Table Of ContentLocalization of charge carriers in monolayer graphene gradually disordered by ion
irradiation
E. Zion,1 A.Haran,2,∗ A. V. Butenko,1 L. Wolfson,3 Yu. Kaganovskii,3 T. Havdala,1
A. Sharoni,1 D. Naveh,2 V. Richter,4 M. Kaveh,3 E. Kogan,3 and I. Shlimak3,†
1DepartmentofPhysicsandInstituteofNanotechnologyandAdvancedMaterials,Bar-IlanUniversity,Ramat-Gan52900,Israel
2Faculty of Engineering, Bar-Ilan University, Ramat-Gan 52900, Israel
5 3Jack and Pearl Resnick Institute, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
1 4Solid State Institute and Physics Department, Technion-Israel Institute of Technology, Haifa 32000, Israel
0
(Dated: May 15, 2015)
2
Gradual localization of charge carriers was studied in aseries of micro-size samples of monolayer
y
graphenefabricatedonthecommonlargescalefilmandirradiatedbydifferentdosesofC+ ionswith
a
M energy35keV.Measurementsofthetemperaturedependenceofconductivityandmagnetoresistance
infieldsupto4Tshowedthatatlowdisorder,thesamplesareintheregimeofweaklocalizationand
antilocalization. Furtherincrease ofdisorderleadstostronglocalization regime, when conductivity
4
1 isdescribedbythevariable-range-hopping(VRH)mechanism. AcrossoverfromtheMottregimeto
the Efros-Shklovskii regime of VRH is observed with decreasing temperature. Theoretical analysis
] of conductivity in both regimes showed a remarkably good agreement with experimental data.
l
l
a PACSnumbers: 73.22.Pr
h
-
s INTRODUCTION trical contacts were deposited directly on the graphene
e
m surface. This sample is marked as 0. The resistivity of
. Graphene, a sheet of sp2 -bonded carbon atoms, for the sample 0 was 380 Ohm/sq which is close to the data
t in the certificate. On the surface of the other specimen,
a severalyearsalreadycontinuestobeinthefocusofatten-
m six groups of mini-samples (200×200 µm) were fabri-
tionofphysicscommunity,duetoitspotentiallytransfor-
cated by means of electron-beam lithography (EBL) as
- mativeimpactacrossawiderangeofapplicationsinclud-
d wellaselectricalcontacts(5 nmTiand45nmPd)for2-
ing advanced electronics and sensing. Carrier scattering
n
probe measurements. The samples from the first group,
o ingraphenecanbeduetoallkindsofdisorder,including
marked as sample 1 were not irradiated, while 5 other
c ripplesinthegraphenelayer,pointdefectsandtheirasso-
[ ciatedshort-rangepotentials,chargedimpuritiesresiding groups were subjected to different doses (from 5×1013
2 in the supporting substrate, and adsorbed atoms on the upto1×1015cm−2)ofirradiationbyC+ionswithenergy
35keV.IonirradiationwasperformedonHVEE-350Im-
v surface. Investigation of the influence of disorder on the
1 properties of graphene is attracting a tremendous inter- planter, C+ ions were obtained from the hollow cathode
8 est due to possibility to modify this novelandpromising ion source by CO2 decomposition.
5
material using weak or strong localization of charge car- The choice of the irradiation conditions was not oc-
4
0 riers. By controllably introducing defects into graphene, casional. They are the same as in the Ref. [6] where
. one may be able to understand how these mechanisms theHalleffectwasmeasuredinmonolayergraphenesub-
1
limit transport. jectedtoionirradiation. Resistancesoftheinitialsample
0
5 Previosly, there were observed separately either weak in Ref. [6] and our initial sample 0 are the same. It was
1 localization (WL) or different kinds of Variable-Range- shown in [6] that both Hall coefficient and mobility are
v: Hopping (VRH) conductivity of strongly localized carri- temperature independent up to high dose of irradiation.
i ers in graphene samples disordered by different methods Moreover,itwasshownthatthesheetHallconcentration
X
like doping, oxidation, ion irradiation (see, for example, of chargecarriers(about 1013 cm−2) does not depend at
ar [1–8]). However, we are not aware of observations of all all on the dose of irradiation. We were based on these
regimes of localizationwith gradualincrease of disorder- resultsinourdecisionto preferthe 2-probegeometryin-
ing in graphene. In this paper we report the results of stead of the Hall bar geometry, because this allows us
study of the localization process in monolayer graphene to measure more samples on the same sample holder.
