ENGI 4537 / ENGI 44O10
Advanced Electronics Measurements -
Microelectronics
Lab manuscript 1 - Hall effect in graphene
1 Introduction
The aim of this set of experimental data is to determine the carrier density and mobility in
graphene, using Hall measurements as a function of gate bias. While graphene is not a semicon-
ductor in the strict sense of the word, it shares some properties with conventional semiconductors,
making both the measurement techniques, and the analysis of results, identical to ones we would
encounter with conventional semiconductors.
In this manuscript you will analyse Hall effect measurements that were done on graphene
as a function of gate bias. You will calculate the charge carrier type and density as well as the
mobility and sheet resistance from raw data acquired in 2016.
2 Background
(a)
E
k
EC
EF
EV
(b)
Figure 1: (a) The crystallographic structure of
a graphene sheet. (b) The dispersion relation
of graphene, showing the Dirac point and the
location of the Fermi level for an intrinsic mate-
rial.
Graphene is an allotrope1 of carbon. It is a two
dimensional (2D), hexagonal crystal of covalently
bonded carbon atoms (See Fig. 1 (a)), that displays
interesting mechanical, optical and electrical prop-
erties, that have unfortunately contributed to specu-
lative and exaggerated approach to its investigation,
by some researchers.
Graphene is classified as a semi-metal. Much
like conventional semiconductors, its electronic
structure is described in terms of a valence and a
conduction band, where electrons act as free charge
carriers in the conduction band and holes as the
free carriers in the valance band. Unlike semicon-
ductors, graphene does not have a band gap, and
the bands coincide at a single point on the disper-
sion relation (E-k ) diagram, marked with a red dot on Fig. 1 (b), that is called the Dirac
1Allotropes are different materials made of the same element, which exhibit different physical properties due to
a difference in the molecular or crystallographic arrangements. Diamond, coal and graphene are all allotropes of
carbon. Molecular oxygen (O2, the gas that we breath) and ozone (O3) are allotrope of oxygen.
Advanced Electronic Measurements - Microelectronics
point2. When the Fermi level is at the Dirac point, the semi-metal is said to be intrinsic and
n = p = ni ≈ 3.5 × 1011cm−2 at room temperature due to the thermal spread of the Fermi-Dirac
distribution. Note that since graphene is a 2D material, the charge density is given in units per
area (instead of per volume). This is equivalent to 2×1017cm−3 for a three dimensional electronic
system, based on the relation that
(
n(2D)
)1/2 = (n(3D))1/3.
The lack of a band gap means that the Fermi level is always within one of the bands (with one
exception of it being at the Dirac point), meaning that graphene acts as a degenerate semicon-
ductor, i.e., that it is rarely truly intrinsic. This explains, in part, the large gate voltages you will
encounter in this experiment. Moving the Fermi level substantially within the band requires the
coincidental accumulation of charges of one polarity and the removal of charges of the opposite
polarity. This, in turn, causes a capacitive-like effect, commonly known as quantum capacitance.
Before graphene was first isolated in 2004 by Andre Geim and Konstantin Novoselov, both
from the University of Manchester, theoretical calculations predicted it to have phenomenal con-
duction properties, due to its theoretical mobility that should have exceeded 105 cm2V−1s−1.
However, the 2D nature of graphene makes it especially susceptible to perturbations from its sur-
roundings. Interface phenomena like phonon and impurity scattering cause a significant reduction
in mobility, and unintentional electrostatic interactions with stray electric fields cause its observed
charge density to fluctuate severely between samples. Therefore, it is of utmost important to
measure the properties of graphene using robust methodologies.
3 Experimental Methodology
1
2
3
4
5
6
G
raphene
A
VH
Vρ
6
1
2
3
4
5
Lock-in
ω
ω
RB
(b)(a)
Figure 2: (a) An optical micrograph of the measured de-
vice. The electrodes are numbered for convenience. The
length of the scale bar at the bottom is 2 µm. (b) The mea-
surement configuration of the device.
