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Supplementary Information for
Batch-fabricated high-performance graphene Hall elements
Huilong Xu,1 Zhiyong Zhang,1 Runbo Shi,1 Honggang Liu,2 Zhenxing Wang,1 Sheng
Wang,1 and Lian-Mao Peng1
1
Key Laboratory for the Physics and Chemistry of Nanodevices and Department of
Electronics, Peking University, Beijing 100871, China
2
Microwave Devices and Integrated Circuits Department, Institute of
Microelectronics, Chinese Academy of Sciences, Beijing 100029, China
1
Part I: Supplementary figures
(a)
(b)
Magnetic field
with coordinate
GHE detector
Magnet
Multimeter
Dry battery
(c)
(d)
(f)
(e)
9.6
B (mT)
00
8.0
5.000
5
0
5
Y (cm)
10.00
10
6.4
15.00
15
20.00
20
4.8
26.00
25
3.2
1.6
0.0
0.0
1.6
3.2
4.8
6.4
8.0
9.6
X (cm)
Figure S1 | A simple Gauss meter based on graphene Hall element (GHE). (a)
Overview image of the Gauss meter. (b) The external magnetic field sampled on top
with grids as positioning coordinates. (c) Macro GHE based magnetic detector on
the grids. (d) Enlarged view of the magnetic detector which is constructed with a
GHE welded on a printed circuit board. (e) Optical microscopic image of the macro
GHE. The scale bar is 100 μm. (f) Map of the external magnetic field using a
commercial Gauss meter.
2
(a)
(b)
G
1200
(c)
1600 2000 2400 2800
-1
Raman Shift (cm )
10
-20
0
20
40
Vgs (V)
(d)
0.1
Dirac Point
0
T = 300 K
-1000
Sv (V/VT)
SI (V)
20
0.2
1000
0.0
-0.1
B = 2 T, V = 0.8 V
-0.2
-0.3
0
5
10
15
20
0
5
10
15
20
15
20
Vgs (V)
Vgs (V)
(e)
(f)
1.0
2000
 (cm /Vs)
-2
n (10 cm )
In
Out
0
-40
3200
2000
-2000
Vds = 50 mV
30
Ids (A)
Intensity (a.u.)
2D
0
12
2
0.5
0.0
-2000
-0.5
0
5
10 15
Vgs (V)
20
-4000
0
5
10
Vgs (V)
Figure S2 | Additional characteristics of graphene and GHEs. (a) Typical Raman
spectrum of the graphene samples in our experiment. The wavelength of the excitation
laser is 633 nm. (b) Transfer characteristics of the device in Fig. 2. with back gate
voltage. The blue curve denotes the transfer characteristics measured with electrodes C1
and C2 as source and drain respectively, and red curve denotes the one measured with
electrodes C3 and C4 as source and drain respectively. (c) Current related sensitivity, and
(d) voltage related sensitivity. (e) carrier density, and (f) Hall mobility versus back gate
voltage of the GHE according to equations (4) and (5) respectively. Carrier mobility was
retrieved under B=2 T.
3
(a)
Device #
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
70
Ids (A)
60
50
40
30
20
10
0
-40
-20
0
20
40
Vgs (V)
(b)
80
Device #
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
70
Ids (A)
60
50
40
30
20
10
0
-40
-20
0
20
40
Vgs (V)
Figure S3 | Transfer characteristics of GHEs. (a) The GHEs were fabricated directly on
the SiO2/Si substrate, and (b) on SiO2/Si substrate with additional APTES modification
process. All devices were measured at a source-drain bias of 50 mV.
4
(a)
(b)
1500
50
T = 300 K
T = 300 K
0
SI (V/AT)
VH (mV)
Ic = 100 A
VH (mV)
50
0
1000
500
-50
-50
-0.5
0.0
0.5
B (T)
-0.5
0.0
0
0.5
B (T)
0
2
4
(d)
(c)
80
10
T = 300 K
0
40
-40
0
-40
0.2
5000
-1 -1
VH (mV)
80
10000
0.1
-80
-0.5
-80
0.0
0.5
B (T)
-0.5
0.0
B (T)
2
SV (V/VT)
Vc = 500 mV
0.3
h (cm V s )
VH (mV)
8
0.4
T = 300 K
40
6
Device #
0.5
0.0
0
2
4
6
8
0
10
Device #
Figure S4 Performance of APTES-treated micro graphene Hall elements in air. (a)
Typical output Hall voltage as a function magnetic field of as fabricated GHEs working
at constant current mode. Inset shows the linear fitting of the Hall voltage. All the
devices were tested using Everbeing Hall effect probe station. (b) Statistics of the
current-related sensitivity of nine GHEs tested. (c) Typical output Hall voltage as a
function magnetic field of as fabricated GHEs working at constant voltage mode. Inset
shows the linear fitting of the Hall voltage. (d) Statistics of the voltage-related
sensitivity of the nine GHEs tested. Also shown is the Hall mobility distribution of the
tested graphene devices. All the devices here have a cross like shape similar to that
