Supplementary information for “Magneto-optical
fingerprints of distinct graphene multilayers using
the giant infrared Kerr effect”
Chase T. Ellis1, Andreas V. Stier1, Myoung-Hwan Kim1, Joseph G. Tischler2, Evan R. Glaser2,
Rachael L. Myers-Ward2, Joseph L. Tedesco2,3, Charles R. Eddy, Jr 2, D. Kurt Gaskill2, and
John Cerne1*
1 Department of Physics, University at Buffalo, SUNY, Buffalo, New York, USA.
2 Electronics Science & Technology Division Code 6800, U.S. Naval Research Laboratory,
Washington, DC, USA.
3 American Society for Engineering Education, 1818 N Street NW, Washington, DC, USA
*Jcerne@buffalo.edu.
Supporting Note 1: Monolayer and multilayer 1/B periodicity
For monolayer graphene LL energies are defined by
Emono, L  sign( L)vF 2e | LB | ,
(S1)
where, v F is the Fermi velocity, L is the Landau level (LL) index, and B is the magnetic field
strength. Thus, the interband cyclotron resonance (CR) condition yields
E ph  E
, 1
 E 1  E  vF 2e BT ,

1 
,
(S2)
where BT , is the magnetic field strength corresponding to the    1 CR transition.
Rearranging equation S2 we find
 v 2e
1
 F
BT ,  E ph



2

1 
,
2
(S3)
Expanding the second parenthetic term in equation S3 we find
 v 2e
1
 F
BT ,  E ph
2


 2  1  2

2

,
(S4)
From equation (S4) it is clear that 1 / BT , is not exactly a linear function of
. However, upon
closer examination of the first and second derivatives of equation (S4) it can be shown that
1 / BT , is approximately a linear function of
for interband transitions where
 1 . Figure S1
shows the nearly linear dependence (Fig. S1a) and nearly constant first derivative (Fig. S1b) of
1 BT , versus
(equation S4) for
features at 1 / BT ,
 1 . The approximately periodic nature of interband CRs
in monolayer graphene can be exploited via Fourier analysis. Similar
treatment of the bilayer graphene CR condition yields similar results; shown in Figure S1c-d.
However, due to the greater complexity of 1 BT , for bilayer graphene1 these calculations must be
done numerically.
From the E ph dependence of the 1 B -frequency we extract fundamental band parameters. This
can be shown analytically for monolayer graphene. For CR transitions that are nearly periodic
versus 1 B (i.e.,  1 ) equation (S4) becomes
2
 v 2e
1
 F
BT ,  E ph

  4  1 ,

(S5)
subsequently, this reveals the relationship between the 1 B -period (and frequency f ) and E ph
1
BT ,
1
2
 vF 2e

 E ph
1
1


BT ,
f mono

 (4) .

