Non-local Transport in Graphene-on-Chromia

Hamed Vakili Author

07/09/2024 Added

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The monthly meeting of IRG 1 held on July 2, 2024.

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[00:00:00.000]Hamed Vakili: I'm Hamed Vakili, and I will be talking about non-local transponding graphene on chromia.
[00:00:10.120]This work is a mix of theoretical and experimental work, which the experimental collaborators are mentioned here.
[00:00:21.440]And for this, I will mainly focus on the theoretical part, but I will also cover some of the more important experimental results.
[00:00:31.680]Yeah, so starting with the geometry that we are using to understand the non-local transport in graphene,
[00:00:46.760]we start with a six-terminal channel,
[00:00:51.160]which is made of a channel made of graphene,
[00:00:54.000]and we connect six terminals to it, as shown in this figure.
[00:00:57.980]And we can then apply a current to one pair of leads.
[00:01:05.120]And, for example, for this case, leads 2 and 3, we apply some current,
[00:01:11.140]and then we measure the voltage between the leads 4 and 5,
[00:01:16.880]and that voltage will be non-local.
[00:01:20.880]And if divided by the current, that will be our non-local resistance.
[00:01:25.040]Now, there can be different ways that this can be realized in a system.
[00:01:29.760]One is that if the bulk of our channel is conductive,
[00:01:36.420]when the current is going from 2 to 3,
[00:01:39.480]some of the current will stray away from the main path
[00:01:44.500]and will basically reach the 4 and 5 leads.
[00:01:50.600]And that will cause some non-local voltage and resistance,
[00:01:55.840]but this will be very small compared to what is seen in the experiment
[00:02:01.520]and also indicates exponentially with the distance
[00:02:04.460]between these two pair of current and voltage leads.
[00:02:08.520]There are other ways of realizing these large non-local voltages,
[00:02:15.280]which would be based on the edge currents.
[00:02:20.320]There are different ways of realizing these edge currents.
[00:02:23.280]One would be to introduce, for example,
[00:02:26.740]a spin-orbit coupling, or by applying magnetic field,
[00:02:31.580]we can go to the quantum Hall effect regime.
[00:02:33.880]But we need this specific type of edge current.
[00:02:38.600]In particular, we need to have both right-moving
[00:02:43.860]and left-moving channels to realize non-zero,
[00:02:47.380]non-local voltage, for example, for the case of
[00:02:50.040]quantum effect regime, we have zero non-local
[00:02:56.080]voltage because only one of the clockwise or
[00:02:59.700]anti-clockwise edge modes are available to us.
[00:03:02.940]There is also another way of realizing this
[00:03:07.320]non-local, which is by having just rush-wise
[00:03:10.280]spin-orbit coupling. That would also introduce
[00:03:13.800]some kind of non-local voltage, which again,
[00:03:16.740]as we will see, is smaller compared to
[00:03:19.760]what the experimental results indicate.
[00:03:23.520]And they also decay with the distance between the leads.
[00:03:29.320]Now, for our simulation setup, we use the graphene
[00:03:35.280]channel and we attach six contacts to it.
[00:03:38.600]And then we calculate the transmission between
[00:03:42.760]all of these contacts.
[00:03:44.400]And then we use the Landor-Butyker formula
[00:03:49.480]to calculate the voltage of each of the contacts
[00:03:53.920]and then subtract five and six to get the voltage difference.
[00:03:58.320]The details of our Hamiltonian is seen here,
[00:04:02.040]which is made of the normal hopping term for graphene,
[00:04:05.120]but Rochebaugh and Kane-Miles spin orbit coupling.
[00:04:08.080]And then we also introduce impurity terms.
[00:04:11.840]Our Hamiltonian is mainly based on this ab initio paper,
[00:04:16.840]which is, sorry, from here.
[00:04:19.200]There is one previous experimental work,
[00:04:24.200]which the same experimental groups,
[00:04:27.800]which showed that at room temperature,
[00:04:31.600]they could see the Rochebaugh spin orbit coupling
[00:04:35.160]exists in the channel.
[00:04:37.120]Now our impurity models, we look at two impurity models.
[00:04:44.360]One is the correlated impurities,
[00:04:47.040]which the correlation is in the form
[00:04:48.920]of a Gaussian function as shown here.
[00:04:51.760]And another one is with the Anderson disorder,
[00:04:54.480]which they are uncorrelated
[00:04:56.680]and they're just onsite random potential.
[00:05:01.160]And in the case of this correlated disorder,
[00:05:05.240]K0 is what determines the strength of the disorder.
[00:05:10.160]Now to get a feeling about the values
[00:05:14.600]of this non-local resistance, so for example here,
[00:05:18.640]we can see for the quantum spin Hall effect edge modes,
[00:05:21.920]which can be realized by K-Millez spin-orbit coupling,
