of the most beautiful ideas of modern Physics is that the laws of nature can have symmetries that are hidden from us laws of nature can have symmetries that are hidden from us because the vacuum need not respect all the symmetries.because the vacuum need not respect all the symmetries.

–– The process by which vacuum hides symmetry is known as The process by which vacuum hides symmetry is known as the Higgs Mechanism. There is a Higgs potential energy in the Higgs Mechanism. There is a Higgs potential energy in the vacuum, e.g. electroweak unification involves a Higgs the vacuum, e.g. electroweak unification involves a Higgs potential energy of 246 potential energy of 246 GeVGeV in vacuum.in vacuum.

•• Higgs potential contributes a field independent Higgs potential contributes a field independent constant term,constant term,

•• Contribution of Higgs field to vacuum energy densityContribution of Higgs field to vacuum energy density

•• It plays the role of a vacuum energy density and is It plays the role of a vacuum energy density and is equivalent to adding a cosmological constant term to equivalent to adding a cosmological constant term to EinsteinEinstein’’s equation.s equation.

•• CAN IT BE VARIFIED?CAN IT BE VARIFIED?

8

22 vmHH ≡ρ

4118 )10(104

eVGeVH =≥ρ

( ) GeVGv F 24621≈=

−

Accelerating UniverseAccelerating Universe

•• The discovery that the Universe is accelerating The discovery that the Universe is accelerating its expansion was made in 1998 by two its expansion was made in 1998 by two independent groups.independent groups.

•• I will present this evidence but first a little I will present this evidence but first a little background.background.

Frame work and cosmic parametersFrame work and cosmic parameters•• The most fundamental discovery of modern The most fundamental discovery of modern

cosmology is the expansion of the universe, cosmology is the expansion of the universe, described by the parameter described by the parameter a(ta(t)), known as scale , known as scale factor. It measures how much the Universe stretches factor. It measures how much the Universe stretches as a function of time. It can be thought of as being as a function of time. It can be thought of as being proportional to the average distance between the proportional to the average distance between the galaxies. The expansion rate is function of time galaxies. The expansion rate is function of time

( )( )tataH

&≡

The present value of the Cosmic Parameter,The present value of the Cosmic Parameter,Hubble constantHubble constant

Because the Universe is expanding, photons emitted long ago are Because the Universe is expanding, photons emitted long ago are red shiftedred shifted

Red shift directly indicates the relative linear size of the uniRed shift directly indicates the relative linear size of the universe verse when the photon was emitted.when the photon was emitted.

( ) 11000

190

1100

101

103

04.073.0100

−

−−

×=

×=

±==

yrhH

kmMpc

hMpcskmhH

( )( )

zzH

aa

tataz

ee

+=

==+

1)(

1

0

00

&

λλ

Evidence:Evidence:•• One of the key evidences for accelerating universe comes One of the key evidences for accelerating universe comes

from supernovae of type from supernovae of type IaIa which are generally thought which are generally thought to be thermonuclear explosions of white dwarfs and to be thermonuclear explosions of white dwarfs and serve as an effective standard candle; one can deduce its serve as an effective standard candle; one can deduce its distance from apparent brightness and duration.distance from apparent brightness and duration.

•• Supernova had actually been detected to be systematically Supernova had actually been detected to be systematically 25% fainter than expected i.e. more distant than one 25% fainter than expected i.e. more distant than one would expect for a cosmic whose expansion has recently would expect for a cosmic whose expansion has recently been showing down, or even coasting.been showing down, or even coasting.

•• They thus provide the first direct evidence of the They thus provide the first direct evidence of the transition from the earlier decelerating phase to the transition from the earlier decelerating phase to the present accelerating expansion.present accelerating expansion.

0≈Ω K

1ayenergy tod fractional

=Ω+Ω+Ω=Ω

Λ KM

supernovae

CMB

galaxies

3 independent 3 independent

data sets intersectdata sets intersect

the improbable, mysterious universethe improbable, mysterious universe

?

??

