1

2

Many Particle Plasma Hamiltonian

I. Kinetic & Particle-Particle Coulomb Interaction

II. Canonical Transformation

3

Interpretation of Canonically Transformed H

Separates Long-Range Part of

Coulomb Interaction{ }

4

Effective Potential V(1) Due to Impressed Potential U(2)

Screening Function K(1,2)

Carrier-Carrier Coulomb Potential ; ρ = perturbed density )(

5

Integral Equation & Screening Function

Density Perturbation Response Function:

6

Solution For Dielectric Function

Polarizability:

Inverse Relation of Screening Fn.& Dielectric Fn.

Uniform Plasma: K(p, ω) ε(p, ω) = 1,

Uniform Plasma:

7

Schrődinger Quantum Dynamics (ħ→1)

Retarded Green’s Fn:

Eigenfn. Expansion:

Uniform Plasma:

Spectral Weight Fn:

8

Statistical Thermodynamic (Equilibrium) Green’s Fn.

= Fermi-Dirac Distribution Fn.

where is chemical potential

and is thermal energy

9

Nonequilibrium Green’s Fn.

Differential Form:

Integral Eq. Form:

“Ring Diag.” Approx. (1st iteration)

Perturbed density:

RANDOM PHASE APPROX. (RPA)

8

10

Quantum Effects in Uniform 3D Solid State Plasma

1. Quantum Statistical: Fermi Fn. at Arbitrary Temp:

2. Quantum Dynamical: denominator term on right hand side. Without this term the result is that of a classical collisionless linearized Vlasov-Boltzmann equation coupled to a Poisson equa., but with initial Fermi averaging.

3. Local Plasmon Disp. Rel:

as in gas plasma: Correspondence Principle

4. Plasmon Disp. Rel. at higher - p in Maxwell statistical limit, with , is exactly that of nonlocal gas plasmas:

5. ALL PHYSICAL FEATURES OF THE PLASMA HAMILTONIAN ARE EMBEDDED IN STRUCTURE OF THE SCREENING FN.

:),(),( 1 ωεω ppK rr −=0),(Re. =→ ωε pPlasmonsi r

0),(Im. ≠→ ωε pDampingii r

)0,()0,(. 1 ppKShieldingiii rr −=→ ε

1][0 ]1[)( −− += βζωω ef

0→h

3D Plasma-Degenerate Quantum Limit (T=0)

11

1. Lindhard Dielectric Fn. (pF is the Fermi wavenumber)

(a) Local plasmon ωp is shifted by wavenumber corrections bearing quantum effects

(b) “Zero Sound” plasma mode (quantum)

12

2. Plasmon Damping at Arbitrary Temperature Determined by :

(a) Nondegenerate Classical Limit Yields Landau Damping(b) Degenerate T=0 Limit Yields Strong Plasmon Decay into Electron-

Hole Pairs by Exciting an Electron out of the Fermi Sea, Creating a Hole-SUBJECT TO CONSERVATION OF ENERGY & MOMENTUM

3. Static Shielding (K(p,0)=ε-1 (p,0) ):(a) Low Wavenumber (p << 2pF)→Debye, Thomas, Fermi:V(r)~exp(-pDTF r)/r(b) Branch Point & Cut at (p~2pF)→Long Range Friedel Oscillatory “Wiggle”

Shielded Potential: V(r)~cos(2pFr)/r3

Low Dimensional Solid State Plasmas:Quantum Confinement

•Heterojunction Interfaces (planar) joining differing energy band edges (eg: Si-SiO2) are subject to band bending that can create a “dip” in the potential profile below the Fermi energy near the interface, trapping conduction electrons in that region. Such electrons are confined near the interface, but can move freely along the planar interface, constituting a 2D plasma in an “inversion layer”.•“Quantum Wells” similarly involve a valley in a planar heterostructure potential profile (eg: GaAs-AlGaAs) giving rise to size quantization leaving only the lowest energy levels for motion across the valley energetically accessible, effectively confining the electrons to the well, where they move freely as a 2D plasma in the plane of the parallel well walls.

