Supplementary Information â Ballistic transport in graphene antidot ...
Supplementary Information – Ballistic transport in graphene antidot lattices Andreas Sandner,† Tobias Preis,† Christian Schell,† Paula Giudici ,†,¶ Kenji Watanabe,‡ Takashi Taniguchi,‡ Dieter Weiss,† and Jonathan Eroms∗,† †Institute of Experimental and Applied Physics, University of Regensburg, D-93040 Regensburg, Germany ‡National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan ¶Present address: Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Buenos Aires, Argentina E-mail: [email protected] Fax: +49 941 943-3196
Fabrication Details Single crystalline hexagonal boron nitride (hBN) S1,S2 was exfoliated onto a stack of PMGI and PMMA polymers spin coated on an oxidized Si-Wafer. S3 Suitable hBN flakes, serving later as the top hBN layer, were located in an optical microscope by using different bandpass filters. The PMGI sacrificial layer was dissolved in photoresist developer and DI water, leaving the PMMA with the hBN floating on DI water. Then the PMMA film was transferred to a microscope slide into which a hole had been cut. S3 Single layer graphene was exfoliated from HOPG (Momentive Performance, ZYA grade) onto oxidized Si, and picked up by vander-Waals interaction using the first hBN flake. S4 Using a home-made setup in an optical microscope, this stack was transferred to a second hBN flake, residing on an oxidized Si S1
substrate with prepatterned markers. Subsequently, the finished stacks were annealed in forming gas flow at 320◦ C for several hours. The hBN/graphene/hBN heterostructure was then patterned into Hall bar shape using electron beam lithography (EBL) and CHF3 /O2 (40 sccm/6 sccm, 60 Watt power) based reactive ion etching (RIE). S4 Cr/Au side contacts were defined with EBL and deposited by thermal evaporation and lift-off after brief oxygen plasma cleaning of the contact areas. S4 Finally, the antidot lattice was defined in a separate EBL and RIE step. Samples were glued into chip carriers with silver filled epoxy to contact the back gate, wire-bonded and measured in a helium cryostat with variable temperature insert, using low frequency lock in techniques with a bias current of 10 nA.
Determination of the intrinsic mean free path Ishizaka and Ando S5 discuss the total mean free path ltot in an antidot sample with intrinsic mean free path le and a square antidot lattice with period a and antidot diameter d. For hard-wall antidots, they calculate the scattering cross section of a single antidot and evaluate the scattering length lsc due to scattering at the antidot lattice only, at B = 0: lsc = 3a2 /(4d). Using Matthiessen’s rule the total mean free path is therefore: 1 ltot
=
1 1 + lsc le
(S1)
We estimate the antidot diameter to be d = 25 . . . 30 nm. For the sample in Fig. 2 in the main text (a = 200 nm) we therefore obtain: lsc = 1000 . . . 1200 nm. Together with the measured total mean free path of ltot = 620 nm we obtain about le = 1300 . . . 1600 nm. For the sample in Fig. 3 (a = 100 nm) we obtain lsc = 250 . . . 300 nm, and together with ltot = 160 nm we estimate le = 340 . . . 440 nm. Ishizaka and Ando also studied the influence of the intrinsic mean free path on the visibility of the antidot peaks in a classical model. S6 They only plotted the components σxx and σxy of the conductivity tensor, while the experiment yields the resistivity components. S2
They show data for very narrow antidots (d/a = 0.15) in the field range that includes the orbits around 4 antidots, as in our experiments. They show data for three different mean free paths, le = 1.9, 3.3 and 6.7 a. Taking the data from their published figure we could recalculate the magnetoresistance in the given situation, see Fig. S1. Comparing this to our data of Fig. 3 (a = 100 nm, d = 25 . . . 30 nm), we notice that the calculated data was produced for much smaller antidots. We estimate that the experimental n = 4 peak shows more clearly than the calculated peak for l = 3.3 a, and therefore we estimate the intrinsic mean free path to be at least le = 4 a = 400 nm, which compares well with the above estimate. A second calculation in their article was done for d/a = 0.3, unfortunately the field range only included the main antidot peak. We also note that due to the details of the potential (e.g. steepness of the potential walls) a 1 : 1 comparison is not possible. Furthermore, also the sample with a = 100 nm shows the symmetry-broken ν = 1 quantum Hall state in a gate sweep at B = 14 T (see Fig. S2). This again clearly shows the high intrinsic electron mobility in this sample.
