Table Of ContentLength dependence of the resistance in graphite: Influence of ballistic transport
P. Esquinazi,1,∗ J. Barzola-Quiquia,1 S. Dusari,1 and N. Garc´ıa2
1Division of Superconductivity and Magnetism, Institut fu¨r Experimentelle Physik II,
Universita¨t Leipzig, Linn´estraße 5, D-04103 Leipzig, Germany
2Laboratorio de F´ısica de Sistemas Pequen˜os y Nanotecnolog´ıa,
Consejo Superior de Investigaciones Cient´ıficas, E-28006 Madrid, Spain
Usingalineararrayofvoltageelectrodeswithaseparationofseveralmicrometersona20nmthick
and30µmlongmultigraphenesampleweshowthatthemeasuredresistancedoesnotfollowtheusual
2 length dependenceaccording to Ohm’s law. The deviations can be quantitatively explained taking
1 intoaccountSharvin-Knudsenformulaforballistictransport. Thisallows ustoobtainwithout free
0 parametersthemeanfreepathofthecarriersinthesampleatdifferenttemperatures. Inagreement
2 with recently reported values obtained with a different experimental method, we obtain that the
n carrier mean free path is of the order of ∼2 µm with a mobility µ∼107 cm2V−1s−1. The results
a indicatethattheusualOhm’slawisnotadequatetocalculatetheabsoluteresistivityofmesoscopic
J graphite samples.
4
1 PACSnumbers:
]
i I. INTRODUCTION to cut part of the sample, i.e. a constriction of the size
c
of the order of the mean free path has to be patterned
s
- inthe sample middle without affectingits internalstruc-
l One of the important parameters that determines the
r ture. Although this is possible using a focused ion beam
t electronictransportinamaterialisthe temperatureand
m and a protecting film [11], one may still doubt whether
magnetic field dependent mean free path ℓ. Its direct
the huge increase in resistance observed for constriction
. measurement, model and parameter-free independent, is
t widths of the order of several micrometers [10] is intrin-
a howeverdifficult and only in some specialcases possible,
m sic and not due to the influence of the ion beam on the
like in materials with relatively large mean free path.
graphite sample. Therefore, it is necessary to obtain the
- Among those materials are the ones, which show ballis-
d mean free path and other transport parameters like the
tic transport, at least for a given device channel length
n mobility using more transparent and less invasive meth-
like in some carbonnanotubes samples [1–3]. The direct
o ods to check whether the obtained mean free path and
measurement of ℓ is possible, for example, with a scan-
c
mobility in multigraphene are really as large as reported
[ ningmicroscopethroughthe scalingofthe channelresis-
previously.
tance [4] or with a multi terminal method [5]. In Ref.6
1 Inthisworkwepresentasimplemethodtoobtainℓin
the electron mean free path of a carbon nanotube was
v thin,mesoscopicgraphiteflakesbasedonthedependence
4 obtainedfromascalingoftheresistancewithlength,get-
of the longitudinal resistance with the distance between
0 tingℓ(300K)=0.2...0.8µmuponthenanotubesample.
thevoltageelectrodesonthesample,amethodsomehow
0 Large values of ℓ were found also in suspended graphene
3 reaching ballistic transport in the micrometer range at related to those used in Refs. [5, 6]. As we show in this
. work,due to the micrometer large mean free path of the
1 low temperatures [7, 8].
carriers in multigraphene, the ballistic contribution to
0 Anothermethodtoobtainℓistheconstrictionmethod,
the measuredresistancebecomes alreadyappreciable for
2
based on the measurement of the longitudinal resistance
1 sample sizes severaltimes largerthanℓ. The experimen-
R as a function of the width W of a constriction located
: tal approach presented in this work represents a more
v between the voltage electrodes [9]. When ℓ & W the
transparent alternative to the constriction method used
i ballistic contribution overwhelms the diffusive ones al-
X in previous works and the obtained results support basi-
lowing to obtain ℓ(T) without the need of free param-
r cally the result that ℓ(300K) & 1 µm in highly ordered
a eters or arbitrary assumptions. This method has been
thin graphite of mesoscopic size and of good quality.
