Charging of graphene by magnetic field and mechanical effect of magnetic oscillations
We discuss the fact that quantum capacitance of graphene-based devices leads to variation of graphene charge density under changes of external magnetic field. The charge is conserved, but redistributes to substrate or other graphene sheet. We derive exact analytic expression for charge redistribution in the case of ideal graphene in strong magnetic field. When we account for impurities and temperature, the effect decreases and the formulae reduce to standard quantum capacitance expressions. The importance of quantum capacitance for potential Casimir force experiments is emphasized and the corresponding corrections are worked out.
pacs:75.70.Ak , 73.22.Pr, 12.20.Ds
Introduction. Graphene is a novel two-dimensional material having unique mechanical and electronic properties. The uniqueness of any two-dimensional material is that it’s electronic properties can be easily tuned by doping or gating. On top of that, graphene is the strongest known Quantum Hall material due to a sharp conical tip of it’s linear dispersion near the Dirac point. Magnetic oscillations can be noticed even at room temperature for magnetic field T (1); (2).
The electronic properties of materials show up in the Casimir effect. There was a flurry of recent theoretical activity devoted to computation of Casimir effect for graphene (3); (4); (5); (6); (7), with several controversies still unresolved, and so it is important to perform experiments to verify these computations. It is hard to make a mechanical measurement of Casimir force in graphene due to it’s two-dimensional nature and since it is almost always electrically charged. The electrostatic force is much stronger than the Casimir force, so, one needs to subtract the electrostatic contribution to single out the fluctuation-induced Casimir force. The first experiments have started to appear only recently (8). Since graphene exhibits a strong Quantum Hall Effect (QHE) it would be of interest to repeat the experiments (8) with strong transverse magnetic field.
A method for subtracting the (clearly dominant) electrostatic force (9) was used in (8): the electrostatic force depends on the gate voltage as , where is a residual graphene voltage due to charged impurities and chemical potential difference with substrate. The formula has allowed the authors of (8) to find the gate voltage where electrostatic force is fully compensated. The above formula does not include the quantum capacitance contribution and, which is included by adding the “quantum capacitor” in a series connection: ( is a charge density, is geometric capacitance per unit area and , where is a density of states (10); (11); (2) is quantum capacitance). Then the electric pressure is , so we get
where , is a Fermi-Dirac distribution, is a density of states for graphene. The chemical potential for graphene depends on the applied voltage and the chemical doping may give a constant shift: . For ideal graphene and so gives a singular contribution near the Dirac cone () at small temperatures. Due to charge puddles in realistic graphene on substrate, the inverse density of states becomes smooth in the vicinity of the Dirac point (2), hence it gives a weakly -dependent quantum capacitance of order , thus the simple fit should work well for small intervals of .
The story gets more interesting with magnetic field. The Casimir force for this case was estimated in Ref.(7), where pronounced dependence on the magnetic field and the chemical potential was shown. Thus it makes sense to scan a wider range of chemical potentials in the experiment. The magnetic field does also influence the electrostatic force, since the charge of ideal graphene is a step-like function of chemical potential with size of the step depending on the magnetic field value. Thus, even if we consider a suspended graphene with only chemical doping, its charge will oscillate when changing magnetic field. The discussion of electrostatic contribution in magnetic field and quantum capacitance effect is the aim of this note.
Below we consider three examples:
Graphene suspended over the wide trench etched in a metallic substrate, or, alternatively, it can be suspended by leaning on crests.
Two sheets of graphene forming a capacitor with fixed voltage applied (such geometry was discussed in Ref.(7) and argued to have a possibility of repulsive Casimir force)
Graphene laying on the insulator-coated semiconductor with given gate voltage and with grounded parallel metallic plate (or sphere) hanging at the distance over graphene (actually, it is attached to vibrating cantilever of atomic-force microscope). Such geometry was used in the recent experiment (8).
With magnetic field the energy levels of conduction band of graphene are where and is an integer, with degeneracy per unit area, where and , accounts for spin and valley degeneracy.
When the levels are quantized, only the levels below the chemical potential would be filled. For undoped graphene one would have half-filled zero LL, this serves as a reference point for summation of formally infinite spectrum of “Dirac sea”. For generic the charge density is quantized and given by
where denotes the integer part (Floor).
Consider the case of graphene suspended over the etched trench of depth in a metallic substrate. In this geometry graphene is connected to a conductor. Another example to which the same computation applies is a piece of pyrographite from which a large graphene flake has exfoliated.
Consider graphene having the chemical potential for mobile carriers with density at zero magnetic field. These are related as
When the magnetic field is switched on, the electronic structure of graphene changes much stronger than the one for the other materials involved, so, we consider the effect of magnetic field only on graphene and thus the chemical potential of the conductor in the bottom of the trench is fixed (here we neglect the electric penetration depth for the conductor). Since the magnetic field may induce changes of the carrier number of graphene, , this creates an extra electric field which shifts the chemical potential of graphene by , so we solve
where depends on and is a capacitance per unit area: is the distance between the plates of the capacitor and is a dielectric permittivity ( for the vacuum). Using Eq.(2) we get the equation:
which has a simple solution in the limit :
where gives the fractional part. The exact solution is also straightforward. This result shows how the charge of graphene oscillates when the magnetic field is changed, see the dotted curve in Fig.1. The corresponding force oscillation follows from .
