Table Of ContentDynamics of the particle - hole pair creation in graphene
M. Lewkowicz1 and B. Rosenstein2,
∗
1Applied Physics Department, Ariel University Center of Samaria, Ariel 40700, Israel
2Electrophysics Department, National Chiao Tung University,Hsinchu 30050, Taiwan, R. O. C.
(Dated: January 12, 2009)
The process of coherent creation of particle - hole excitations by an electric field in graphene
is quantitatively described. We calculate the evolution of current density, number of pairs and
9 energyafterswitchingontheelectricfield. Inparticular,itleadstoadynamicalvisualizationofthe
0 universalfiniteresistivitywithoutdissipation inpuregraphene. WeshowthattheDCconductivity
0 of puregrapheneis πe2ratherthantheoften cited valueof 4e2. Thisvaluecoincides with theAC
2 h π h
2 conductivitycalculatedandmeasuredrecentlyatopticalfrequencies. Theeffectoftemperatureand
random chemical potential (charge puddles) are considered and explain the recent experiment on
n
a suspendedgraphene. Apossibility ofBloch oscillations is discussedwithinthetightbindingmodel.
J
PACSnumbers: 81.05.Uw 73.20.Mf 73.23.Ad
1
1
1. Introduction. It has been demonstrated recently sheets on Si substrates provided values roughly 3 times
]
l thatagraphenesheet,especiallyonesuspendedonleads, larger than σ [8]. Recent experiments on suspended
l 1
a is one of the purest electronic systems. Electronic mo- graphene [2] demonstrated that the DC conductivity is
h bility reaches values of 2 105cm2/(Vs) and might be lower, 1.7σ , as temperature is reduced to 4K. Hence
- · 1
s yet improved [1, 2] indicating that transport in samples one still faces the question of what is the proper theo-
e
of submicron length is most likely ballistic. In a simpli- retical value. Since the conductivity of clean graphene
m
fied model of a single graphene sheet (neglecting scat- in the infinite sample is a well defined physical quantity
t. tering processes and electron interactions) the chemical there cannot be any ambiguity as to its value.
a
m potential is located right between the valence and con- In contrast both the experimental and the theoreti-
ductance bands and the Fermi ”surface” consists of two cal situation for the AC conductivity in the high fre-
-
d Dirac points of the Brillouin zone [4]. A lot of effort quency limit is quite different. The theoretically pre-
n has been devoted to the question of transport in pure dicted value in the Dirac model is σ independent of
2
o graphenedue to the surprisingfact that the DC conduc- frequency under condition ω >> T/~ [6, 10]. The
c
[ tivityisfinitewithoutanydissipationprocesspresent. A Dirac model becomes inapplicable when ω is of order
widely accepted value of the ”minimal conductivity” at of γ/~ = 4 1015Hz or larger, where γ = 2.7eV is the
1 zero temperature, hopping ene·rgy of graphene. It was shown theoretically
v
using the tight binding model and experimentally in [3]
6 4e2
7 σ1 = , (1) that the optical conductivity at frequencies higher or of
4 π h order γ/~ becomes slightly larger than σ . Moreover, in
2
1 was calculated very early on using the Kubo formula in light transmittance measurements at frequencies down
1. a simplified Dirac model as well as in the tight binding to 2.5 1015Hz it was found equal to σ2 within 4%. The
·
0 model [4, 5, 6, 9]. Within this approach one starts with model does not contain any other time scale capable of
9 the AC conductivity and takes a zero frequency limit changing the limiting value of AC conductivity all the
0
typically with certain ”regularizations” (like finite tem- way to ω > 0. Therefore one would expect that the
: −
v perature, disorder strength η etc.) made and removed DC conductivity even at zero temperature is σ2 rather
i
X at the end of the calculation. As noted by Ziegler [7] than σ1. As we show in this note this is indeed the case.