usedtoobtainℓ(T)andthemobilityµ(T)inbulk highly
oriented pyrolytic graphite (HOPG) samples [9] as well
as in some tens of nanometers thick and micrometers
II. EXPERIMENTAL DETAILS
large multigraphene samples [10]. The obtained results
indicate that the mean free path of the carriers within
thegraphitelayersinsidethegraphitestructureisindeed A ≃ 30 µm long, (5...8.5) ± 0.3 µm wide and
large, reaching the micrometer range even at room tem- (20±2) nm thick multigraphene sample was contacted
perature [10]. However, the constriction method needs with 14 electrodes prepared by electron-beam lithogra-
phy and Pd/Au deposition, see Fig. 1. For the determi-
nation of the absolute value of the mean free path the
distance between electrodes is of importance; each elec-
∗Electronicaddress: [email protected] trodehadawidthof(1.36±0.1)µm. Takingthedistance
2
((11))
((33)) ((1144)) 500
((44)) ((1133)) CH1
((55)) ((1122)) CH2
450
((66)) ((1111)) CH4
((77)) ((1100)) CH6
((88)) ((99)) 400 CH8
) 2*CH1
((22)) 3*CH1
(
R 350 4*CH1
II++ II-- VV++ VV-- CCHH
11 22 33 44 CCHH11 e 5*CH1
11 22 33 55 CCHH22 10 (cid:181)m c
11 22 33 66 CCHH44 n 300
a
11 22 33 77 CCHH66 t
s
11 22 33 88 CCHH88
i
s 250
e
R
FIG. 1: The upper left sketch shows the current and volt-
200
age electrodes configuration for the different channels. The
middle right picture is an optical microscope photo of the
measured sample with all its electrodes. The bottom left ta-
150
bledefinesthechannelswiththecorrespondingconfiguration
for current and voltage electrodes.
100
between the middle point of the electrodes we have then
50
L0 =(4.8±0.2)µm. Themultigraphenesamplewaspre- 0 50 100 150 200 250
pared by a rubbing and a ultrasound method described
◦ Tempera ture T(K)
in Ref. 12 and from a HOPG sample with 0.35 rock-
ing curve width. Micro-Raman measurements indicate
thatevensamplesof10nmthicknessareofgoodquality FIG. 2: Temperature dependence of the resistance of the
without showing any contribution of the defect-related multigraphenesampleatfivedifferentchannelsdefinedinthe
D-peak at 1350 cm−1 [13]. inset of Fig. 2. The continuous lines represent the expected
resistance if it would be just only proportional to the length
With an AC bridge from Linear Research(LR700) we
between electrodes.
measured the resistance R(T,L) at different tempera-
tures T and at different lengths L between voltage elec-
trodes. The inset in Fig. 1 shows the different config-
expected. This disagreement is independent of the ef-
uration channels. For example, channel 1 means input
fective value of L , i.e. whether we take it between the
0
current at electrodes 1 and 2 and voltage measurements
middlepointsoftheelectrodewidthsorjustbetweenthe
at electrodes 3 and 4. The results we discuss below are nearest edges (≃3.5 µm).
independent whether we take the electrodes arrayat the
Taking into account that the width of the sample
right or left of the sample, i.e. 3 and 4 or 14 and 13 (see
is not constant but decreases with the channel num-
insetinFig.1)indicatingahomogeneousbehaviorofthe
ber, see Fig. 1, the expected R (T) and according
sample and the currentdistribution. The used input AC CHi
to Ohm’s law should be even larger, i.e. R (T) =
current was I =2 µA in all the measurements. CHi
(L /L )(W /W )R (T), where W = 8.3±0.2 µm is
i 0 1 i CH1 1
thewidthatchannel1andW anaveragewidthbetween
i
the measured width at electrode 3 (W ) and the width
1
III. RESULTS ∗
W at the corresponding end electrode of the channel
i
i. This width W can be estimated either by the geo-
i
Figures 2 and 3 show the temperature dependence of metric mean or a simple average, i.e. Wi = pW1Wi∗ or
∗
the absolute and normalized resistance of the sample at (W1+Wi )/2, The small differences between the two ef-
differentchannels,respectively. InFig.2wealsoshowthe fective widths change only slightly the absolute estimate
expected R(T) of the channels CH2 to CH8 if the resis- of the mean free path and are not relevant.