shown in figure 2(b) in the main text, but with a relatively larger width-to-length
dimension of W / L = 15 µm / 24 µm. Detailed description the W and L is in figure S9.
5
(a)
10
4
Ic (A)
10
3
10
2
200
100
60
0.5
NV (nV/Hz )
0.5
NV (nV/Hz )
500
Ic = 500 A
40
20
10
1
4.5M
T = 300 K
5.0M
Frequency (Hz)
100
1k
Frequency (Hz)
(b)
4
Ic = 500 A
0.5
Bmin (nT/Hz )
10
150
0.5
3
Bmin (nT/Hz )
10
Ic = 500 A
100
50
4.5M
10
2
Frequency (Hz)
5.0M
100
T = 300 K
1k
Frequency (Hz)
Figure S5 | Noise spectral density (NV) and Resolution (or the minimum detectable
magnetic field Bmin) of GHE under varying frequency at 300 K. (a) NV is measured
via a spectrum analyzer (Agilent N9020A) and a Stanford Low-noise voltage
preamplifier SR560 together with a lakeshore probe station TTP-4 under normal air
atmosphere condition. The terminals C1 and C2 [see Fig. 2(b) in the main text] are biased
with constant current from 100 µA to 500 µA by Keithley 4200SCS. The terminals C3
and C4 are connected to the two inputs of SR560 respectively, where the differential
voltage between C3 and C4 is amplified by SR560. The output of SR560 is connected to
the spectrum analyzer. Voltage noise at low frequency shows 1/f behavior. Inset shows
the noise measured at high frequency, where the noise of the measurement setup
dominates. (b) Calculated Bmin by dividing the frequency dependent voltage noise
with absolute sensitivity SA, i.e. Bmin=
/ SA, where SA = SI · I is 0.4 V/T for
the device with a bias current of 500 µA and current related sensitivity of 800 V/AT
measured in air. Inset shows the Bmin at high frequency of up to 5M. Since the total noise
measured at high frequency is dominated by the noise of the measurement setup, real
Bmin of graphene device at high frequency should be smaller than that shown in the inset.
6
(a)
0
VH0 (mV)
-2
-4
-6
Ic = 200 A
-8
-10
0
100
200
300
400
300
400
T (K)
(b)
0
VH0 (mV)
-2
-4
-6
Vc = 2 V
-8
-10
0
100
200
T (K)
Figure S6 | Temperature dependent offset voltage of the GHE. The measurements
were carrired out at temperatures range from 1.8 K to 400 K working in (a) current
mode, and (b) voltage mode.
7
InAs QW
SI (V/AT)
600
Graphene
400
InSb
200
GaAs
InAs
0
0
100
200
300
400
T (K)
Figure S7 | Current-related sensitivity and operating temperature range of
different Hall elements. Here, the GaAs Hall element is HG-106C-2U, supplied by
Asahi Kasei Microdevices (AKM) Corporation. Others devices, also supplied by
AKM Corp., are HQ-0222, HS-0111, HZ-116C for the InAs quantum well (QW) Hall
element, InSb and InAs Hall elements respectively.
Part 2: Derivation of relevant equations about sensitivity of
Graphene Hall elements
In this part, we will deduce general relations between the sensitivity of the Hall element and the
basic material parameters in GHE regardless of the detailed geometry. We then conside and
discuss GHE with special geometry, including the Hall plate with infinite long channel and a
cross-shaped Hall plate.
The magnetic field is perpendicular to the 2D graphene plane as shown in Figure S8. To
simplify discussions, we will only consider the situation of low electric field and low magnetic
field here. Low electric field means that the carriers transport in graphene through diffusive
method and velocity of carriers is much smaller than the saturation velocity vsat, i.e. vd =µE << vsat.
In the GHE on Si/SiO2 substrate, vsat may be estimated by
vsat   /
surface phonon energy of SiO2, which is about 55 meV, and
 n ,S1 where, Ω is the
is the Plank constant, n is the
8
carrier density. Low magnetic field is defined here as µ2B2 << 1.S2 Supposing µ~ 0.3 m2/Vs and
n~1×1012 cm-2 as shown in Fig. 5(a) in the main text, we have E << 1.6 V/μm and B << 1 T.
1. General case
Generally, both holes and electrons contribute to the conductivity of graphene, and the mobility of
the charged carriers may vary with their energy, i.e., µ is spread around the Fermi energy.S2,S3 Thus,
at zero magnetic field, the total conductivity may be regarded as the sum of the conductivities
contributed from subgroups of carriers with different mobility, that is  