(MG) samples subjected by different doses of ion irradi- As a result, we were able to enhance the reproducibil-
ation. ity of the obtained results. Comparison of the graphene
resistance measured by 2-probe method and by 4-probe
The initial large size (5×5 mm) specimens were sup-
method presentedin [6]showedthat in this system, con-
pliedby”Graphenea”company. Monolayergraphenewas
tact resistance is insignificant.
produced by CVD on copper catalyst and transferred to
a300nmSiO /Sisubstrateusingwettransferprocess. It In our previous work [9], a concentration of structural
2
is specified in the certificate, that the sheet resistivity of defectsN wasdeterminedforeachgroupofsamplesus-
D
thespecimenis350Ohm/sq. Ononespecimen,goldelec- ingmeasurementsoftheRamanscattering. Thesevalues
2
areshownintheinsetinFig. 1. (Itturnsoutthatsample
1 is also slightly disordered due to EBL process). Mea-
surements of the current-voltagecharacteristics (I−V ) 12 1 5.8 KK 1
1.0 3 15 K
for all samples, performed at room temperature, showed 4 25 K 2
thatforsamples5and6,I−V isstronglynon-lineareven 5 50 K
atverysmallcurrent. Thatiswhyinthispaper,thetem- 3
0.8
perature dependences of resistance R(T) are shown only
for samples 0 - 4.
4
The resistance was measured by two-probe method in 0.6
)
heliumcryostatdownto1.8Kinzeromagneticfieldand h
inmagneticfieldsupto4Tesla. Fig. 1showsthegeneral 2 /
picture R(T) for all samples. / (e0.4 5
The sample 0 shows typical metallic behavior, when
R slightly decreases with decrease of T. For sample 1 0.2 6
R slightly increases with decreasing T, which is charac-
7
teristic for ”dirty” metals. For other samples R changes
with T exponentially, whichis characteristicfor strongly 0.0
8
localized carriers. 6 75 K
7 100 K 9
8 150 K
-0.2 9 200 K 10
10 10 250 K
10 Sample # ND(1012cm-2) 11 300 K 11
109 1 1 -4 -2 0 2 4
4 2 3
108 3 6 B (T)
4 12
m) 107
h FIG.2: Magnetoconductanceofsample1asfunctionofmag-
O
R ( 106 3 neticfield;solidlines–experiment,dashedlines–formula(1)
with fitted parameters.
5
10 2
4 1
10
to restorationof WL because fermions in K and K′ val-
3
10
0 leys have opposite chiralities.
10 100 Quantum corrections to the conductivity of graphene
T (K)
have been intensely studied theoretically [11–19]. It was
FIG. 1: Resistivity of disordered monolayer graphene sam- predicted that atrelatively hightemperatures WAL cor-
ples as a function of temperature. Inset shows thedensity of rections will dominate, while with decreasing T the WL
structural defects in samples [9]. correctionswilldominate. Therewereseveralexperimen-
tal papers reporting logarithmic dependence of conduc-
tivity on temperature and magnetic field at low temper-
atures [1–5, 7]. However, in our sample 1, the logarith-
WEAK LOCALIZATION micdependenceisobservedinwidetemperatureinterval,
startingfrom300K,whichgivesanopportunitytocheck
in a very detailed way the theoretical predictions.
Fig. 2 shows the experimental dependences of mag-
For the MC the theory [15] predicts
netoconductance (MC) of sample 1 in wide temperature
interval, from 300 K down to 1.8 K. Plot of the temepa-
e2 B B
ture dependence of conductivity on the scale σ vs. lnT ∆σ(B,T)= F −F
πh(cid:20) (cid:18)B (cid:19) (cid:18)B +2B (cid:19)
(Fig. 3) shows the logarithmic temperature behavior of ϕ ϕ i
σ at low T, characteristic for regime of WL [10], with B
−2F (1)
tendency to saturation at very low temperatures. (cid:18)Bϕ+B∗(cid:19)(cid:21)
WL regime ofconductivity inmonolayergraphenehas 1 1 ~c
F(z)=ln(z)+ψ + , B = τ−1 ,
important features due to the facts that charge carriers (cid:18)2 z(cid:19) ϕ,i,∗ 4De ϕ,i,∗
arechiralDiracfermions,whichareresideintwoinequiv-
alent valleys at the K and K′ points of the Brillouin where ψ is the digamma function, τ is the coherence
ϕ
zone. Due to chirality, Dirac fermion acquires a phase time, τ−1 is the intervalley scattering rate, τ−1 is the
i ∗
of π upon intravalley scattering, which leads to destruc- combined scattering rate of intravalley and intervalley
tive interference with its time-reversed counterpart and scattering and of trigonal warping.
weakantilocalization(WAL).Intervalleyscatteringleads Fitting Eq. (1) to experimental data for magnetocon-
3
length L increases from 7 nm to 70 nm and then sat-
ϕ
0.6 3
urates. The maximal value of L allows us to estimate
B ϕ
140 (T)B0, i.4 Bi 2 B (T)* the density of dephasing centers as 2×1010 cm−2.