•Quantum Wires

•Quantum Dots13

14

2D Degenerate Q-Well/Inversion LayerSolid State Plasmas I

]2[ 2 densityDisDρ

15

2D Degenerate Inversion Layer/Q-Well Solid State Plasmas II

A. 2D Plasmons:

Low Wavenumber

Quantum Effects in corrections

B. Strong Plasmon Damping from when energy & momentum conservation permit plasmon decay into 2D electron-hole pairs.

C. 2D Static Shielding:Debye-Thomas-Fermi Wavenumber

Friedel Osc.-Wiggle Shielding

{

),(Im ωε pr

{

0),( =ωε pr

→

Fppfrom 2−

16

1D Degenerate Quantum Wire Plasmas ( )1→h

1D-Fourier transform of Coulomb/potential is divergent!

, where is Q-wire radius

Plasmon:

Higher order wavenumber terms bear quantum corrections.

a

]1[ 1 densityDisDρ

Bulk Quantum Magneto-PlasmaExact RPA Dielectric Fn.

where

,

and the 3D equilibrium density, is given by

Two Local Magnetoplasmons :

,

,0ρ

.

17

Low Wavenumber Magnetoplasmon Dispersion Relation

Intermediate Mag. Fields, →DHVA OSC.→

where

and

18

)/2cos( cn ωζπ h

Degenerate Quantum Strong Magnetic Field LimitOnly Lowest Landau Eigenstate Populated: (no spin)

Magneto-Plasma SummaryPlasmons: (a) 2 Local Magnetoplasmons, (as above)

(b) 2 quantum magnetoplasmon resonances near each branch point of

one damped, other undampedand becomes quantum-type “Bernstein” mode as .

Static Shielding:

(a) Low Wavenumber Debye-Thomas-Fermi Shielding becomes anisotropic when wavenumber corrections are included.

(b) Long Range Friedel Oscillatory “Wiggle” from is strongly anisotropic and is destroyed for directions not exactly parallel to B.

(c) Electron confinement to lowest Landau eigenstate prevents DHVA-oscillations.

Fz pp 2log −

0→zp

2/~ cFc E ωζω hh =>

19

20

QUANTUM PLASMA PHENOMENOLOGY IN GRAPHENE

A DEVICE FRIENDLY MATERIAL• High Mobility at Room Temperature• High Electron Density: 1013cm-2 in single subband• Long Mean Free Path at Room Temperature-

Ballistic Transport• Temperature Stability of Graphene• Quantum Hall Effect at Room Temperature• Convenience of Planar Form (NOT Tube)• Sensors-Detection of Single Adsorbed Molecule• Actuator for Electromechanical Resonator• Spin Valve• Graphene FET

21

Introduction: Structure

• Graphene is composed of a single 2D layer of carbon atoms

( )aa 2/1,2/31 −=r

( )aa 1,02 =r

The honeycomb lattice as a superposition of two triangular lattices. The basic vectors are and , and the sublattice is connected by ( )ab 2/1,32/11 =

r

( )ab 2/1,32/12 −=r ( )ab 0,3/13 −=

r

22

Introduction: Structure; Massless Dirac Spectrum• In Graphene, low energy electrons behave as

massless relativistic fermions due to their linear energy spectrum around two nodal, zero-gap points (K and K') in the Brillouinzone.

[γ=3αa/2, α is the hopping parameter in tight-binding approx.; a is the lattice spacing; note that γ is also the constant Fermi velocity (i.e. independent of carrier density)]

k

εk Εk=γkConduction Band

Valence Band

23

• In experiments involving Graphene, many intriguing transport phenomena have been observed The existence of a “residual”

conductivity at zero gate voltage

Conductivity varies almost linearly with the electron density; High mobility at room temperature

Science 306, 666 (2004); Nature 438, 197 (2005);Nature 438, 201 (2005).

he /4 2min ≈σ

Quantum Hall effect at Room TemperatureScience, 315, 1379 (2007).

Introduction: Unusual Properties

24

Kinetic Equation: Formulation-Hamiltonian I

• HAMILTONIANHamiltonian of free electrons with 2D

momentum p near s=K or s=K' (Dirac points) in pseudo-spin basis ( represents the Paulispin matrices; is the Fermi velocity)

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=⋅=

0)sgn()sgn(0)(

0yx

yx

ipspipsp

phs

γσγ rr(

⎩⎨⎧

′=−=

=Ks

Kss

11

)sgn(

σr

γ

25

• In our study two basis sets are used: the pseudo-spinbasis and the pseudo-helicity basis

• Pseudo-helicity is the component of pseudo-spin on the momentum direction.