k
0
, where
k
 0k  enk  k is the conductivity originated from the carriers with density nk and mobility  k at
zero field. The current density under electric field E is then given by J   E . Under a weak
magnetic field, the relation between the total current density and the total electric field may be
written as equation (S1) according to the Boltzmann transport theory,S2-S4
E  B J  RH ( J  B)
(S1)
where, the average magnetoresistivity
 B and average Hall coefficient RH are respectively
given by
2

 



k 
k
k 
 B    B  1     k B k /  B  B 2  
k
   k

 

 k

RH 
 
k
k
B
2
(S2)
k
k

k 
  B 
 k

1
2
 

k
k 
2
1     k B k /   B  B 
k

  k

,
(S3)
 0k
k
in which  
, where  0  enk  k is the conductivity originated from the group of
2 2
1  k B
k
B
carriers with density nk and mobility
 k at zero magnetic field, and  k  1, with the positive
sign for holes and negative sign for electrons.
The Hall scattering factor rH  H /  , where µH is the Hall mobility and µ is the drift
9
mobility, is considered to be unit here,S2,S3 therefore the Hall mobility and drift mobility are
considered as the same value here. The Hall angle
 H of the total current with respect to the
external electric field can be derived from equations (S2) and (S3) as,
tan  H  H B
where,  H  (
(S4)
 
k
k
B
k ) /   Bk . The second term in equation (S1) unambiguously gives the
k
Hall electric field
k
EH in the presence of magnetic field B ,
EH   RH ( J  B)
(S5)
Thus, the Hall voltage is given by
VH    RH ( J  B)ds
s
.
(S6)
Figure S8. Diagram of a two dimensional Hall plate with length L and width W in the
presence of magnetic field B perpendicular to the Hall plate. θ is the Hall angle, which
describes the deflection of the current density J to the external electric field E.
2. Infinite long Hall plate
For a infinite long plate with L / W   , equation (S6) simply reduces to
VH  RH IB ,
(S7)
where I=J∙W is the current flow along the plate. Then the current related sensitivity can be
obtained as
SI 
R
1 VH
 RH  B H .
I B
B
(S8)
Writing I=V/R=(W/L)B, we obtain the voltage related sensitivity
SV 
R
 