Antidot diameter In addition to the estimates presented in the main text, we also discuss the comparison between experimental and theoretical data. In Ref., S6 various antidot diameters were considered. Comparing our data of Fig. 3 in the main text to the calculations, a reasonably good match is achieved between d/a = 0.2 and d/a = 0.3, again confirming the estimated antidot diameter of d = 25 nm. In addition, Ishizaka and Ando also reported on the comparison between simulations based on classical Kubo formula and a quantum model. S7 They find that when the Fermi wavelength λF is of the order of the antidot diameter d, the scattering cross section of the antidots is enhanced. Therefore, the calculated curves in the quantum simulation match the classical curves for a slightly larger antidot diameter. We find a similar behavior in our
S3
0.08
l l l
0.06
e
e
e
= 1.9 a = 3.3 a = 6.7 a
xx
2
(h/e )
expt
0.04
0.02
0.5
1.0
1.5
2.0
2.5
3.0
3.5
a/R
C
Figure S1: Resistivity curves were calculated from the conductivity data taken from Ref. S6 Those curves were calculated for d/a = 0.15 (smaller than in experiment) and for the intrinsic mean free paths given in the legend. The experimental data is included in pink.
S4
100
20 T = 1.4K
R
xx
B = +14T
R
xy
10
0
(k
25
R
R
xx
(k
)
= 1
)
50
xy
15
75
-25
= 2
5
-50 -75
0
-100 -40
-30
-20
-10
0
V
g
10
20
30
40
(V)
Figure S2: A back gate sweep at 14 Tesla, taken on a sample with a = 100 nm. The symmetry-broken ν = 1 quantum Hall state is clearly visible in the Rxx and Rxy graphs.
S5
data in a sample with a = 100 nm (see Fig. S3). At the two higher densities, the antidot peak corresponding to an orbit around 2 antidots is clearly visible. This is only possible in a lattice with small antidots. For lower densities, this feature disappears, and also the feature around a/RC ≤ 1 moves to lower fields and loses its structure. This all is compatible with an increase of the effective antidot diameter, as expected in a quantum model, when λF approaches the antidot diameter.
6000 12
n
S
= 0.4 x 10
4000
12
n
= 0.8 x 10
-2
cm
R
xx
(
)
S
a = 100 nm
-2
cm
2000
n
S
n
S
12
= 2.0 x 10
12
= 3.0 x 10
n = 2
-2
cm
-2
cm
0
-3
-2
-1
0
a/R
1
2
3
C
Figure S3: Data for a sample with a = 100 nm at various low densities. The x-axis is normalized to the cyclotron radius at the given density. While in the high density data the feature for an orbit around 2 antidots is clearly visible, as expected for a system with small antidots (d/a ≤ 0.3), this feature disappears in the low density data.
S6
Additional data Here we present additional magnetoresistance curves for samples with a = 100 nm, a = 50 nm and a = 250 nm. In Fig. S4, we show data for a sample with a = 100 nm which shows a clear transition from the classical to the quantum regime at a density of nS = 2.2 × 1011 cm−2 (λF = 75 nm).