FriedmanFriedman’’s s expanding universeexpanding universe

each time instant = each time instant = 3D space 3D space of constant curvatureof constant curvature

Friedman equation:Friedman equation:

expansion rate = matter/radiation + dark energy + curvatureexpansion rate = matter/radiation + dark energy + curvature

22

22

3)(

38

aKG

aaH rm −

Λ++=≡ ρρπ&

a(ta(t))

Critical Density:

Then the first Friedman equation

Equation of state:Equation of state:

The second Freidman equation gives the deceleration The second Freidman equation gives the deceleration parameterparameter

2

20

20

2)(HH

aHa

aHazq

&&&&−=−=

( ) ( ) ⎥⎦

⎤⎢⎣

⎡+Ω+⎟⎟

⎠

⎞⎜⎜⎝

⎛= +∑ )1(33

20 131

21

)(iz

zHH

iii

ωω

( ) ⎥⎦⎤

⎢⎣⎡ Ω−+Ω⎟⎟

⎠

⎞⎜⎜⎝

⎛= Λ

32

0 121

)(z

zHH

m

Since and is negligible, and It becomes cleSince and is negligible, and It becomes clear ar that acts as that acts as ““repulsiverepulsive”” (anti) gravity, tending to (anti) gravity, tending to speed up the expansionspeed up the expansion

We also note that matter density scales with scale factor asWe also note that matter density scales with scale factor as

Λ

3amm

∝=ΩΩ ΛΛ

ρρ

0=mω 1−=ΛωrΩ

•• Thus at early times we have that the Vacuum Thus at early times we have that the Vacuum Energy much more suppressed compared to that of Energy much more suppressed compared to that of matter or radiation while it dominates at late times. matter or radiation while it dominates at late times. Hence one could expect a transition from Hence one could expect a transition from deceleration to acceleration at some z. We have deceleration to acceleration at some z. We have already seen that there is evidence for this in the already seen that there is evidence for this in the present data. present data.

The deceleration parameter

28.0=Ωm 72.0=ΩΛThe present data, ,

Leads to i.e. the accelerating universe.,00 <q

•• CDM part explains structure formation CDM part explains structure formation and and ΛΛ explains the late time explains the late time acceleration of the Universe:acceleration of the Universe:

•• itit’’s the simplest models the simplest model•• compatible with all data up to nowcompatible with all data up to now•• no other model gives a better statistical fitno other model gives a better statistical fit•• but but ……. theory cannot explain it . theory cannot explain it

•• why so small? and why so small? and …… why so finewhy so fine--tuned?tuned?

( ) ( )obs

44susy

4lfundamentatheory

obs

411

43obs

TeV) 1(~~

10

eV) 10(~8

ΛΛ

ΛΛ

−Λ

>>≥

>>>

Λ=

ρρ

ρρπ

ρ

MM

eVG

Higgs

LCDM fits the data wellLCDM fits the data well…… but we cannot explain itbut we cannot explain it

300

whilebut

formation structurefor crucial:~−

Λ

Λ

∝∝ aa mρρ

ρρ

•• What we make of this mismatch? The fact thatWhat we make of this mismatch? The fact that

means that the smallness of the cosmological constant means that the smallness of the cosmological constant needs to be explained. In a unified theory of strong, needs to be explained. In a unified theory of strong, weak, electromagnetic interaction, other (heavy!) Higgs weak, electromagnetic interaction, other (heavy!) Higgs fields have nonfields have non--zero vacuum expectation values that zero vacuum expectation values that may give rise to still greater mismatches.may give rise to still greater mismatches.

•• At a fundamental level, we can therefore conclude that a At a fundamental level, we can therefore conclude that a spontaneously broken gauge theory of the strong, weak, spontaneously broken gauge theory of the strong, weak, and electromagnetic interactionsand electromagnetic interactions--or merely of the or merely of the electroweak interactionselectroweak interactions--cannot be complete.cannot be complete.

vacH ρρ >>

•• Either we must find a separate principle to zero the Either we must find a separate principle to zero the vacuum energy density of Higgs field, or we may vacuum energy density of Higgs field, or we may suppose that a proper quantum theory of gravity, in suppose that a proper quantum theory of gravity, in combination with the other interactions, will resolve the combination with the other interactions, will resolve the puzzle of the cosmological constant. puzzle of the cosmological constant.

•• Dirac Equation has recently found an Dirac Equation has recently found an interesting application in a system interesting application in a system which is essentially non relativistic.which is essentially non relativistic.