r the order of limits makes a difference and several other The basic physical effect of the electric field is a
a values different from σ1 were provided for the same sys- coherent creation of electron - hole pairs mostly near
tem. The standard value σ1 is obtained using a rather Dirac points. To effectively describe this process we de-
unorthodoxprocedurewhenthe DClimitω 0ismade velopa dynamicalapproachto chargetransportin clean
→
before the zero disorder strength limit η 0 is taken. If grapheneusingthe”firstquantized”approachtopaircre-
→
the order of limits is reversed one obtains [7] ation physics similar to that used in relativistic physics
[12]. To better visualize the phenomenon of resistivity
πe2 without dissipation, we describe an experimental situ-
σ2 = . (2) ation as closely as possible by calculating directly the
2 h
time evolution of the electric current after switching on
When the limit is taken holding ω =η one can even ob- an electric field. In this way the use of a rather formal
tainavalueofσ =πe2 [7],thussolvingthe”missingπ” Kubo or Landauer formalism is avoided and as a result
3 h
problem. Indeed, atleastearlyexperiments ongraphene noregularizationsareneeded. Theeffectsoftemperature
2
and of charge fluctuations or ”puddles” are investigated A physical quantity is usually conveniently written in
andexplainthe temperature dependence of conductivity terms of ψ. For example the current density (multiplied
measured recently in suspended graphene [2]. Although by factor 2 due to spin) is
weconsideraninfinitesamplethedynamicalapproachal-
lowsus to obtainqualitativeresults for finite samples by
∂H(p)
iontthreordfuaccitnogrstidmeetecrumtoinffisnglikterabnaslpliostritccflainghbteticmone.veVnaierinotulys Jy =−2eZBZψ†k(t) ∂py ψk(t) (5)
characterized by time scales like the relaxation time for
scattering of phonons or impurities. Tofirstorderinelectricfieldψk =eℏi|h|t(uk+Eξk+...)
2. Time evolution of the current density at zero tem- and consequently Jy =J0+Eσ, where
perature. Electronsingraphenearedescribedsufficiently
well by the 2D tight binding model of nearest neighbour
2e ∂H(k) 2e ∂ h
interactions [4], namely with (second quantized) Hamil- J0 = − ℏ u†k ∂k uk = ℏ ∂k| |; (6)
tonian being sum over all the links connecting sites on ZBZ y ZBZ y
twotriangularsublatticesA,B. TheHamiltonianinmo- σ(t) = σ (t);
k
mentum space is ZBZ
2e ∂H(k) ∂H(k) e ∂2H(k)
σk(t) = − ℏ u†k ∂k ξk+ξ†k ∂k uk− ℏtu†k ∂k2 uk .
c 0 h (cid:20) y y y (cid:21)
H =ZBZ c†A c†B H(cid:18)cBA(cid:19); H =(cid:18)h∗ 0(cid:19), (3) ThesolutionoftheSchroedingerequationforthecorrec-
(cid:0) (cid:1) tion ξ is
b k
where h(k) = −γ αeik·δα with γ being the hop-
ping energy; δ = a 0,√3 , δ = a 3, √3
are the locatio1ns ofP3n(cid:0)earest(cid:1)neigh2,b3ours 3o(cid:16)n±t2he−h2on(cid:17)- ξk = 2iℏe2t2∂∂k|h|uk (7)
eycomb lattice separated by distance a 3A˚. In y
tphoeintBsriKllo−uin=zo2anπe 31o,f√1t3he,laKtt+ice=the2arπe a32r,e0≃twino wDhiricahc +8|ihe|3 (cid:18)h∗∂∂khy −cc(cid:19)(cid:18)1−e−2i|h|t/ℏ− 2iℏ|h|t(cid:19)vk,
the energy gap be(cid:16)tween(cid:17)the valence an(cid:0)d th(cid:1)e conduc- 1
tion band vanishes. Expansion around K , h(k) = where v = 1 . Substituting this into Eq.(6)
ℏvgexp −iπ3 (∆kx+i∆ky), where the grap−hene veloc- the condkuctiv√i2ty(cid:18)bhe∗c/o|mh|e(cid:19)s
ity is v = √3aγ, leads to relativistic equations for the
g(cid:0) 2(cid:1) ℏ
Weyl field constructed as ψ1 =ψW1 , ψ2 =e−iπ3ψW2 .
Let us first consider the system in a constant and ho- e2 ∂2 h 2t 1 ∂h 2 2 h
monogaetnteo=us0.elIetcitsridcefiscerldibeadlobnyg tthheemyindiimreactliosunbsswtititucthioend σk(t)= ℏ "− ∂k|y2| ℏ − 4|h|4 (cid:18)h∗∂ky −cc(cid:19) sin(cid:18) ℏ| |t(cid:19)#.