• The pseudo-helicity basis is the basis of Hamiltonian eigenstates (diagonal)

• The pseudo-spin of carriers in graphene near Diracpoints is parallel or antiparallel to momentum. Correspondingly, the pseudo-helicity of carriers, equal to 1 or -1, is characterized as a left-handed or right-handed state in the pseudo-helicity basis. This feature enables us to describe the low-energy electrons in graphene as massless relativistic fermions.

Kinetic Equation: Formulation-Hamiltonian II

26

Kinetic Equation: Formulation-Hamiltonian III

Introducing a unitary transformation to go from a pseudo-spin basis to a pseudo-helicity basis

0h(

[ ] )](),([diagˆ21

)()(0

)()(0 ppUhUh ssss εε==

+

pp

(

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

ppipspipsp

pU yxyxs

γγγ)sgn()sgn(1)(

p

can be diagonalized as [ ]pγε μμ

1)1( +−=

The carriers experience scattering by impurities. In the pseudo-helicity basis, the corresponding potential takes the form

pp kpkp UVUT |)(|),( −= +

27

Conclusions I

• We have investigated transport in graphene in the diffusive regime using a kinetic equation approach. The contribution from electron-hole interbandpolarization to conductivity was included (it was ignored in all previous studies).

• We found that the conductivity of electrons in graphene contains two terms: one of which is inversely proportional to impurity density, while the other one varies linearly with the impurity density.

28

• Our numerical calculation for the RPA screened Coulomb scattering potential and our analytical results for a short-range scattering potential indicate that the MINIMUM (rather than “residual”) conductivity in the diffusive regime is in the range 4-5e2/h. We also obtained linear dependence of the conductivity on electron density for higher Ne/Ni values.

• Ref: S.Y. Liu & N.J.M. Horing, J. Appl. Phys., in press.

Conclusions II

29

Graphene Dielectric Response Dynamic, Nonlocal RPA Polarization

Hamiltonian: ;Green’s Fns.:

A. RETARDED (pseudospin rep.; )

B. THERMODYNAMIC

ph s rr(⋅= σγ0

)(),()( 0 ttItpGhtIi s ′−=−∂∂ δ

trt(t

++→ οωω i

;)()(),(),(

;)(),(),(222*

222

pipppGpG

ppGpG

yxRyx

Rxy

Ryy

Rxx

γωγωω

γωωωω

−−==

−==rr

rr

)],(Im[2),( ωω pGpA rtrt−=Spectral weight

{ } { } ),(),( )()(1

0

0ωω ω

ω pAipG ff

rtrt+−=

<>

30

Dielectric Screening Function (on the 2D sheet):

Polarizability:

, where is

[ ] 11 ),(1),(),( −++−+ ++=+=+ οωαοωεοω ipipipK

),(2),(2

++ +−=+ οωποωα ipRpeip

effVR δδρ=

)];();(.[)2(

)];();(.[)2(

,),(

2

2)(

0

2

2)(

0

*

tpqGtqGTrqdedt

tpqGtqGTrqdedt

ipR

tii

tii

−−=ℑ

−−=ℑ

ℑ−ℑ=+

>>+−

∞−<

><−−

∞

>

><+

∫∫

∫∫+

+

tt

tt

π

π

οω

οω

οω

31

Graphene Polarizability I: Degenerate Limit (T=0°K)

Density Perturbation Response Fn., “Ring” Diagram R:

where and

( are pseudo-spin andvalley degeneracies; is the Heaviside Fn.)

πγνδδρν nggDxRDVxR vs

eff1);,(~),( 00 ===

),,(~),(~),(~ ννν xRxRxR −+ +=

)(zθ

),(),(~)(),(~),(~21 νθννθνν −+−= +++ xxRxxRxR

vsFF gppxE ,;; == ων

Wunch, et al., New Journ. of Phys. 8, 318 (2006)

Hwang-Das Sarma, Phys. Rev. B75, 205418 (2007)

Kenneth W.-K.Shung, Phys. Rev. B34, 979 (1986)