GRAPHENEGRAPHENEGrapheneGraphene consists of a consists of a honecombhonecomb lattice of Carbon lattice of Carbon

atoms. Graphite stack of graphene layer.atoms. Graphite stack of graphene layer.

GrapheneGraphene is a unique system in many ways:is a unique system in many ways:A):A): It is a perfect mixture of a semiIt is a perfect mixture of a semi--conductor (zero density of states) and a conductor (zero density of states) and a metal (it is gapless) and has properties of metal (it is gapless) and has properties of soft material.soft material.Many of Many of graphenegraphene’’ss properties are currently properties are currently subject of intensive research and graphene subject of intensive research and graphene can be easily chemically and/or structurally can be easily chemically and/or structurally modified in order to change its functionality modified in order to change its functionality and henceforth its potential applications.and henceforth its potential applications.It can be relatively easily obtained from It can be relatively easily obtained from graphite, abundantly available on the graphite, abundantly available on the earthearth’’s surface.s surface.

B):B): Its low energy excitations are Its low energy excitations are masslessmasslesschiralchiral Dirac Fermions as we shall see:Dirac Fermions as we shall see:It mimics physics of QED for It mimics physics of QED for masslessmasslessfermions except that in graphene the Dirac fermions except that in graphene the Dirac Fermions move with the speed ~1/300 the Fermions move with the speed ~1/300 the velocity of light which implies many of velocity of light which implies many of unusual properties of QED can show up in unusual properties of QED can show up in graphene but at much smaller speeds.graphene but at much smaller speeds.

The electronic properties of graphene can be described by a The electronic properties of graphene can be described by a tight binding model with only one orbital per atom. Within tight binding model with only one orbital per atom. Within this approximation a basis set is provided by the Bloch this approximation a basis set is provided by the Bloch functions made of 2pfunctions made of 2pz z orbitalsorbitals from the 2 from the 2 inequivalentinequivalent carbon carbon atoms A and B which form the unit cell of the graphene atoms A and B which form the unit cell of the graphene hexagonal lattice.hexagonal lattice.

a): Tight Binding Modela): Tight Binding ModelGraphene is made out of carbon atoms arranged in Graphene is made out of carbon atoms arranged in a hexagonal structure shown above.a hexagonal structure shown above.For a hexagonal layer the unit cell contains two For a hexagonal layer the unit cell contains two atoms A and B, belonging to the 2 sub lattices. The atoms A and B, belonging to the 2 sub lattices. The lattice vectors arelattice vectors are

is the carbonis the carbon--carbon distance. The carbon distance. The reciprocal lattice vectors are given by reciprocal lattice vectors are given by

The first The first BrillouinBrillouin zone is a hexagon. Of particular zone is a hexagon. Of particular importance for the physics of graphene are 2 pointsimportance for the physics of graphene are 2 points

)3,3(2

),3,3(2 21 −==

aaaa rr

042.1 Aa =

)3,1(32),3,1(

32

21 −==a

ba

b ππ rr

and at the corners of the and at the corners of the graphenegraphene’’ssBrilliouinBrilliouin zone.zone.

and are named Dirac points. Their and are named Dirac points. Their positions in momentum space are positions in momentum space are

K K′

K K′

)3

1,1(32),

31,1(

32

−=′=a

Ka

K ππ rr

The Dirac cones sit at the , points.The Dirac cones sit at the , points.For an atom A of a For an atom A of a sublatticesublattice, the 3 nearest , the 3 nearest neighbors B which belong to the other neighbors B which belong to the other sublatticesublatticehave vectorshave vectors

The nearest tight binding approach reduces the The nearest tight binding approach reduces the problem to the problem to the diagonalizationdiagonalization of the Hamiltonianof the Hamiltonian

where the sum is over pairs of nearest neighbor where the sum is over pairs of nearest neighbor atoms, i, j on the lattice. annihilates (creates) atoms, i, j on the lattice. annihilates (creates) an electron on site on an electron on site on sublatticesublattice A (an A (an equivalent definition is used for equivalent definition is used for sublatticesublattice B). B).

is the nearest hopping energy (hopping is the nearest hopping energy (hopping between different between different sublatticessublattices).).