Fig. 2 (a) shows the device on which
this lab is based. The dark shaded mate-
rial in the middle is the graphene flake, to
which six gold electrodes are connected.
The graphene flake was mechanically
exfoliated (peeled by sheering) from a
small crystal of graphite and deposited
on a highly doped silicon chip, covered
with a thermal silicon oxide layer, 30 nm
in thickness. The silicon chip acts as the
gate electrode and the oxide as the gate
dielectric. As the silicon is highly doped,
we can assume that none of the gate
voltage drops over the silicon itself.
Following the deposition of graphene on the silicon oxide, the metal electrodes were deposited
by means of lithography and metal evaporation. The two wide electrodes (#1 and #4) serve as
current injection electrodes, and the four narrow electrodes serve as voltage probes. The device
was measured at cryogenic temperature of 4.2 K . The cryogenic conditions mean that you can
2Named after Paul Dirac
Dr I. Amit 2
Advanced Electronic Measurements - Microelectronics
regard the Fermi-Dirac distribution as a step function, thus relieving you of the use of the ambi-
polar expression for RH (Eq. 2.17 in the third volume of the lecture notes).
Fig. 2 (b) shows the measurement configuration. Electrode #1 was connected to an AC volt-
age source through a ballast resistor. The ballast resistor is chosen to have resistance that is
significantly higher than that of the measured device, and so the device is supplied with a con-
stant current. The ammeter connected to electrode #4 measured a constant Ix = 10 nA. The
Hall voltage, VH was measured between electrodes #2 and #6, and the longitudinal voltage, Vρ
was measured between electrodes #2 and #3. The distance between the edges of the trans-
verse electrodes (e.g., #2 and #6) is 1 µm; the distance between the edges of the longitudinal
electrodes is 4 µm.
The supplied voltage (and therefore, the developed current) was an AC signal at a frequency
of 77.6 Hz. This was done to reduce the noise in the measurement. For this reason, both the Hall
and the longitudinal voltage were AC signals at the same frequency, and were collected through a
lock-in amplifier. However, since there are no reactive components in the current path, operating
at an AC configuration does not change the result or the method of analysis.
A DC gate bias was applied to the device, in the range of −60 to +60 V at 1 V increments.
Your library of data contains a single file for each gate bias value. The gate bias range corre-
sponds to a transition from a p-type device to an n-type device, but may not be symmetric. A DC
magnetic field was applied at a direction normal to the carrier chip surface. The magnetic field
strength was between −8.9 and +8.9 T .
4 Required Analysis
Volume 3 of the lecture notes details the equations needed to analyse a Hall effect experiment,
but is based on three dimensional materials. For two dimensional materials, the equation for the
Hall voltage is simply
VH =
IB
qp
(1)
For a p-type material, where I is the longitudinal current, B is the magnetic field, q is the ele-
mentary charge and p is the area density of holes in units of cm−2. The resulting Hall coefficient
is:
RH =
VH
IB
=
1
qp
(2)
With similar, but opposite sign expressions for n-type materials. Likewise, the extraction of the
Hall mobility is done by using the sheet resistance of the material (with units of Ω/) rather than
the resistivity:
µH =
|RH |
Rs
(3)
For your report, complete the following tasks:
• Show, from the principles detailed in Section 2.2 of the third volume of the lecture notes
(Eq. 2.6 onwards) that the expressions given in Eqs. 1-3 in this manuscript are correct for
a 2D case.
Dr I. Amit 3
Advanced Electronic Measurements - Microelectronics
• Calculate the type and area density of charge carriers in graphene as a function of gate
bias. You may want to automate the procedure by writing a script, but be mindful of mea-
surements close to the Dirac point that are harder to analyse.
• Discuss your findings in light of the band structure of graphene. Tie it to your knowledge
of semiconductor devices. At what gate bias is the Fermi level at (or closest to) the Dirac
point? What can you conclude from that?
• Calculate the mobility of the appropriate charge carrier as a function of gate bias.
• Discuss your findings in light of the scattering mechanism discussed in the lecture.
Dr I. Amit 4