(8)
p=ℏk+ eA with vector potential (choosing a gauge in
c The zero field current J0 and the first term (linear in
which the scalar potential is zero) A= (0, cEt). Since
time) in the conductivity vanish upon integration over
−
the crucial physical effect of the field is a coherent cre-
the Brillouin zone, since one can choose it to be =
ation of electron - hole pairs mostly near Dirac points π/a 2π/(31/2a) BZ
a convenient formalism to describe the pair creation is π/adkx 2π/(31/2a)dky and the integrand is a dReriva-
the ”first quantized” formulation described in detail in Rti−ve of a pRe−riodic function.
[12]. The second quantized state at T =0 which evolves The integralof the secondpart (oscillatoryin time) of
from the zero field state in which all the negative en- σ (t) gives the result shown in Fig.1. After an initial
k
ergy( h(k))statesareoccupiedisuniquelycharacter- increase over the natural time scale t ℏ/γ = 2.5
ized b−y t|he fir|st quantized amplitude ψ (t)= ψk1(t) 10−16s, it approaches σ2, Eq.(2), via γosc≡illations. The·
obeying the matrix Schroedinger equatkion iℏ∂(cid:18)ψψk=2(Ht)(cid:19)ψ amplitude of oscillations decays as a power
t
in sublattice space with the initial condition σ sin(2t/t )
γ
=1+ (9)
σ 2t/t
2 γ
1 1
ψ (t=0)=u ; u = . (4)
k k k √2 h∗/ h for t >> tγ. The limiting value is dominated by contri-
(cid:18)− | |(cid:19)
butions from the vicinity of the two Dirac points in the
Hereu isfoundasanegativeenergysolutionofthetime integral of Eq.(6). The contribution of a Dirac point is
k
independentSchroedingerequationpriortoswitchingon obtained for t >> t by integrating to infinity (in polar
γ
the electric field, Hu = h u . coordinates centered at the Dirac point)
k k
−| |
3
Σ 5.0
Σ2
1.4
4.5
1.2
4.0
1.0
0.8 2H(cid:144)LΣeh 3.5
0.6
3.0
0.4
2.5
0.2
0 20 40 60 80 100 120 140
t
10 20 30 40 50 tΓ THKL
FIG.1: Evolutionofthecurrentdensityσ(t)=J(t)/E after FIG. 2: The minimal conductivity as a function of tempera-
a DC electric field is switched on at t = 0. Unit of time ture for time tbal =500tγ is compared with measured in the
is tγ = ~/γ. Conductivity is compared to its ”dynamical” 0.5µm long sample in ref.[2]. Values for the random Fermi
value σ2 = π2eh2. The zero temperature conductivity (blue) energy are also taken from ref.[2].
approaches σ2, while finite temperature depresses the pair
creation and eventually thecurrent density vanishes as 1 .
tT the thermal factor
h
σ (t)= tanh | | σ (t). (11)
T k
T
σ e2 1 2π ∞ 2 sin(2vgqt) e2π ZBZ (cid:18) (cid:19)
= sin(ϕ) = , The first term still vanishes, while the second gives
2 ℏ (2π)2 Zϕ=0Zq=0 q h 4 a depressed value compared to that at T = 0, see Fig.
(10)
1. Moreover,the conductivity vanishes at the large time
does not influence the result.
limit. Thisiseasytoappreciatequalitatively: thecontri-
A physical picture of this resistivity without dissipa-
butions from the vicinity of the Dirac points, h << T,
tion is as follows. The electric field creates electron - | |
whichwerethe maincontributorsto σ(T)areeffectively
hole excitations in the vicinity of the Dirac points in
suppressed. Physically this suppression can be under-
which excitations are massless relativistic fermions. For
stood as follows. As mentioned above the finite resistiv-
suchparticlesthe absolutevalueofthe velocityisv and
g ityofpuregrapheneisduetopaircreationbyanelectric
cannot be altered by the electric field and is not related
field near Dirac points. The pair creation is maximal
to the wave vector k. On the other hand, the orienta-
when in the initial state the valence band is full and the
tionofthevelocityisinfluencedbytheappliedfield. The
conductance band is empty. Thermalfluctuations create
electric current is ev, thus depending on orientation, so
pairsaswell. Intheformalismweadoptedthefinitetem-
that its projection on the field direction y is increased
perature initial state is described by the density matrix
by the field. The energy of the system (calculated in a
whichspecifiedthenumberofincoherentpairspresentin
way similar to the current) is increasing continuously if
the energy range near the Dirac points. Therefore pair
nochannelfordissipationisincluded. Obviouslyatsome
creation by an electric field is less intensive due to the
time the system goes beyond linear response into Bloch
diminished phase space available and the conductivity
oscillations which are briefly discussed below. We have
vanishes at large times.