tancewouldbejustproportionaltothedistancebetween The failure of the diffusive Ohm’s law to describe the
voltage electrodes, i.e. R (T) = (L /L )R (T), data canbe alsorecognizedinFig. 3. In the scale ofthis
CHi i 0 CH1
with L /L = 2...5. As one can clearly recognize in figurethe largerthedistancebetweenelectrodesthe bet-
i 0
that figure, none of the measured curves follows the ex- ter is the normalization. As we will see below this is ex-
pected diffusive Ohmic behavior but are below the one actlywhatweexpectiftheballistictransportplaysarole
3
450
250 K
400
1.8 200 K
) 350 150 K
1.7 e ( 300 100 K
c 50 K
n
250 20 K
a
1.6 st 200 2 K
) CH1 si
K e
0 CH2 R 150
5 1.5 CH4
100
2
= CH6
50
T 1.4 CH8
(
R 0
/ 0 1 2 3 4 5 6
)
T 1.3
( L/L0 (electrode pair position)
R
FIG. 4: Length dependence of the measured resistance. The
1.2
unitlengthdistanceL0betweenelectrodesisforallelectrodes
the same. The dashed and dotted (20 K) lines are obtained
1.1 from the fits to Eq. (2) with only the ballistic resistance R0
as fittingparameter.
1.0
0.9 3.0
)
0 50 100 150 200 250 m
(cid:181)
Tempera ture T(K) l (
h
at
p
FatIGth.e3:diTffeemrepnetrcahtuanrendelesp.eTnhdeenccoenotifntuhoeunsolrinmeasliazreedfirtessifsotlalonwce- ee 2.5
r
ing theparallel resistor model with a semiconducting and an n f
interface contribution. All curves are fitted with the same a
e
semiconducting energy gap of Eg =350[K]/kB. M
2.0
intheobservedbehavior. Note thatthe semiconducting-
0 50 100 150 200 250
like behavior with a saturation at low temperatures is
Temperature T(K)
observed at all distances between the voltage electrodes
indicating that this dependence is intrinsic and not re-
lated to the sample size, supporting the conclusions of FIG. 5: Temperature dependence of the mean free path ob-
Ref.13. Infact,usingthe sameparallelresistancemodel tained from Eq. (4). The error bar is estimated through the
and the exponential temperature dependence appropri- error in thedetermination of R0(T) from thefits of the data
ateforsemiconductorsasinRef.13,theobtainedenergy to Eq. (2), see Fig. 4. The continuous line follows the equa-
gap E /k ≃ 350 K is independent of the distance be- tion ℓ(T)=((2.93)−1+((6.4×105)/T2)−1)−1 (T in K andℓ
g B
in µm).
tween electrodes.
Figure 4 shows the absolute resistance as a function
of the channel length at different temperatures. The ex-
where ℓ is the mean free path, t the thickness, W the
perimental points appear to be consistently non-linear,
widthofthesampleandLthedistancebetweenthevolt-
curving up with the distance to the first channel. The
age electrodes for each of the selected channels. Defin-
reason for the observed dependence is explained below
ing the ballistic resistance R = R(L = 0) as the first
0
and allows us to obtain the carrier mean free path.