11
n
= 1.4x10
n
= 2.2x10
S
20
11 S
-2
cm
T = 1.5 K
15
-2
cm
R
xx
(k
)
n = 1
15
10 -4
10
WL
-2
0
2
4
B (T)
Figure S4: Data for a sample with a = 100 nm at two low densities. In the lower density curve, there is no antidot peak visible at the expected position. The pronounced peaks at higher fields are Shubnikov-de Haas oscillations. For higher densities, the antidot peak is clearly visible. The red box marks the region where weak localization is observed. Data from a sample with a = 50 nm is shown in Fig. S5. The density here is nS = 2.5 × 1012 cm−2 , with λF = 22 nm, which is just at the transition to the classical regime. Therefore, the antidot peak (marked with an arrow) is very weak. Due to back gate leakage, S7
we could not explore higher densities in this sample. Finally, we also fabricated a sample with a lattice period a = 250 nm (see Fig. S6). In this sample, we could only observe the fundamental antidot peak. Unlike the other samples in this study, this sample also had a moir´e superlattice from the interaction between the hBN substrate and the graphene lattice, S8,S9 whichs probably limited the mobility in this case.
1.4 K 80 K
2 R
C
= a
60
R
xx
(k
)
80
40
-10
-5
0
5
10
B (T)
Figure S5: Magnetoresistance of a sample with a = 50 nm at T = 1.4 K and T = 80 K. The density was nS = 2.5×1012 cm−2 . The arrow marks the expected position of the fundamental antidot peak.
S8
400
1
200
R
xx
(
)
300
100
T = 1.4 K 12
n
s
0 -1.0
= 1.2
-0.5
10
-2
cm
0.0
0.5
1.0
1.5
2.0
B (T)
Figure S6: Data from a sample with a = 250 nm. The arrow marks the position of the main antidot peak, corresponding to an orbit around one antidot.
S9
Validity of the classical picture The classical limit of quantum system applies when the quantum mechanical wavelength is much shorter than a characteristic length scale of the problem. In our case, the wavelength is the Fermi wavelength λF . Ando’s group considered explicitly λF ≈ 0.3 a, S7 still well away from the breakdown of the classical approximation. They find only minor differences between a classical and a quantum simulation, in agreement with our experimental data at high densities. To our knowledge, they did not cover the regime of λF ≈ a. Rotter et al. performed quantum magnetotransport calculations in two-dimensional superlattices with λF ≈ a, S10 but here the modulation strength was weaker than in the antidot case. Therefore, a direct comparison is not possible. To the best of our knowledge, no quantum transport calculations for antidot lattices in the regime λF ≈ a were published.
References [S1] Kubota, Y.; Watanabe, K.; Tsuda, O.; Taniguchi, T. Science 2007, 317, 932–934. [S2] Taniguchi, T.; Watanabe, K. J. Cryst. Growth 2007, 303, 525 – 529. [S3] Dean, C. R.; Young, A. F.; Meric, I.; Lee, C.; Wang, L.; Sorgenfrei, S.; Watanabe, K.; Taniguchi, T.; Kim, P.; Shepard, K. L.; Hone, J. Nat. Nanotechnol. 2010, 5, 722–726. [S4] Wang, L.; Meric, I.; Huang, P. Y.; Gao, Q.; Gao, Y.; Tran, H.; Taniguchi, T.; Watanabe, K.; Campos, L. M.; Muller, D. A.; Guo, J.; Kim, P.; Hone, J.; Shepard, K. L.; Dean, C. R. Science 2013, 342, 614–617. [S5] Ishizaka, S.; Ando, T. Phys. Low-Dim. Struct. 1999, 5/6, 5–12. [S6] Ishizaka, S.; Ando, T. Phys. Rev. B 1997, 55, 16331–16338. [S7] Ishizaka, S.; Ando, T. Phys. Rev. B 1997, 56, 15195–15201.
S10
[S8] Dean, C. R.; Wang, L.; Maher, P.; Forsythe, C.; Ghahari, F.; Gao, Y.; Katoch, J.; Ishigami, M.; Moon, P.; Koshino, M.; Taniguchi, T.; Watanabe, K.; Shepard, K. L.; Hone, J.; Kim, P. Nature 2013, 497, 598–602. [S9] Ponomarenko, L. A. et al. Nature 2013, 497, 594–597. [S10] Rotter, P.; Suhrke, M.; R¨ossler, U. Phys. Rev. B 1996, 54, 4452–4455.
