How to calculate potential difference using Poisson's eq?

[wasong]

In the paper "Controlling the Electronic Structure of Bilayer Graphene", they did 5x10^(18) cm^-3 of nitrogen doping on a 6H-SiC substrate. They assumed a very thick graphite layer and a junction, and calculated the potential difference between the first and second layers of graphene near the substrate through a Schottky barrier of 0.4 eV. The screening length of the graphene layers is 4 angstroms. How can they calculate the potential difference between the two graphene layers? Could you please tell me the calculation process? Thank you.

 

[berkeman]

Welcome to PF. Could you provide a link to that paper please? Thanks.

 

[wasong]

Welcome to PF. Could you provide a link to that paper please? Thanks.

https://www.science.org/doi/10.1126/science.1130681 Thanks.

 

[renormalize]

Mentor, can you help me? This is too difficult.

The paper you referenced states that the potential $U$ is found by solving Poisson's equation:
"The variation of carrier concentration has a marked influence on the band structure as derived from a comparison of the experimental results with the tight-binding calculation (Fig. 4). The Coulomb potential difference U displays a sign change at the electron concentration where the gap closes. It is expected that U increases with an increase of the charge difference in either graphene layer induced by the fields at the respective interfaces. We have calculated the potential of each graphene layer from Poisson’s equation based on the Schottky barrier height of 0.4 eV (15) assuming infinitely thick graphene multilayers, and find that for the as-prepared sample the potential difference between the first and second layers shows reasonable agreement with the Coulomb potential difference U estimated from the size of the gap evaluated in the tight binding model." (emphasis added)
But lacking the details of the charge distribution and boundary conditions assumed by the authors, it's anyone's guess how they arrive at their Fig. 4B:

Nevertheless, you can at least approximate their result by modeling the two layers of graphite as a simple parallel-plate capacitor, for which the voltage $V$ between the plates is given by the well-known formula $V=\frac{Qd}{\varepsilon_{0}A}+V_{0}\,$. Here $Q$ is the charge-per-unit-cell (in $\text{coulombs}$, found from the horizontal axis of Fig. 4B), $\text{d}$ is the separation distance (in $\text{m}$) between the graphite layers, $\varepsilon_{0}$ is the permittivity of vacuum (in $\text{farads per m}$), $A$ is the unit-cell area (in $\text{m}^2$) and $V_0$ is a constant background voltage. Then $U=q_e V$, where $q_e$ is the coulomb-charge of the electron. (Note that this simplified equation predicts a linear relation between $U$ and $Q$ and thus it fails to model the "roll off" in the upper right corner of Fig. 4B.)

 

[wasong]

The paper you referenced states that the potential $U$ is found by solving Poisson's equation:
"The variation of carrier concentration has a marked influence on the band structure as derived from a comparison of the experimental results with the tight-binding calculation (Fig. 4). The Coulomb potential difference U displays a sign change at the electron concentration where the gap closes. It is expected that U increases with an increase of the charge difference in either graphene layer induced by the fields at the respective interfaces. We have calculated the potential of each graphene layer from Poisson’s equation based on the Schottky barrier height of 0.4 eV (15) assuming infinitely thick graphene multilayers, and find that for the as-prepared sample the potential difference between the first and second layers shows reasonable agreement with the Coulomb potential difference U estimated from the size of the gap evaluated in the tight binding model." (emphasis added)
But lacking the details of the charge distribution and boundary conditions assumed by the authors, it's anyone's guess how they arrive at their Fig. 4B:
View attachment 354580
Nevertheless, you can at least approximate their result by modeling the two layers of graphite as a simple parallel-plate capacitor, for which the voltage $V$ between the plates is given by the well-known formula $V=\frac{Qd}{\varepsilon_{0}A}+V_{0}\,$. Here $Q$ is the charge-per-unit-cell (in $\text{coulombs}$, found from the horizontal axis of Fig. 4B), $\text{d}$ is the separation distance (in $\text{m}$) between the graphite layers, $\varepsilon_{0}$ is the permittivity of vacuum (in $\text{farads per m}$), $A$ is the unit-cell area (in $\text{m}^2$) and $V_0$ is a constant background voltage. Then $U=q_e V$, where $q_e$ is the coulomb-charge of the electron. (Note that this simplified equation predicts a linear relation between $U$ and $Q$ and thus it fails to model the "roll off" in the upper right corner of Fig. 4B.)

This paper says that screening length in graphene is 4 anstroms. I think that electric field is decreased at 4 anstroms. I don't understand your answer.

 

[renormalize]

This paper says that screening length in graphene is 4 anstroms. I think that electric field is decreased at 4 anstroms. I don't understand your answer.

My answer describes a simplified model of bilayer graphene that does not incorporate screening. Even so, it's quite accurate for low doping concentrations. Rewriting the parallel-plate capacitor equation from post #4 in a form that approximates the electrostatic potential $U$ of bilayer graphene gives:$$U=\frac{\left(n-n_{0}\right)\,\text{e}^{-}d}{\varepsilon_{r}\,\varepsilon_{0}\,A}\tag{1}$$where the parameters appearing on the right are defined as:

• Permittivity of vacuum: $\varepsilon_0=8.85\times10^{-12}\text{ F/m}$
• Elementary charge: $\text{e}^-=1.60\times10^{-19}\text{ C}$
• Bilayer graphene thickness: $d=3.4\text{ Angstroms}$ (Ohta et al. pg. 5)
• Graphene hex unit-cell area: $A=0.051\text{ nm}^2$ (https://wiki.aalto.fi/display/SSC/Graphene)
• Bilayer graphene dielectric constant: $\varepsilon_r=6$ (https://pubs.rsc.org/en/content/articlepdf/2019/na/c8na00350e)
• Number $n$ of elementary charges per unit-cell: $0.005\le n\le0.035$
• Number of elementary charges per cell yielding zero band-gap (and hence $U=0$): $n_0=0.0125$ (Ohta et al. fig. 3B)

Note that every value is from the literature; i.e., there are no free parameters in eq.(1). So let's see how eq.(1) compares to the model of Ohta et al.:

The capacitor model agrees remarkably well up to about $0.0175\text{ e}^-$ or so. The deviations at higher doping may well be due to ignoring screening. But the only way to know for sure is for you to examine the detailed derivation of the model of Ohta et al., either from some other paper or by contacting the authors themselves.