It is clear that temperature and disorder would reduce the effect we discuss. For the case of very clean suspended graphene we expect the disorder to be weak and choose a simplified model of equal-shape broadening of all the Landau levels. It is clear that the actual result would depend mostly of the shape of the level that is nearest to , so, it’s the width of that level that we should take as our broadening. The broadening is computationally equivalent to smearing of the chemical potential, see Fig. 1.
The effect we discuss is another manifestation of integer Quantum Hall Effect. Qualitatively, if the last filled Landau level (LL) is less than half-filled, then the chemical potential is higher than the one without magnetic field, so, graphene wants to get rid of carriers and gets positively charged; the opposite happens for more-than-half filled level. This also shows that the upper bound for magnetic charging of graphene is half the population of one LL: , this bound is never achieved due to non-infinite geometric capacitance and level broadening. For example, with nm and we get in the denominator: , which is a typical quantum to geometric capacitance ratio for graphene experiments on thin insulator layers.
The magnetic oscillations of charge have a mechanical effect, creating the attraction between the plates of charged capacitor. We see that a typical variation of electron density could be of order , which translates into the electric pressure
This pressure is of the same order of magnitude as the Casimir pressure at distance nm between the plates ( estimated to be roughly . Note that the electrostatic force from the magnetic charging effect falls off as due to linearly decreasing capacity, while the Casimir force falls off as for small temperatures, so these effects are comparable.
Let us see if it is feasible to measure the effect. Consider a trench of width nm. Then the membrane has a parabolic form with tension and the central deflection is where is a 2D Young modulus (note that it is possible that for small deformations the Hookes law is invalid for graphene due to microscopic out of plane buckled form of graphene (13); (14), thus, for small deformations the effective Young modulus could be lower). For nm trench we get , which can be measured in STM or in Bragg diffraction experiments. For wider trenches the deflection grows as .
Having the possibility to measure the electric attraction, one can apply voltage to graphene and tune it to minimize the attraction, analogously to Ref.(8).
Now consider a geometry of two parallel sheets of graphene, that are electrically connected. This may be imagined as a drum made of two grapehene sheets. Let these sheets be doped to and without magnetic field and have carrier densities . An interest in such type of geometry stems from the prediction of possible repulsive Casimir force in magnetic field when and are of opposite signs (7). In magnetic field we assume carrier density redistribution and solve:
together with Eq.(2) for both graphene sheets. The solution in the approximation is
where . Note that the effect of magnetic oscillations cancels out if the two graphene sheets are at equal chemical potentials (and are of equal quality).
Now we turn to a much more flexible experimental setup used in (8) and discuss gated graphene laying on the insulator-coated semiconductor with a grounded parallel metal plate (or sphere) hanging over it, the upper plate is an atomic force microscope used in frequency-shift regime (15); (16); (8). The presence of substrate and a larger distance to the metal plate (of order 300 nm) makes the quantum capacitance effects much weaker, but these are still important to improve precision. Remarkably, this experimental setup allows for an excellent direct mechanical measurement of magnetic oscillations together with QHE.
Now we assume only weak magnetic oscillations, so the density of states is a smooth function and it is convenient to reformulate the solution in terms of continuum density of states: we have a series connection of two capacitors: the standard geometric one with (per unit area) and a quantum one with , where is a density of states (10); (11); (2). So, the total capacitance is . We see that the relative effect of quantum capacitance decreases as due to decreasing of , so, for fixed voltage its contribution to force decreases as , which is small, but may still compete with the Casimir force that behaves as at low temperatures (7).
To study magnetic oscillations and Casimir effect at strong magnetic field one needs to extend the experiment of Ref.(8) by extra bottom-gating, so that a wide range of Landau level filling factors could be scanned.
For the electrostatic force acting on the unit area of graphene we may use Eq.(1) and follow the model of Ref. (2) to get the density of states. The model consists of Lorentz and temperature level broadening superimposed on the Gaussian carrier number broadening due to charge puddles, see Fig. 2.
With the improved fit the value of residual potential difference can be mechanically measured with fabulous precision. is the potential difference between graphene and a metal plate when there is no electric field between them, so, it equals to graphene chemical potential:
The electron doping of graphene is a linear function of bottom gate voltage (one can also easily write the quantum capacitance correction, but it is small for relatively thick insulator layer): . So, the experiment allows for precise measurement of both and . Knowing this for the particular sample is also helpful for theoretical refinement of Casimir force computations. Importantly, the known can be plugged back into Eq.(1) () to improve the fit and hence the precision. To get a more pronounced Quantum Hall physics, the above experiment may be repeated with gated graphene suspended over thin layer of insulator. Then one may hope to get a strong evidence for interaction effects.
To conclude, we have elaborated on the two possible experimental schemes to measure the Casimir effect for graphene with magnetic field. As a by-product, we note that the newly-developed mechanical method (15); (16); (8) may lead to the precise measurement of density of states if sample is additionally gated.
Acknowledgements: I am grateful to Pablo Rodriguez-Lopez, Ignat Fialkovsky, Galina Klimchitskaya and Feo Kusmartsev for useful discussions. This work has been supported by EPSRC through the grant EP/l02669X/1.
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