K′K

)0,1(

)3,1(2

),3,1(2

3

21

a

aa

−=

−==

δ

δδr

rr

iRr

)8.2( eVt ≈

H −t∑i,jaibj h.c.

aiai

The The eigeneigen functions and functions and eigeneigen vectors of the vectors of the Hamiltonian are obtained from the equationHamiltonian are obtained from the equation..

(A)(A)

where is a set of triad of vectors connecting an where is a set of triad of vectors connecting an atom A with its B nearest neighbors, in our case , atom A with its B nearest neighbors, in our case ,

, and the triad of their respective , and the triad of their respective opposites. is the opposites. is the 2p2pz z energy level, taken as the energy level, taken as the origin of the energy.origin of the energy.

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−

∑∑

B

A

B

A

i

vkii

uki

CC

ECC

et

et

i

i

ε

εrr

rr

.

.

iur

ivr1δr

2δr

3δr

ε

The The eigenvalueseigenvalues give the energy bands: give the energy bands: [Wallace, 1947][Wallace, 1947]

in which two signs define 2 energy bands:in which two signs define 2 energy bands:

)(3

)23cos()

23cos(4)3cos(3)(

2/1

kft

akakaktkE yxy

+±=

⎭⎬⎫

⎩⎨⎧

++±=±

the +the +veve sign applies to the upper ( ) band and sign applies to the upper ( ) band and ––vevesign to the lower ( ) band.sign to the lower ( ) band.Band structure of graphene:Band structure of graphene:

Instructive to calculate Instructive to calculate f(kf(k) ) close to one of the Dirac close to one of the Dirac pointspoints

is the momentum measured relative to the is the momentum measured relative to the Dirac pointsDirac points

π*π

qKk rrr+=

qr

Thus the energy band around the Dirac Thus the energy band around the Dirac point ispoint is

where represents the Fermi velocity.where represents the Fermi velocity.

It is seen that the dispersion is conical.It is seen that the dispersion is conical.

)()()( 2 BqOqvqE frr

+±≈±

23tavf =

smvf /101 6×=

The energy dispersionThe energy dispersion

resembles the energy of ultraresembles the energy of ultra--relativistic particles, relativistic particles, (although is 300 times smaller than the speed (although is 300 times smaller than the speed of light); these particles are quantum mechanically of light); these particles are quantum mechanically described by the described by the masslessmassless Dirac Equation.Dirac Equation.The most interesting aspects of the graphene is that The most interesting aspects of the graphene is that Low energy excitations are Low energy excitations are masslessmassless, , chiralchiral Dirac Dirac Fermions.Fermions.Cyclotron mass:Cyclotron mass:The cyclotron mass is defined, within the classical The cyclotron mass is defined, within the classical approximation asapproximation as

is area in kis area in k--space enclosed by the orbit.space enclosed by the orbit.

qvqE fr

±≈± )(

fv

EEAm

∂∂

=)(

21*

π

)(EA

Using the Energy dispersionUsing the Energy dispersion

Contrast, Schrodinger dispersion would imply Contrast, Schrodinger dispersion would imply a constant cyclotron mass.a constant cyclotron mass.The electronic density, The electronic density, nn, related to the Fermi , related to the Fermi momentum ismomentum is

2)( qEA π=

qvqE fr

±≈± )(

f

f

f

vqm

vEEA

qvE

=

=

=

*

2

2

222

)( π

mqE2

2

=

nv

m

nk

f

F

ππ

=⇒

=

*

2

Fitting the above equation provides an estimation to Fitting the above equation provides an estimation to the Fermi velocity [ ] and hopping parameter the Fermi velocity [ ] and hopping parameter ttrespectively.respectively.

Cyclotron mass of charge Cyclotron mass of charge carriers as a function of their concentration carriers as a function of their concentration nn..

Nevertheless the experimental observation of Nevertheless the experimental observation of dependence of the cyclotron mass provides evidence dependence of the cyclotron mass provides evidence for the existence of for the existence of masslessmassless Dirac quasiDirac quasi--particles in particles in graphene.graphene.

atv f 3=

cv

eVtmsv

F

F

3001

310 16

≈

=≈ −

Dirac EquationDirac Equation

Let us expand the HamiltonianLet us expand the Hamiltonian

at any of the two independent lattice (Fermi) points at any of the two independent lattice (Fermi) points given by the lattice vectorsgiven by the lattice vectors

i.e. Expandi.e. Expand

wherewhere

around the Dirac pointaround the Dirac point

a1 a2 3, 3 , a2 a

2 3,− 3

H −t∑i,jaibj h.c.