(S6)
Using the dependence of f on the photon energy E ph , we can determine vF . The band
parameters of multilayer graphene can also be determined in this manner. However, the more
complicated LL energies of multilayer graphene do not yield analytical results for the band
parameters that must be solved for numerically.
0.25
0.20
0.15
0.10
0.5
0.05
d
b 3.3
monolayer
graphene
d(1/BT)/dn
3.0
d(1/BT,)/dn
non-linear
linear
1.0
bilayer
graphene
0.30
1/BT (1/T)
1.5
c
monolayer
graphene
2.0
non-linear
linear
1/BT, (1/T)
a
0.5
0.4
3.34
3.32
3.30
3.28
3.26
bilayer
graphene
0.08
0.06
0.04
0.02
0.3
0
1
2
3
4
5
Landau level index 
Figure S1: Linear behavior for 1 / BT , versus
0
1
2
3
4
5
Landau level index 
. (a) The nearly linear dependence of 1 / BT ,
versus
results in a nearly constant first derivative for monolayer graphene, shown in (b).
Similar behavior is found for (c,d) bilayer graphene.
Supporting Note 2: Association of 2 f AB , bi and f ABC , tri FTKS components.
As discussed in the main text (and shown in Fig. S2a), for E ph  114.5 meV , the FTKS peak at
f  36.3 T is consistent with expectations for the second harmonic AB bilayer ( 2 f AB , bi ) and
ABC trilayer ( f ABC , tri ) frequency components. As seen in Fig. S2b, at E ph  133.8 meV the two
components separate, which is completely predicted by theory2, as shown in Fig. S2c. For
E ph  117 meV the two components are separated in the 1 B -frequency space; however, at
E ph  117 meV the two components are coincident. Due to the broadening of the two
components, they are indistinguishable at E ph  114.5 meV . The E ph dependence of the 2 f AB , bi
and f ABC , tri enables us to correctly identify the features.
SiC graphene
Eph=133.8 meV
fABC,tri
2fAB,bi
magnetic frequency (T)
(arb.)
0
1
2f 2 - 1
44
fABC,tri 2f
AB,bi
(arb.)
b
c
SiC graphene
Eph=114.5 meV
3
FT Re{qKerr}
FT Re{Kerr}
a
i
f ABC,tr
40
36
32
0
10
20
30
40
1/B-magnetic frequency (T)
50
60
100
110
120
130
photon energy (meV)
Figure S2: (a) SiC graphene FTKS spectrum measured at E ph  114.5 meV , showing the 2 f AB , bi
and f ABC , tri components, which occur at the same 1 B -frequency. (b) FTKS measurement at
E ph  133.8 meV , showing the splitting of the two frequency components, as expected from
theory shown panel c. (c) Calculated1,2 f ABC ,tri and 2 f AB , bi versus E ph . Gray shading denotes
measured photon energy range. Theory shows these two frequency components coinciding at
E ph  117 meV and separating away from this energy, as observed in panels a and b. Insets
show the theoretical separation of the two peaks.
Supporting Note 3: Theoretical overestimation of Re[ K ] for monolayer graphene.
In this work we detect changes in the polarization of light reflected by the sample using
photoelastic modulation (PEM) and lockin amplification techniques. As outlined by Ref. 3, the
Kerr rotation ( Re[ K ] ) is proportional detected lockin signal ( I 2 ), referenced to the second
harmonic ( 2 ) of the PEM driving frequency (  ). The relationship between Re[ K ] and I 2 is
given by
Re[ K ]  CRe
I 2
I0 R
where CRe is a constant related to the measurement system, I 0 is the intensity of the light
incident upon the sample, and R is the reflectance of the sample. For a heterogeneous sample,
such as those discussed in the main text, the total rotation ( Re[ K ,Total ] ) is related to the intensity
of light reflecting from each section of the sample. For example, considering a heterogeneous
sample with two regions containing two distinct types of multilayer graphene (A and B), the total
2 signal  I 2 ,Total  measured by the lockin is
I 2 ,Total  Re[ K , A ]
f A I 0 RA
f I R
 Re[ K , B ] B 0 B ,
CRe
CRe
(S7)
where f A and f B are the fractional areas corresponding to the two types of graphene that are
being probed and RA and RB represents the reflectances of the individual graphene regions.
Unfortunately, our measurement system cannot separate the light reflected by the two regions;
thus, we are forced to analyze the result using the total measured intensity I mea.  f A I 0 RA  f B I 0 RB
Re[ K ,Total ]  CRe
I 2 ,Total
I mea.
 Re[ K , A ]
f A RA
f B RB
.
 Re[ K , B ]
f A RA  f B RB
f A RA  f B RB
(S8)
For this situation, equation (S8) reveals that the heterogeneous nature of sample will yield errors
in the determination of the true magnitude of Kerr features, with a scaling error for the
1
magnitude of Kerr features associated with layer A given by  f A RA f B RB  1 . This is the most
probable reason for the difference between the measured and theoretical magnitudes of the
monolayer Kerr angle discussed in the text.
Supporting Note 4: Fermi Energy limits for monolayer and bilayer graphene and their
relationship to the photon energy for electron-hole band symmetry
For electron-hole symmetric bands the condition for observing a Kerr response due to an
interband LL transition at a particular photon energy is given by E < E F < E 1 . This condition
ensures that   1  transitions are Pauli blocked while    1 are allowed. Since the
two equal-energy transitions are activated by opposite handedness of circularly polarized light,
meeting this condition will create an imbalance between   and   . This imbalance yields CR
Kerr features corresponding to interband LL transitions since the Kerr angle is proportional to
the difference between   and   . Using the interband CR condition for monolayer graphene
(equation S2), we solve for the B -value where the CRs occur:
BT ,
 E ph 


 vF 2e 
2

1

1 
2
.
(S9)
Evaluating the LL energy E 1 at BT , yields the upper Kerr activation Fermi energy limit
EF ,max  E 1 ( BT . )  E ph
1
1 
.
(S10)
Graphene layers with EF  EF ,max have both    1 and  1  LL transitions Pauli
blocked, yielding no Kerr angle response, as well as no photon absorption. Similarly, the lower
limit is given by
EF ,min  E ( BT , )  E ph
.
1 
(S11)
For layers with EF  EF ,min both degenerate transitions are allowed equally; thus, chiral
symmetry is not broken, and no Kerr angle response is observed, despite photon absorption.
However, for EF ,min  EF  EF ,max chiral symmetry is broken by Pauli blocked  1 
transitions. As
increases the difference between EF ,max and EF ,min decreases; therefore to
observe LL transitions in the Kerr spectrum with large transition index , the Fermi energy must
lie between the two limits for large values of (gray shaded region in Fig. S3a). For very large
the upper and lower limits approach the value
lim E

1
( BT , )  lim E ( BT , ) 

E ph
2
,
(S10)
which is depicted in Fig. S3b. This makes EF  E ph / 2 the ideal Fermi energy, which yields the
most LL transition Kerr features in a magnetic field sweep. The relationship between the Fermi
energy limits and photon energy shows that CR Kerr features can be turned on and off in three
ways: 1) Tuning the Fermi energy away from the ideal value E ph / 2 , 2) tuning the photon energy
away from 2 EF . and 3) tuning B or E ph away from CR. As shown in Fig. S3b, this behavior is
preserved for bilayer graphene,
a)
b)
L=1
L=2
EF,max
Energy
EF,min
L=0


B
Fermi Energy Limits (meV)
100
Eph=170 meV
90
EF=85 meV
80
ABC trilayer EF,max
AB bilayer EF,max
70
mono EF,max
mono EF,min
AB bilayer EF,min
60
ABC trilayer EF,min
50
EF=50 meV
Eph=100 meV
40
L=-2
L=-1
0
5
10
15
LL transition index
20

Figure S3: (a) Schematic of band symmetric monolayer graphene LL structure and Fermi
energy limits that activate CR Kerr features. (b) Expected Fermi energy limits EF ,max and EF ,min
for both monolayer and bilayer graphene at two different probe photon energies (upper black/red
symbols E ph ,1 =170 meV, lower black/red symbols E ph ,2 =100 meV). Both upper and lower limits
converge on E ph / 2 as
1
2
3
increases.
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