[00:05:25.520]we have very large non-local resistance.
[00:05:29.640]But for example here, from the S-ray current,
[00:05:33.360]if we turn off the K-Millez
[00:05:36.000]and we don't have any edge current,
[00:05:37.920]we see that it's very much smaller,
[00:05:40.040]the one order of magnitude is smaller
[00:05:41.560]and decays exponentially with the separation
[00:05:45.080]between the leads, which as we will see
[00:05:48.360]this stray current or even the
[00:05:52.240]Rochefort spin orbit coupling alone
[00:05:54.360]cannot explain the experimental results.
[00:05:56.880]When we introduce the correlated disorder model
[00:06:06.440]to our channel, we see that our non-local resistance
[00:06:11.560]from this kind of clean peak changes
[00:06:14.320]into multiple peaks, sharper peaks,
[00:06:18.080]but separated from each other, as shown in this top figure.
[00:06:22.640]And interestingly, we saw that when
[00:06:24.920]we apply the magnetic field, it doesn't
[00:06:27.640]change the fingerprints of the non-local signatures that much.
[00:06:33.320]In comparison, if you use only the Anderson model,
[00:06:38.840]we were not able to really see any separation of the peaks,
[00:06:43.960]although still some kind of peaks were introduced, but it
[00:06:47.800]wasn't similar to the correlated disorder model.
[00:06:52.680]Another point here is that if we increase the magnetic field
[00:06:57.920]enough, we will eventually reach a regime that the non-local
[00:07:04.640]voltage is suppressed.
[00:07:06.800]This would be mainly because with large enough magnetic field,
[00:07:13.800]we will be in the quantum Hall effect regime, and quantum Hall
[00:07:17.520]effect regime because only one of the HMOD channels is available.
[00:07:21.800]Generally, we will have zero non-local resistance.
[00:07:26.400]Now, the reason that we use this very large magnetic field in the
[00:07:30.200]simulation is because our simulation area is significantly
[00:07:35.080]smaller than what's available in the experiment.
[00:07:38.280]So to come to achieve similar magnetic field flux, we need to
[00:07:44.760]increase the magnetic field strength.
[00:07:47.240]Now, to see how this correlated disorder plays this role and how
[00:07:57.000]it affects the edge currents, we looked at, we simplified our
[00:08:02.280]channel a little bit.
[00:08:03.160]We only looked at four, we only attached four terminals and we
[00:08:07.920]looked at the transmissions from this lead number three.
[00:08:13.560]And if we look at the
[00:08:16.960]transmission from lead number three, without inclusion of the
[00:08:21.840]chain mills spin orbit coupling, we see that this
[00:08:25.600]transmission is going through the whole bulk of the channel.
[00:08:30.400]And this would correspond to these black dots.
[00:08:35.290]which show basically zero non-local resistance.
[00:08:39.790]But in comparison, if we introduce the K-Miles spin orbit coupling
[00:08:45.410]and look at this energy point where we have large non-local resistance,
[00:08:52.570]which is shown in this B and C, we see that we have the non-local,
[00:08:58.730]sorry, we have the edge current, although the edge current is modified
[00:09:03.050]due to the presence of the edges, but the transport is mainly dominated by the edge current.
[00:09:08.750]Now B and C, B is for the zero magnetic field, C is for the non-zero magnetic field,
[00:09:15.410]but we see that still the transport, the shape of the current,
[00:09:21.250]the edge current doesn't change that much.
[00:09:23.490]Now in comparison, if we look at some of the points that we have very small non-local resistance,
[00:09:30.030]for example, for this green,
[00:09:32.890]a triangle in D and E,
[00:09:35.650]we see that the edge currents are modified so much that they are basically transporting through the bulk of the channel.
[00:09:44.370]And they are realizing this kind of cross transmission between lead number three and lead number four.
[00:09:52.250]In the six terminal setup, this will decrease the non-local resistance significantly.
[00:10:01.990]We also map the current density to the impurity distribution.
[00:10:05.930]We see that the current generally tries to go through the borders of this region of impurities.
[00:10:15.030]Although there is some smearing of the current, which makes them not completely limited to those areas.
[00:10:23.830]But it maps generally well to the impurity profile.
[00:10:31.090]If we compare the local and non-local conductance, we see that these fingerprints of the local and non-local conductance aren't generally overlapping each other completely.
[00:10:44.090]Although some of them show very good overlaps, like the peaks for example here or here.
[00:10:52.090]So it just shows that this random behavior of the impurities can make it different for local and non-local conductance.