H 0 HAB

HAB∗ 0

HAB −teik .a 1 ieik .a 2

k K ′ q

K′ 23a 1,− 3

This givesThis gives

H 0 iqx qy

−iqx qy 0

v f − q 1 2 q 2 1

H v fi31q1 i32q2

0 . qv f

are Dirac matrices.are Dirac matrices.The above is the Dirac Hamiltonian for The above is the Dirac Hamiltonian for masslessmasslessfermionfermion in (2+1) dimension near the Dirac point . in (2+1) dimension near the Dirac point . Corresponding Dirac Equation isCorresponding Dirac Equation is

(II)(II)

oror

oror

0 3 , 1 i 1 ,2 i 2

K′

H E E v f |q|

i ∂∂t 0−i∇v f

i0∂0 1∂1 2∂2 0

i ∂ 0

This is the Dirac Eq. in (2+1) dimension for This is the Dirac Eq. in (2+1) dimension for masslessmassless fermionfermion. The energy relation (II) resembles . The energy relation (II) resembles the energy of ultrathe energy of ultra--relativistic particles. These relativistic particles. These particles are described by the particles are described by the masslessmassless Dirac Dirac Equation in (2+1) dimension.Equation in (2+1) dimension.If we expand around the Dirac point If we expand around the Dirac point we would obtainwe would obtain

so thatso that

Now it is known that in 3 spaceNow it is known that in 3 space--time dimensions, time dimensions, there exist 2 there exist 2 inequivalentinequivalent representations for the representations for the γγ--matrices.matrices.

K 23a , 2

3 3 a

HAB 3at2 iqx − qy

H v fi 3 T 1 Tq1 i 3 T 2 Tq2

We have used the first representation for the We have used the first representation for the expansion around the Dirac point . We take the expansion around the Dirac point . We take the second representation for the expansion around the second representation for the expansion around the Dirac point , which is obtained from by the Dirac point , which is obtained from by the parity operation which is defined asparity operation which is defined as

So thatSo thatThus we see the parity operatorThus we see the parity operator

0 3 , 1 i1 ,2 i2

K′

K′K

det −1

q1 q1 , q2 −q2 and HK HK′

0 3 3 T, 1 i1 i1 T,2 −i2 −i2 T

1 0 0

0 1 0

0 0 −1

Application:Application:MasslessMassless Dirac Equation in 2+1 dimensions: Dirac Equation in 2+1 dimensions:

TunnelingTunneling

0)()( 22

11

0

0

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂+∂+−

∂∂ xiiVt

iv f

ψγγγ

The The transmisiontransmision through the barrier isthrough the barrier is

Notice:Notice:1): For , 1): For , nn integer, integer, and , independent of the value of and and , independent of the value of and barrier becomes completely transparent.barrier becomes completely transparent.2): For normal incidence , 2): For normal incidence , for any value of .for any value of .This result is the manifestation of Klein paradox This result is the manifestation of Klein paradox and does not occur for nonand does not occur for non--relativistic particles. In relativistic particles. In this latter case for normal incidence, T is always this latter case for normal incidence, T is always smaller than one.smaller than one.

( )[ ] ( )

222

22

sinsin'1sin)cos(coscos

coscos

⎥⎦⎤

⎢⎣⎡ −+

=

θφφθ

φθφ

ssDqDq

T

xx

Dqx n T T−T 1

→ 0, → 0 T0 1Dqx

3): In the limit3): In the limit |V 0 | |E |, → 0

T → co s2

co s 2 D q x co s 2 sin 2 D q x

c o s 2

1 − c o s 2 D q x s in 2

Electrons moving through a barrier separating pElectrons moving through a barrier separating p--and nand n-- doped graphene, a doped graphene, a pp--nn junction are junction are transmitted as holes. The relation between the transmitted as holes. The relation between the velocity and the momentum for a hole is the velocity and the momentum for a hole is the inverse of that for an electron. This implies, that if inverse of that for an electron. This implies, that if the momentum parallel to the barrier is conserved, the momentum parallel to the barrier is conserved, the velocity of the quasi particle is inverted. When the velocity of the quasi particle is inverted. When the incident electrons emerge from source, the the incident electrons emerge from source, the transmitted holes are focused into an image of the transmitted holes are focused into an image of the source. This behavior is the same as that of source. This behavior is the same as that of photons moving in a medium with negative photons moving in a medium with negative reflection index, and it can be used in filtering reflection index, and it can be used in filtering electron beams. Similar effects occur in graphene electron beams. Similar effects occur in graphene quantum dots, where the inner and outer regions quantum dots, where the inner and outer regions contain electrons and holes separately.contain electrons and holes separately.