S11
Fabrication Details Single crystalline hexagonal boron nitride (hBN) S1,S2 was exfoliated onto a stack of PMGI and PMMA polymers spin coated on an oxidized Si-Wafer. S3 Suitable hBN flakes, serving later as the top hBN layer, were located in an optical microscope by using different bandpass filters. The PMGI sacrificial layer was dissolved in photoresist developer and DI water, leaving the PMMA with the hBN floating on DI water. Then the PMMA film was transferred to a microscope slide into which a hole had been cut. S3 Single layer graphene was exfoliated from HOPG (Momentive Performance, ZYA grade) onto oxidized Si, and picked up by vander-Waals interaction using the first hBN flake. S4 Using a home-made setup in an optical microscope, this stack was transferred to a second hBN flake, residing on an oxidized Si S1
substrate with prepatterned markers. Subsequently, the finished stacks were annealed in forming gas flow at 320◦ C for several hours. The hBN/graphene/hBN heterostructure was then patterned into Hall bar shape using electron beam lithography (EBL) and CHF3 /O2 (40 sccm/6 sccm, 60 Watt power) based reactive ion etching (RIE). S4 Cr/Au side contacts were defined with EBL and deposited by thermal evaporation and lift-off after brief oxygen plasma cleaning of the contact areas. S4 Finally, the antidot lattice was defined in a separate EBL and RIE step. Samples were glued into chip carriers with silver filled epoxy to contact the back gate, wire-bonded and measured in a helium cryostat with variable temperature insert, using low frequency lock in techniques with a bias current of 10 nA.
Determination of the intrinsic mean free path Ishizaka and Ando S5 discuss the total mean free path ltot in an antidot sample with intrinsic mean free path le and a square antidot lattice with period a and antidot diameter d. For hard-wall antidots, they calculate the scattering cross section of a single antidot and evaluate the scattering length lsc due to scattering at the antidot lattice only, at B = 0: lsc = 3a2 /(4d). Using Matthiessen’s rule the total mean free path is therefore: 1 ltot
=
1 1 + lsc le
(S1)
We estimate the antidot diameter to be d = 25 . . . 30 nm. For the sample in Fig. 2 in the main text (a = 200 nm) we therefore obtain: lsc = 1000 . . . 1200 nm. Together with the measured total mean free path of ltot = 620 nm we obtain about le = 1300 . . . 1600 nm. For the sample in Fig. 3 (a = 100 nm) we obtain lsc = 250 . . . 300 nm, and together with ltot = 160 nm we estimate le = 340 . . . 440 nm. Ishizaka and Ando also studied the influence of the intrinsic mean free path on the visibility of the antidot peaks in a classical model. S6 They only plotted the components σxx and σxy of the conductivity tensor, while the experiment yields the resistivity components. S2
They show data for very narrow antidots (d/a = 0.15) in the field range that includes the orbits around 4 antidots, as in our experiments. They show data for three different mean free paths, le = 1.9, 3.3 and 6.7 a. Taking the data from their published figure we could recalculate the magnetoresistance in the given situation, see Fig. S1. Comparing this to our data of Fig. 3 (a = 100 nm, d = 25 . . . 30 nm), we notice that the calculated data was produced for much smaller antidots. We estimate that the experimental n = 4 peak shows more clearly than the calculated peak for l = 3.3 a, and therefore we estimate the intrinsic mean free path to be at least le = 4 a = 400 nm, which compares well with the above estimate. A second calculation in their article was done for d/a = 0.3, unfortunately the field range only included the main antidot peak. We also note that due to the details of the potential (e.g. steepness of the potential walls) a 1 : 1 comparison is not possible. Furthermore, also the sample with a = 100 nm shows the symmetry-broken ν = 1 quantum Hall state in a gate sweep at B = 14 T (see Fig. S2). This again clearly shows the high intrinsic electron mobility in this sample.