terminEq.(1),takingintoaccountthe effectivechannel
According to Sharvin-Knudsen formula and Ohm’s
width W , assuming homogeneous resistivity and thick-
i
law, the resistance of the sample in terms of geometri-
ness through the whole sample length, this equation can
cal parameters and the resistivity ρ, is given by
be rewritten as:
R= ρ(T)[ℓ(T)+L] , (1) Ri(T)=R0(T)+(R1(T)−R0(T))W1 Li , (2)
tW Wi L0
4
where the width atchannel1 is W =8.3±0.2µm, W ’s With the knowledge of the mean free path and the
1 i
are estimated as explained above and L /L =0...5. resistivity of the sample we can calculate the Fermi
i 0
With the measured widths the experimental data wavelength λ and the mobility µ of the carriers using
F
shown in Fig. 4 can be very well fitted using Eq. (2) the equations λ = 2πe2ρℓ/ha and µ = eλ ℓ/h where
F F
having R (T) as the only free parameter. To obtain the a is the distance between graphene planes in Bernal
0
mean free path from R (T) we need to calculate the re- graphite and the other parameters are the usual natural
0
sistivity ρ(T). This last is obtained from: constants. At 2 K (250 K) we obtain λ = 4.5(2.3) µm
F
andµ=3×107(1.4×107) cm2V−1s−1 with a maximum
ρ(T)= tW1(R (T)−R (T)), (3) error of 50% in the absolute values.
1 0
L
0
and therefore the mean free path is calculated from:
R (T)L
0 0
ℓ(T)= . (4)
R (T)−R (T)
1 0
IV. CONCLUSION
UsingEqs.(3)weobtainρ(2K)=(208±40)µΩcm,where
the errorisdue to the errorsinthe geometryparameters
Theaimofthisexperimentalworkwastoshowinasin-
and R (T). The mean free path obtained from Eq. (4)
0
glemultigraphenesampleoflargeenoughlengththatthe
and as a function of the temperature is shown in Fig. 5.
resistance does not follow the Ohm law. It does not in-
Note that the obtained mean free path is independent
creasestrictly proportionalto the distancebetweenvolt-
of the values of the width and thickness of the sample
age contacts, as one would expect from the usual diffu-
and the statistical error is that of R (T) from the fits in
0
sive Ohm law. Instead, we show experimentally that a
Fig. 4. Within experimental error the temperature de-
length independent contribution to the resistance due to
pendence follows a simple parallel resistance model, i.e.
ℓ(T)=((2.93)−1+((6.4×105)/T2)−1)−1(T inKandℓin the ballistic transport of the carries needs to be consid-
µm), with a temperature dependent term given by T−2, ered. From our measurements and with the help of the
Sharvin-Knudsen formula for ballistic transport we are
similartothatfoundinHOPGbulksamples[9]. Thisde-
able to obtain the mean free path of the carriers at dif-
pendence suggests that electron-electron interaction can
ferent temperatures and without free parameters. The
be the main temperature dependent scattering process.
obtained results support previous results and indicate
The error in the absolute value of the mean free path
that the graphene layers within the graphite structure
obtained in this work resides mainly in the uncertainty
have micrometer large mean free path and Fermi wave-
of L . One may speculate that the Pd/Au deposited
0 lengthandmobility&107 cm2V−1s−1 at300K.Wealso
electrodes are invasive and may affect the current distri-
bution and in this case L ∼ (4.8−1.36) µm. On the conclude that for multigraphene samples ofgood quality
0 and of size in the micrometer range, a significant error
otherhandtheelectrodesmayaffectonlytheuppermost
graphene layer from the ∼60 graphene layers inside the in the estimate of the resistivity is done if the ballistic
contribution is not taken into account. Clearly, the re-
rest of the sample. Also, the effective resistance of the
sultsindicatethattheBoltzmann-Drudeapproachtoob-
polycrystalline Pd film electrode contacting the upper
mostgraphenelayeralongits≃1.4µmwidthisnotnec- tain electronic parametersfrom transportmeasurements
is not adequate.
essarilysmallerthanthemainlyballisticresistanceofthe
graphene layer. Therefore,taking the electrode width as ThisworkwassupportedbytheDFGunderES86/16-
the maximum uncertainty in L we estimate ∼ 30% as 1. S.D. is supported by the Graduate School of Natural
0
the largest absolute error in the mean free path. Sciences “BuildMona”.
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