0.2 1
STRONG LOCALIZATION
B*
130
) 0.0 0
S 0 100 200 300
T (K) Letsdiscussnowsamples2-4withpronouncedinsulat-
( ing behaviour. Plotting the data on the Arrhenius scale
lnR vs. 1/T showed that energy of activation continu-
120
ouslydecreaseswith decreasingT whichis characteristic
for the variable-range-hopping (VRH) conductivity [25].
Calculation
There are two kinds of VRH depending on the structure
Experiment
of the density-of-states (DOS) g(ǫ) in the vicinity of the
110
Fermi level (FL) µ: when g(ǫ) = g(µ) =const, R(T) is
described by the Mott (T−1/3) law in the case of two-
0 1 2 3 4 5 dimensional (2d) conductivity:
ln(T(K))
R(T)=R exp(T /T)1/3, T =C [g(µ)a2]−1. (3)
FIG. 3: Conductivity of sample 1 as function of tempera- 0 M M M
ture. Circles present experimental data, squares present Eq.
Here C =13.8is the numericalcoefficient [25], a is the
(2)withparametersdeterminedfromfittingthemagnetocon- M
ductance (A was chosen to be 11.7·e2/πh). The inset: The radius of localization.
values of Bϕ (solid line) and Bi (short-dashed line) and B∗ The Coulomb interaction between localized carriers
(long-dashed line, right axis). leadstoappearanceofthesoftCoulombgapinthevicin-
ity of FL which in the case of 2d has a linear form
ductance of sample 1 at different temperatures is illus- g(ǫ)∼|ǫ−µ|(e2/κ)−2, (4)
trated on Fig. 2. In the process of fitting we were able
to extract all three parameters Bϕ,i,∗ entering the equa- where κ is the dielectric constant of the material. This
tion. TheseparametersshownininsetinFig. 3. Itturns leads to the Efros-Shklovskii(ES) VRH or T−1/2 law:
out,thattheyaretemperature–dependentwhichwasnot
predicted by theory. Using these parameters we can cal- R(T)=R exp(T /T)1/2, T =C (e2/κa), (5)
0 ES ES ES
culate conductance at zero magnetic field according to
Eq. (10)of Ref. [15], whichcanbe rewritteninthe form where the numerical coefficient CES =2.8 [25].
Coulomb interaction can alter the DOS only near the
e2 Bi B∗ FL.FarfromFL,the DOSisrestoredtoits initialvalue,
σ(B =0,T) = − ln 1+2 +2ln 1+
πh(cid:20) (cid:18) B (cid:19) (cid:18) B (cid:19) whichis approximatelyequaltog(µ), seeinsetinFig. 4.
ϕ ϕ
B Denoting the half-width of the Coulomb gap as ∆ one
ϕ
+2ln +A, (2)
can conclude, therefore, that T ≪ ∆, ES law has to be
(cid:18)1T(cid:19)(cid:21)
observed, while in the opposite case (T ≫ ∆), the Mott
where A is a constant, dependent upon the unit of mag- law should dominate.
netic field (chosenas 1T). We compareEq. (2)with the There is a number of reports about observation of ei-
experimentally measured conductivity σ(B =0,T). The ther Mott or ES laws in different disordered graphene-
comparison is presented on Fig. 3. The good agreement based materials [26–29]. We show that in samples 3 and
proves the correctness of the chosen parameters. 4, both VRH laws are observable at different tempera-
The inset to Fig. 3 shows that, apart from the low- tures. (For sample 2, the VRH regime will be observed
est temperatures, Bϕ ∼ 1/τϕ ∼ T. Mechanisms that atlowertemperatures). InFigs. 4and5,logRisplotted
cangivethe dependenceτϕ ∼1/T are: electron–electron versus T−1/3 and T−1/2. At high temperatures, depen-
scatteringindirtylimit[10],electron–phononscattering, dences R(T) are straightened on the scale T−1/3, while
[20, 21], and electron-flexural phonon interaction [22]. at low temperatures they are straightened on the scale
Thesaturationofτϕatlowtemperaturesiswellknownin T−1/2. The latter shows the approach to the ES law
classical2dsystemsandmaybeconnectedwithexistence which should be observed at the lowest temperatures.