Next

32

where the real parts are (notation: )

and the imaginary parts are

Graphene Polarizability II

22

2

22

2

8)(

8)(),(~

xxxi

xxxxR

−

−+

−

−=− −

ννθπ

ννθπνand Next

Π~~ −≡R

33

Graphene Polarizability III: Definitions

34

Static Screening Dielectric Function

• RPA screened model (Notation: )

[Kenneth W.-K.Shung, Phys. Rev. B34, 979 (1986); T. Ando, J. Phys. Soc. Japan 75, 074716 (2006); B. Wunsch et al. New J. Phys. 8, 318(2006); E.H. Hwang et al.PRL 98, 186806 (2007) ]

is the Thomas-Fermi screening wave vector ( is the static background dielectric constant)

FF kpqp ↔↔ ;

35

• RPA screened Coulomb scattering model

Conductivity Results and Discussion I

qqeqV

)(2)(

0

2

κεε=

• Zero Temperature Conductivity (rs = e2/(4π ))γ

where •Conductivity minimum for finite , not “residual”.

eN

where G(x) is

and F(x) is given by

Conductivity Results and Discussion II

36

37

Graphene Plasmon & E.M. Modes

Low-Wavenumber Plasmon: 0);( =→ ωοε p

γκ

κω

ωω

ν

ν

Fso

Fs

Fo

peggq

nEegg

ppqp

2

4/12/120

22/10

)2(

]81[

=

→=

=⇒

where

(not as in normal 2D plasma)

and = (GrapheneThomas-Fermi wavevector)

2/1n

_

38

New Graphene Transverse Electric Mode in TeraHertz Range

[Mikhailov & Ziegler, Phys. Rev. Lett. 99, 016803 (2007)]

New Graphene TE Mode

TM Mode: 2D Graphene plasmon-polariton

2667.1 << FEωh

667.1<FEωh

THzfTHz 1815 ≤≤

39

Graphene Quasiparticle Self-Energy,A. Screened Coulomb e-e Self-Energy, :

(Das Sarma, Hwang & Tse, Phys. Rev. B75, 121406(R)(2007))

Conclusions: Intrinsic Graphene is a marginal Fermi liquid (quasiparticle

spectral weight vanishes near Dirac point)Extrinsic Graphene is a well-defined Fermi liquid

(Doping induces Fermi liquid behavior)

Σ

)2,1()21()2,1( GiVeffCoul

−−⇒∑

);13()32(3)21( −−=− ∫ Couleff UKdV

where the screened potential is

∑Coul

40

B. Phonon Induced Self Energy[Tse & Das Sarma, Phys. Rev. Lett. 99, 236802(2007)]

measures electron-ion interaction strength; is the free phonon Green’s function, and

),();()()2,1(~2211 xttDxV epepeffvv o ΓΓ=

)( 1xepvΓ

),( 21 ttDo

)2,1()2,1(~)2,1( GVi effphonon

−⇒∑

Conclusions: Phonon mediated e-e coupling has a large effect on Graphene band structure renormalization

Graphene Self-Energy (continued)

41

Graphene Quantum Dots & Superlattice

A. Graphene Q-Dots [Matulis & Peeters, arXiv0711.446v1[cond-mat.mes-hall]28 Nov. 2007]Dirac fermions in a cylindrical quantum dot potential are not fully confined, but form quasi-bound states. Their line-broadening decreases with orbital momentum. It decreases dramatically for energies close to barrier height due to total internal reflection of electron wave at the dot edge

42

B. Electronic Superlattices in Corrugated Graphene [Isacsson, et al., arXiv:0709.2614v1[cond-mat.mes-hall]17Sep.(2007)]

Theory of electron transport in corrugated Grapheneribbons, with ribbon curvature inducing an electronic superlattice having its period set by the corrugation wavelength. Electron current depends on SL band structure, and for ribbon widths with transverse level separation comparable to band edge energy, strong current switching occurs.