Antidot diameter In addition to the estimates presented in the main text, we also discuss the comparison between experimental and theoretical data. In Ref., S6 various antidot diameters were considered. Comparing our data of Fig. 3 in the main text to the calculations, a reasonably good match is achieved between d/a = 0.2 and d/a = 0.3, again confirming the estimated antidot diameter of d = 25 nm. In addition, Ishizaka and Ando also reported on the comparison between simulations based on classical Kubo formula and a quantum model. S7 They find that when the Fermi wavelength λF is of the order of the antidot diameter d, the scattering cross section of the antidots is enhanced. Therefore, the calculated curves in the quantum simulation match the classical curves for a slightly larger antidot diameter. We find a similar behavior in our
S3
0.08
l l l
0.06
e
e
e
= 1.9 a = 3.3 a = 6.7 a
xx
2
(h/e )
expt
0.04
0.02
0.5
1.0
1.5
2.0
2.5
3.0
3.5
a/R
C
Figure S1: Resistivity curves were calculated from the conductivity data taken from Ref. S6 Those curves were calculated for d/a = 0.15 (smaller than in experiment) and for the intrinsic mean free paths given in the legend. The experimental data is included in pink.
S4
100
20 T = 1.4K
R
xx
B = +14T
R
xy
10
0
(k
25
R
R
xx
(k
)
= 1
)
50
xy
15
75
-25
= 2
5
-50 -75
0
-100 -40
-30
-20
-10
0
V
g
10
20
30
40
(V)
Figure S2: A back gate sweep at 14 Tesla, taken on a sample with a = 100 nm. The symmetry-broken ν = 1 quantum Hall state is clearly visible in the Rxx and Rxy graphs.
S5
data in a sample with a = 100 nm (see Fig. S3). At the two higher densities, the antidot peak corresponding to an orbit around 2 antidots is clearly visible. This is only possible in a lattice with small antidots. For lower densities, this feature disappears, and also the feature around a/RC ≤ 1 moves to lower fields and loses its structure. This all is compatible with an increase of the effective antidot diameter, as expected in a quantum model, when λF approaches the antidot diameter.
6000 12
n
S
= 0.4 x 10
4000
12
n
= 0.8 x 10
-2
cm
R
xx
(
)
S
a = 100 nm
-2
cm
2000
n
S
n
S
12
= 2.0 x 10
12
= 3.0 x 10
n = 2
-2
cm
-2
cm
0
-3
-2
-1
0
a/R
1
2
3
C
Figure S3: Data for a sample with a = 100 nm at various low densities. The x-axis is normalized to the cyclotron radius at the given density. While in the high density data the feature for an orbit around 2 antidots is clearly visible, as expected for a system with small antidots (d/a ≤ 0.3), this feature disappears in the low density data.
S6
Additional data Here we present additional magnetoresistance curves for samples with a = 100 nm, a = 50 nm and a = 250 nm. In Fig. S4, we show data for a sample with a = 100 nm which shows a clear transition from the classical to the quantum regime at a density of nS = 2.2 × 1011 cm−2 (λF = 75 nm).