of dephasing centers (for example, magnetic impurities) These plots allow us to determine both parameters T
M
[23, 24]. and T (Table 1) and calculate the temperature T of
ES c
ObtainedvalueofBϕ allowus to determine the values deviation from T1/3 law to T1/2 law in the case of 2d
ofdephasinglengthL = Dτ = ~/4B e. Whenthe conductivitysimilarlytothecalculationofT for3dcon-
ϕ ϕ ϕ c
temperature decreasesfropm300K tpo 3 K,the dephasing ductivity [30]. In VRH, only localized states in an opti-
4
S # T (K) T (K) T (K) ∆(K) T (K)
M ES c
3 308±54 50±7 11.4±0.8 12±1.2 10 300 75 25 8 3
10
4 5962±305 490±12 29±5 60±6
g( )
4
TABLE I: Hoppingconductivity parameters for samples 3,4. 109 g( )
8
10
malbandofwidthǫ(T)nearthe Fermilevelareinvolved m) 0
in the hopping process. The band becomes continuously h
O 7 2
narrowerwithdecreasingT. Hoppingresistanceisdeter- R ( 10
mined by the critical parameter ξ :
c
6 3
10
R=R0expξc, ξc =(2r/a)+(ǫ/T). (6) Tc
5
Here energy and temperature are measured in the same 10 2
units, r is the mean distance of hopping. In the Mott
regime, g(ǫ) = g(µ) = const and, therefore, the total 4
10
number of states in the optimal band is N(T) = g(µ)ǫ
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
and the meandistance betweenstates in two dimensions
-1/3 -1/3
is r ≈[g(µ)ǫ]−1/2. Substituting in (6), one can find ǫ(T) FIG. 4: logR for samplesT2-4 (Kplot)ted versus T−1/3.
fromtheminimalvalueofξ determinedfromdξ /dǫ=0:
c c
ǫ(T)=T2/3[g(µ)a2]−1/3. (7)
T (K)
10 100 25 10 5 3
This gives the relationship between T and the width of 10
the optimal band: T =[g(µ)a2]1/2ǫ3/2. At the crossover 9
10
temperature Tc, ǫ=∆, so 4
m) 108
T =[g(µ)a2]1/2∆3/2. (8) h
c O
R ( 107
BeinginsidetheCoulombgap,thecrossovertemperature
6
can be determined from g(ǫ = ∆) = g(µ) which gives 10
∆ = g(µ)(e2/κ)2. Substituting into Eq. (8) and using 3
5
expressions for T and T , Eqs. (3) and (5), we get 10 2
M ES
4
Tc=(C2 /C3 )(T3 /T2 )≈8.6(T3 /T2 ) (9) 100.0 0.1 0.2 0.3 0.4 0.5 0.6
M ES ES M ES M -1/2 -1/2
T (K )
The functional proportionality of T to the ratio FIG. 5: logR for samples 2-4 plotted versus T−1/2.
c
(T3 /T2 )has beenobtainedearlierinRef. [31,32], but
ES M
withsignificantlydifferentnumericalcoefficient. Thelat-
ter is, however, crucial for comparison with experiment. ofamphotericimpurityaction[33],i.e. theycanbeeither
The values of T calculated from Eq. (9) for samples 3 acceptors or donors and compensate each other. Thus,
c
and 4 are given in Table 1 and shown as arrows in Fig. increaseofND leadstotherandompotentialreliefampli-
4. The good agreement shows the correctness of the ob- tudeincrease. Onecanalsoassumethatthis phenomena
tained numerical coefficient. We can also estimate the may be connected with the fact that some point defects
width of the Coulomb gap. In calculation, the value of (say, vacancies) may produce complex associative cen-
κ=2.45forthemonolayergrapheneonthe SiO surface ters with other defects. These associations can show an
2
was used as κ = (κ +κ )/2, where κ = 3.9 (for SiO ) amphotericimpurityaction,whereastheindividualcom-
1 2 1 2
and κ = 1 (for air). One can see that indeed, the ES ponents are not amphoteric [34].
2
law is observed when T <∆. In conclusion, a gradual transformation of conductiv-
Comparison of samples 3 and 4 shows that increase itymeasuredinawideintervaloftemperatures(300-1.8
of the density of defects N leads to stronger localiza- K) and magnetic fields (up to 4 T) was observedin a se-
D
tion which manifests in a significant decrease of a and riesofmonolayergraphenesamplessubjectedbydifferent
increase of the energy band needed for hopping. This dose of ion irradiation. Increasing the density of struc-
looks like increase of the amplitude of the random po- turaldefectsN inducedbyirradiationhasledtochange
D
tential relief in classical semiconductors induced by ran- the mechanism of electron transport from metallic con-
domly distributed positively charged donors and nega- ductivity to the regime ofWL andfinally to the VRH of
tively charged acceptors in the case of compensation. stronglylocalizedcarriers. ItisshownthatWLregimein
Hence we assume that structuraldefects in grapheneare slightly disordered sample starts from high (room) tem-
5
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