43

Graphene Energy Loss SpectroscopyNJM Horing, V. FessatidisPower loss =

Parallel: Stopping Power---High Velocity

Perpendicular: Total Work ( )

⎢⎣

⎡∇−= +⋅−+⋅−⋅∫ ∫ ∫ )()(

2

2

212 00121

2)2()(4 zpRpitvppizipzrpi zzzz eeedpepddzZef ν

πππ

r

νrr

⋅f

011

2221 );,,(

Rvtrz

zz

ppvpppzzK

+=

⎥⎦

⎤+

+⋅=×

νω

( ) ( ) ( )⎟⎠⎞⎜

⎝⎛ −−= −

−−

cccZeD cvW z

2422 1

2

222 cos40 πκγπ

FpDec κπ /2 02=

44

Atom/Graphene van der Waals Interaction NJM Horing & JD Mancini

where

distance between the atom and the 2D Graphene sheet; is the energy difference of the atomic electron levels,

; is the matrix element of the atom’s dipole moment operator between atomic electron levels n, o ; is the dynamic, nonlocal polarizability of the Graphene sheet. The prime on denotes omission of the n=0 term.

∑ ∫∫ ++= −

∞∞'

20

22

0

2

022

2

0

)2(

),(),(

234

n D

DZp

n

nnvdW iup

iupedppu

DduEαε

αω

ω

πε ο

οο

r

h

=Zοωn

ao

anno EE −=ω onD

r

),(2 ωα pD

Σ′

[ ]22222 )16(),( pupggiup vsD γα +−= h

45

Device Friendly Features of Graphene IBecause carbon nanotubes conduct electricity with virtually no resistance, they have attracted strong interest for use in transistors and other devices.

BUT serious obstacles remains for volume production:inability to produce nanotubes of consistent sizes and consistent electronic properties; difficulty of integrating nanotubes into electronic devices;high electrical resistance at junctions between nanotubes and the metal wires connecting them that produces heating and energy loss.

46

In graphene, the carrier mobilities at room temperature can reach 3,000-27,000 cm²/Vs [Science 306,666(2004); 312, 1191(2006)] making graphene an extremely promising material for future nanoelectronic devices. Graphene mobilities up to 200,000 cm2/Vs have been reached at low temperature [P.R.L. 100, 016602 (2008) “Electron-Phonon scattering is so weak that, if extrinsic disorder is eliminated, room temperature mobilities~200,000 cm2/Vs are expected over a technologically relevant range of carrier concentration”]Since the mean free path for carriers in graphene can reach L = 400 nm at room temperature, graphene-based ballistic devices seem feasible, even at relaxed feature sizes compared to state-of-the-art CMOS technology.

Stable to High Temperatures~3,000K

The planar form of graphene over carbon nanotubes generally allows for highly developed top-down CMOS compatible process flows.

Device Friendly Features of Graphene II

47

Device Friendly Features of Graphene III• Schedin, et al. reported that graphene-based

chemical sensors are capable of detecting minute concentrations (1 part per billion) of various active gases and allow us to discern individual events when a molecule attaches to the sensor’s surface. [cond-mat/0610809]

High 2D Surface/Volume Ratio maximizes role ofadsorbed molecules as donors/acceptors

High ConductivityLow NoiseHigh Sensitivity, Detects SINGLE Molecule

48

Device Friendly Features of Graphene IV• A simple spin valve

structure has already been fabricated using graphene to provide the spin transport medium between ferromagnetic electrodes [cond-mat/0704.3165]

Long Spin Lifetime-Low Spin/Orbit Coupling-High ConductivityInject Majority Spin Carriers-Increase Chem. Pot. of Majority SpinsResistivity Changes-Signal Varies with Gate Voltage[Hill, Geim, Novoselov and Cho, Chen, Fuhrer ]

49

• J. Scott Bunch, et al. demonstrated that graphene in contact with a gold electrode can be used to electrostatically actuate an electromechanical resonator. [Science 315, 490 (2007) ]

Device Friendly Features of Graphene V

2D Graphene Sheet Suspended over Trench in SiO2 SubstrateMotion activated by rf-Gate-Voltage superposed on dc-Vg, applied to Graphene sheetElectrostatic Force between Graphene & Substrateresults in Oscillation of Graphene SheetAlso, Optical Actuation by Laser focused on sheet causing Periodic Contraction/Expansion of Graphenelayer

Suspended GrapheneSheet

SiO2Substrate

Trench

50

• Using graphene, “proof-of-principle”FET transistors, loop devices and circuitry have already been produced by Walt de Heer’s group.[http://gtresearchnews.gatech.edu/newsrelease/graphene.htm] [Also, Lemme, et al.]

• Quantum interference device using ring-shaped Graphene structure was built to manipulate electron wave interference effects.

Device Friendly Features of Graphene VI