11
n
= 1.4x10
n
= 2.2x10
S
20
11 S
-2
cm
T = 1.5 K
15
-2
cm
R
xx
(k
)
n = 1
15
10 -4
10
WL
-2
0
2
4
B (T)
Figure S4: Data for a sample with a = 100 nm at two low densities. In the lower density curve, there is no antidot peak visible at the expected position. The pronounced peaks at higher fields are Shubnikov-de Haas oscillations. For higher densities, the antidot peak is clearly visible. The red box marks the region where weak localization is observed. Data from a sample with a = 50 nm is shown in Fig. S5. The density here is nS = 2.5 × 1012 cm−2 , with λF = 22 nm, which is just at the transition to the classical regime. Therefore, the antidot peak (marked with an arrow) is very weak. Due to back gate leakage, S7
we could not explore higher densities in this sample. Finally, we also fabricated a sample with a lattice period a = 250 nm (see Fig. S6). In this sample, we could only observe the fundamental antidot peak. Unlike the other samples in this study, this sample also had a moir´e superlattice from the interaction between the hBN substrate and the graphene lattice, S8,S9 whichs probably limited the mobility in this case.
1.4 K 80 K
2 R
C
= a
60
R
xx
(k
)
80
40
-10
-5
0
5
10
B (T)
Figure S5: Magnetoresistance of a sample with a = 50 nm at T = 1.4 K and T = 80 K. The density was nS = 2.5×1012 cm−2 . The arrow marks the expected position of the fundamental antidot peak.
S8
400
1
200
R
xx
(
)
300
100
T = 1.4 K 12
n
s
0 -1.0
= 1.2
-0.5
10
-2
cm
0.0
0.5
1.0
1.5
2.0
B (T)
Figure S6: Data from a sample with a = 250 nm. The arrow marks the position of the main antidot peak, corresponding to an orbit around one antidot.
S9
Validity of the classical picture The classical limit of quantum system applies when the quantum mechanical wavelength is much shorter than a characteristic length scale of the problem. In our case, the wavelength is the Fermi wavelength λF . Ando’s group considered explicitly λF ≈ 0.3 a, S7 still well away from the breakdown of the classical approximation. They find only minor differences between a classical and a quantum simulation, in agreement with our experimental data at high densities. To our knowledge, they did not cover the regime of λF ≈ a. Rotter et al. performed quantum magnetotransport calculations in two-dimensional superlattices with λF ≈ a, S10 but here the modulation strength was weaker than in the antidot case. Therefore, a direct comparison is not possible. To the best of our knowledge, no quantum transport calculations for antidot lattices in the regime λF ≈ a were published.
References [S1] Kubota, Y.; Watanabe, K.; Tsuda, O.; Taniguchi, T. Science 2007, 317, 932–934. [S2] Taniguchi, T.; Watanabe, K. J. Cryst. Growth 2007, 303, 525 – 529. [S3] Dean, C. R.; Young, A. F.; Meric, I.; Lee, C.; Wang, L.; Sorgenfrei, S.; Watanabe, K.; Taniguchi, T.; Kim, P.; Shepard, K. L.; Hone, J. Nat. Nanotechnol. 2010, 5, 722–726. [S4] Wang, L.; Meric, I.; Huang, P. Y.; Gao, Q.; Gao, Y.; Tran, H.; Taniguchi, T.; Watanabe, K.; Campos, L. M.; Muller, D. A.; Guo, J.; Kim, P.; Hone, J.; Shepard, K. L.; Dean, C. R. Science 2013, 342, 614–617. [S5] Ishizaka, S.; Ando, T. Phys. Low-Dim. Struct. 1999, 5/6, 5–12. [S6] Ishizaka, S.; Ando, T. Phys. Rev. B 1997, 55, 16331–16338. [S7] Ishizaka, S.; Ando, T. Phys. Rev. B 1997, 56, 15195–15201.
S10
[S8] Dean, C. R.; Wang, L.; Maher, P.; Forsythe, C.; Ghahari, F.; Gao, Y.; Katoch, J.; Ishigami, M.; Moon, P.; Koshino, M.; Taniguchi, T.; Watanabe, K.; Shepard, K. L.; Hone, J.; Kim, P. Nature 2013, 497, 598–602. [S9] Ponomarenko, L. A. et al. Nature 2013, 497, 594–597. [S10] Rotter, P.; Suhrke, M.; R¨ossler, U. Phys. Rev. B 1996, 54, 4452–4455.
S11