[00:11:00.190]Now if we look at the experimental results, in the experiments they had two set of devices or two devices which they had the non-local measurements for it.
[00:11:17.290]In one of them they managed to get this very large non-local resistance, in the other one it was very small.
[00:11:24.290]And in the case that they saw this very small non-local resistance,
[00:11:29.290]they see that the local current is very large,
[00:11:32.290]and for the case that we have large non-local resistance,
[00:11:36.290]the local current or local conductance will be very small,
[00:11:42.290]which is what we also see in the simulation.
[00:11:46.290]Now, this is also the experimental results,
[00:11:50.290]which show that the non-local resistance survives up to some good temperatures,
[00:11:58.390]and they also see that this fingerprint of the non-local resistance
[00:12:05.390]doesn't really change that much as they increase the temperature.
[00:12:12.390]The general picture of what's going on when we have a small or large non-local resistance
[00:12:20.390]is still as was described that because of the presence of impurities,
[00:12:27.490]we can have this type of cross transmission,
[00:12:30.590]not just between the adjacent leads that would normally be connected through the edge current.
[00:12:38.090]Now, another relevant topic on this presence of the impurities,
[00:12:45.390]we see that in the case of a smooth disorder with graphene,
[00:12:51.690]we have two limits of universal conductance fluctuation.
[00:12:56.590]As we see here, these two limits are due to graphene having two separate cones,
[00:13:04.990]and one limit would be four times larger than the other one.
[00:13:09.290]The first limit is reached when the impurity is weak,
[00:13:12.890]and if we increase the impurity strength enough,
[00:13:17.090]eventually there will be significant cross-cone transmission that the cones basically mix with each other,
[00:13:25.690]so that it goes from one university class to the other one.
[00:13:30.390]Now with these university classes, we can also control them or change them based on introducing of other terms from the Hamiltonian,
[00:13:41.890]for example, the Rochebaugh or Kahn-Mille, they all have different universality class.
[00:13:46.890]And in conclusion, we see that we need Kahn-Mille spin orbit coupling to realize the non-local spin orbit,
[00:13:54.790]the non-local signature in the six-terminal setup for graphene correlated impurity,
[00:14:01.590]as we saw, can modify the edge current to effectively go through the bulk of the channel to realize this kind of cross transmission,
[00:14:10.990]which then decreases or introduces fluctuations in the non-local resistance.
[00:14:16.690]And as we saw, that non-local resistance doesn't change much from the magnetic field.
[00:14:23.190]It shows that the,
[00:14:24.390]it is energy pass that they go is energy dependent.
[00:14:28.690]And alternative to this kind of non-local measurement,
[00:14:33.290]we can also use the universal conductance fluctuations to check
[00:14:37.590]if these other terms of like chemically rush wise or with coupling is present in the system.
[00:14:44.190]Thank you for listening.
[00:14:47.190]Okay. Thank you Ham. Interesting talk.
[00:14:50.290]And let's see if there's any questions
[00:14:53.990]from the audience.
[00:14:56.390]Yeah, can I just ask?
[00:15:07.590]Yeah, one question I have is,
[00:15:15.290]so you showed the leads and this was basically a six lead system and two
[00:15:23.590]where the current is flowing to where the non-local voltage is measured.
[00:15:28.090]What is the physical significance of the other two leads in this image
[00:15:34.090]one and two that are not connected to anything?
[00:15:38.090]So technically, if we remove one and two, we would still have non-local
[00:15:46.190]voltage, just the value of it will be different for this case.
[00:15:53.190]In this case of six terminals, six leads, for the pristine graphene, it will be two-thirds of the quantum resistance.
[00:16:03.190]I don't remember for the four terminal, for the four leads, but it will give a different result.
[00:16:09.190]But another benefit is that it kind of limits the scattering from these edges so that they can go away, basically, do not interrupt.
[00:16:21.990]I see.
[00:16:22.790]Okay.
[00:16:23.290]Any other questions?
[00:16:29.990]Okay.
[00:16:43.790]Okay, that concludes today's meeting.
[00:16:47.290]Let's thank the speaker and also thank you, everyone,
[00:16:52.390]for attending.
[00:16:53.390]And our next meeting will be in August and Ham will be the host.
[00:16:59.790]Yeah.
[00:17:00.590]Thank you.
[00:17:01.290]Yeah.
[00:17:02.090]Thank you.
[00:17:02.690]Bye.
[00:17:03.690]Bye.
[00:17:04.190]Bye.
[00:17:05.190]Bye.
[00:17:06.190]Thank you.

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Non-local Transport in Graphene-on-Chromia

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