Dirac Fermions in a magnetic fieldDirac Fermions in a magnetic field

In a static magnetic field In a static magnetic field one can show thatone can show that

where we have usedwhere we have used

In this way, we may define canonical variables In this way, we may define canonical variables

i∂ 0∂ → D ∂ ie

c AiD 0

b

s

(Ao 0, Ai 0)

−D12 − D2

2 − ec Bb 2

v f2 b

Q, PD1 ,D2 − ie

c B

− D2

eB/c Q

iD1 P

Harmonic oscillatorHarmonic oscillator

(the cyclotron frequency)(the cyclotron frequency)new length scale.new length scale.

The above are Landau levels near the Dirac points .The above are Landau levels near the Dirac points .The Landau levels at the opposite Dirac points have exactly The Landau levels at the opposite Dirac points have exactly the same spectrum and hence each Landau level is doubly the same spectrum and hence each Landau level is doubly degenerate. This particular Landau spectrum has been degenerate. This particular Landau spectrum has been obtained by many different experimental probes.obtained by many different experimental probes.

E N 12

eBc

N c N

c

Q, P i

12 P2 1

2 2Q2 b 1

2E2

vf2 eB

c b

Eb

eBc

c 2 v f eB/c 2 v f/lB , lB ceB

K ′

SUMMARYSUMMARY

•• Two sets of equations distinctly standout:Two sets of equations distinctly standout:•• MaxwellMaxwell’’s Equationss Equations, displacement current, displacement current•• No experiment at that time needed it. Maxwell No experiment at that time needed it. Maxwell

introduced it purely from symmetry considerations, to introduced it purely from symmetry considerations, to remedy the lack of symmetry between Faraday and remedy the lack of symmetry between Faraday and AmpereAmpere’’s Law.s Law.

•• Far reaching consequences:Far reaching consequences:•• MarrigeMarrige of Maxwell equations with Quantum of Maxwell equations with Quantum

Mechanics brought the communication revolution.Mechanics brought the communication revolution.•• Domesticated the High Technology.Domesticated the High Technology.

•• DiracDirac’’s Equations Equation: Derived purely from logic.: Derived purely from logic.•• One could anticipate spin and hyperfine One could anticipate spin and hyperfine

structure but not prediction of anti matter.structure but not prediction of anti matter.•• ElectronElectron--positron tomography, positron tomography, •• New concept of vacuum: Particles can float in New concept of vacuum: Particles can float in

and out of existence in a kind of and out of existence in a kind of ““foamfoam”” even in even in empty space.empty space.

•• Far reaching consequences: Vacuum Far reaching consequences: Vacuum polarization, namely quantum fluctuations of the polarization, namely quantum fluctuations of the vacuum have been observed in atoms. vacuum have been observed in atoms.

•• Due to similar fluctuations, a tiny universe could have Due to similar fluctuations, a tiny universe could have appeared seemingly out of nothing. A furious appeared seemingly out of nothing. A furious expansion (inflation) takes from a size of a proton to expansion (inflation) takes from a size of a proton to the size of grapefruit in a tiny fraction of second. The the size of grapefruit in a tiny fraction of second. The energy of inflation becomes a hot fire ball that energy of inflation becomes a hot fire ball that continues growing into a dense soup of matter. The fire continues growing into a dense soup of matter. The fire ball eventually cools and expands into our universe, ball eventually cools and expands into our universe, during which todayduring which today’’s galaxies were formed: s galaxies were formed:

Thus we treat the universe as a Thus we treat the universe as a quantum fluctuations of the vacuumquantum fluctuations of the vacuum--state of nothingness; in other words state of nothingness; in other words nothing is everything.nothing is everything.

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