1 VH

 (W / L)  RH  B   B B H  RH B B  ,
V B
B
B 

(S9)
10
where
 B  1/  B . As can be seen from equations (S8) and (S9), either the current related
sensitivity or the voltage related sensitivity generally is not a constant but a variable which is
dependent on magnetic field. The non-constant current or voltage related sensitivity will result in
non-linearity (NL) of the Hall element, which may be related to the sensitivity asS2
NL  (S  SB~0 ) / S B ~0 ,
(S10)
in which S is the current or voltage related sensitivity at a given magnetic field, and SB~0 is the
corresponding sensitivity at near zero magnetic field.
This non-linearity, originated from the basic material properties, is named as the physical
magnetoresistance effect,S2 to be distinguished from the geometrical magnetoresistance effect
which has been used in the extraordinary magnetoresistance devices based on graphene
recently.S5,S6 To be noticed, in an infinite long Hall plate the geometrical magnetoresistance effect
is absent but only the physical magnetoresistance effect may exist.S2
To further uncover the relationship between sensitivities of GHE and basic material
parameters, we consider two special cases as the followings.
(ⅰ) Sensitivity in strongly extrinsic graphene in Coulomb scattering range
In strongly extrinsic p-type graphene with n p
ne , carriers around the Fermi level have an
energy independent mobility µ(E)=Const, if considering Coulomb scattering as the dominant
scattering mechanism, which is usually true in most of the as-fabricated graphene devices on
SiO2.S7. Then equation (S2) and (S3) are simplified to
 B  (enh  ) 1 ,
(S11)
RH  (enh ) 1 .
(S12)
Now we have  B   B 0   0 , and VH  (enh ) IB . This means absence of the physical
1
1
magnetoresistance effects but only the Hall electric field exists. As a result, the current and voltage
related sensitivities are respectively given by
11
S I  1/ enh ,
(S13)
SV  (W / L)  .
(S14)
Thus, the output Hall voltage of the GHE, either under constant current or voltage operation mode,
increases with magnetic field in an exactly linear relationship, suggesting that graphene has its
potential in building ultra-linear Hall element, which may be important in precise measurement
applications.S8
It should be emphasized that the linearity of GHE at constant voltage operation mode keeps
well in a wide magnetic field range up to 1 T, which is hard to be realized in Hall elements made
from conventional semiconductors. In conventional Hall elements, physical magnetoresistance
effects usually exists due to the mobility spread around Fermi energy and is usually in
proportional to the square of magnetic field as (  B   B 0 ) /  B 0 ~ (   B) , where
2
 ~   3  /   3   2  2 /   4   describes the mobility spread over energy and τ is the
1/2
momentum relaxation time.S2 As a result, the non-linearity(NL) in conventional Hall elements at
constant voltage mode quickly increases with magnetic field following NL ~ (  B) . For
2
example, in a typical semiconductor in which   /   ~ 1.8 and   /   ~ 1.1 , µ~0.4
3
3
2
2
m2/Vs and B~1 T, NL can reach up to approximately 10 %.S2 While scattering factor
 3  /   3 and the Hall scattering factor rH   2  /   2 is approaching unit in a strongly
extrinsic graphene with Coulomb scattering being the main scattering mechanism, since the
momentum relaxation time τ is independent on carrier energy.S3
(ⅱ) Sensitivity in mixed-conductivity case
Considering both holes and electrons contribute to the conductivity, and all the holes and electrons
have the same mobility µh and µe respectively, then Equations (S2) and (S3) can be simplified as

   h    e  2  
h
e
2
B e
 B   B   B  1   B hh
 B 
e
   B   B 
 

1
(S15)
12
RH 
 Bh h   Be e

h
B


e 2
B
   h    e  2 
2
B e
1   B hh
 B 
e
   B   B 

.
If the magnetic field is close to zero, we will have  B ( B
RH ( B
(S16)
0)  (enh h  ene e ) 1 , and
nh h2  ne e2
0) 
. Similarly, the current and voltage related sensitivities are
e(nh h  ne e )2
respectively calculated with,
nh h2  ne e2
,
e(nh h  ne e )2
SI ( B
0) 
SV ( B
nh h2  ne e2
0)  (W / L)
nh h  ne e
(S17)
.
(S18)
Near the Dirac point voltage, electron-hole puddles prevails in real graphene devices. Suppose
nh  ne and h  e , thus we have  B 
1   2 B2
, and RH  0 ,  H  0 . Therefore near
 0h   0e
the Dirac point in real graphene devices with electron-hole puddles, the Hall voltage and Hall
angle
 H are both absent, and only the magnetoresistance effect exists.
3. Geometry considerations for a real Hall element with finite size
Figure S9 Diagram of a cross-shaped graphene Hall element. Current flows from
electrode C1 to electrode C2, and the Hall voltage generated under magnetic field B is
detected between electrodes C3 and C4. The distances between C1 and C2 is L, and
similarly L’ represents the distance between C3 and C4.
For a cross-shaped thin film Hall device with only one type of carriers, SI and SV can be expressed
13
asS2,S9, S10
SI  G  (en) 1
,
SV  G  W / L .
(S19)
(S20)
Here, n and µ are the sheet carrier density and Hall mobility respectively, W/L is the
width-to-length ratio of the rectangular activation region, and G is the geometry correction factor.
For a symmetrical cross-shaped graphene Hall element with W = W’ (see figure S9) and L = L’, G
approaches unit if L/W > 2.S10 That is, at such dimensional constrains, the cross-shape Hall plate
may be practically equal to an infinite long Hall plate discussed above. Here we use L (or L’) to
denote the distance between C1 & C2 (or C3 & C4). Decrease of W’ will further make the
geometrical factor G even closer to unit,S9,S10 which is exactly the situation in our GHEs with L /
W = 5 µm/ 2 µm and L’/W’ = 5 µm / 1 µm. As a result, the current related and voltage related
sensitivities in our cross-shaped GHE can be respectively given by
SI  1/ en ,
(S21)
SV  W / L .
(S22)
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14
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