performed a similar calculation for the evolution of the
Under assumption of Dirac point dominance, T << γ
current density for an AC electric field switched on at
(definitelycoveringthetemperaturerangeT <200K be-
t = 0. After a short transient one obtains the value of
yondwhichscatteringisnotnegligible[1]),theexpression
the DC conductivity σ independent of frequency. This
2 can be simplified in the same way as Eq.(10),
isconsistentwithboththeKuboformuladerivations[10]
and optical experiments [3]. e2 ∞ ~vgq sin(2vgtq)
σ (t)= tanh , (12)
3. The temperature dependence and effect of charge T
h T q
”puddles”. AtfinitetemperatureT withinthefirstquan- Zq=0 (cid:18) (cid:19)
tizedformalismoneaddsthecontributionsofalltheener- andisamonotonicallydecreasingfunctionoftheproduct
giesincludingpositiveonesweightedwiththeBoltzmann tT. For t>>t σ (t)= e2 ~ .
γ, T h tT
factor. Due toelectron-holesymmetrythe contribution Assuming ballistic transport in a finite sample of sub-
to conductivity of a positive energy state with momen- micron length determining an effective ballistic time t ,
b
tum k is minus that of the contribution of the negative this contribution cannot explain the increase of conduc-
energy state with the same wave vector. This results in tivity with temperature in suspended graphene reported
4
in [2]. However, there is an important source of pos- aswellastogobeyondlinearresponse. Ofcoursetheen-
itive contribution to conductivity even in the ballistic ergycontinuouslyincreaseswithtimeandatcertaintime
regime. It was clearly demonstrated that a sample close approachesthe conductionbandedgeatwhichstagelin-
to minimal conductivity consists of positively and nega- ear response breaks down. We calculated the evolution
tively chargedpuddles. This means effectively that even ofcurrentdensity,energyandpairnumberbeyondlinear
at minimal conductivity the chemical potential µ locally response and found that Bloch oscillations set in with a
is finite, ratherthanzero,albeit smallonaverage. Phys- period of t = ℏ = γ t . The range of appli-
Bloch eaE eaE γ
ically this implies that in addition to the novel constant cability of the linear response was also determined. The
contribution due to pair creation, there is an ordinary averagecurrentoverlargertimesiszero. Thismeansthat
contribution due to acceleration of electrons like in or- atveryhighfieldstheminimalconductivityphenomenon
dinary metal. In ballistic regime it grows linearly with disappears. However in order to reachthe conditions for
time. observationoftheBlochoscillationsingrapheneallother
The experiment [2] shows that the amplitude of the time scales τ,t ,t ,2π/ω should be larger than t .
T bal Bloch
randomFermienergyincreaseslinearlywithtemperature Additional phenomena beyond linear response as well as
µ =µ +αT. For example, for the 0.5µm long sample their relation to the Schwinger’s calculation of the pair
T 0
µ =8meV andα=0.1meV/K. Thedifferencebetween creation rate [ [12, 14]] is under investigation.
0
σ andσ isequaltotheintegralinEq.(6)overthetwo
µ µ=0
regions around the Dirac points determined by h(k) < WearegratefultoH.C.Kao,E.Kogan,E.Sonin,W.B.
| |
µ. That way one obtains for t>>tγ Jian, E. Andrei, R. Krupke and V. Zhuravlev for dis-
cussions. Work was supported by NSC of R.O.C. grant
#972112M009048 and MOE ATU program. M.L. ac-
e2 √3µt √3µt
σ (t) σ (t)= Si , (13) knowledges the hospitality and support at Physics De-
µ − µ=0 h " ℏ − ℏ !# partment of NCTU.
whichisamonotonicallyincreasingfunctionoftheprod-
uctµtonly(Si isthe sineintegralfunction). InFig.2we
fitthevalueoftheballisticeffectivetimet 2 10 13s,
bal −
≃ · ∗
whichisofthesameorderofmagnitudeasforthe0.5µm Electronic address: [email protected]
long sample, L/v 5 10 13s. [1] S.V.Morozovetal,Phys.Rev.Lett,100,016602(2008).
g −
≃ · [2] X. Du, I. Skachko, A. Barker and E. Y. Andrei, Nature
4. Discussion and summary.
Nanotechnology 3, 491 (2008).
To summarize, we studied the dynamics of the parti-
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