# What is the approximate electrical conductivity $\sigma$ of graphene in S/m or S/cm?

I am trying to find an approximate value of the electrical conductivity $\sigma$ of graphene in units of S/m or S/cm. This table on Wikipedia gives $\sigma$ values for a variety of materials (including references), but I do not see graphene.

In the classic 2004 paper on graphene by Novoselov and Geim (Novoselov et al., Science 2004, 306, 666: available here), I see a plot of $\sigma \text{ (m} \Omega^{-1})$ versus $V_g \text{ (V)}$, where $V_g$ is the gate voltage: Based on that plot, what can we say about the electrical conductivity $\sigma$?

One confusing thing about the above plot is that it is in $\text{m} \Omega^{-1}$. But $1 \text{ S} = 1 \text{ } \Omega^{-1}$, so in the plot is given in units proportional to S, not S/m. Why is this?

• Just to leave a note: normally we discourage people from asking for reference information like this, but since you did some research and weren't able to find the value you're looking for, this is a fine question. – David Z Jul 17 '12 at 17:54

Resistivity is the relevant parameter for three-dimensional materials. Sheet resistance (less commonly called "sheet resistivity") is the relevant parameter for two-dimensional materials, and its inverse is called "sheet conductance" or "sheet conductivity". In the Novoselov paper you cited, they talk about sheet resistance and sheet conductance. Please forgive them for the bad habit of using the words "resistivity" and "conductivity" when they mean "sheet resistance" and "sheet conductance" respectively! (When they first define it they do actually use the word "sheet", but then they start leaving it out to be concise.)

The units of sheet resistance are ohms, and sheet conductance is siemens. (Sometimes the unit of sheet resistance is written $\Omega/\Box$, "ohms per square", so that you don't mistakenly think it's an ordinary resistance.)

For most purposes the sheet conductance or sheet resistance of graphene is a far more relevant parameter than the "bulk" conductivity or resistivity. But if you really have to convert one to the other, you would multiply or divide by the thickness of graphene. [The thickness of graphene is not very well defined ... therefore the bulk conductivity and resistivity are not very well defined either. The sheet resistance, on the other hand, is perfectly well defined.]

• Thanks so much. But in the figure in the question, it appears that sheet conductance ($\sigma$ in units of $(\text{m}\Omega/\square)^{-1}$) depends on the gate voltage $V_g$. If this is true, then what gate voltages do the values in the Wikipedia table correspond to? I would like to somehow compare the conductivity of graphene and, say, copper. According to Wikipedia, the 3-D conductivity of copper is $5.96 \times 10^7 \text{ S/m}$ at $20^\circ \text{C}$. – Andrew Jul 17 '12 at 20:35
• P.S. When I say "I would like to somehow compare the conductivity of graphene and, say, copper," I am assuming that I know the thickness of the graphene in the Novoselov paper. I need to check this in the paper, but it is probably there. Thus, I guess my real question now is: Why doesn't the Wikipedia table list $\sigma$ as a function of $V_g$, as the figure in the Novoselov paper implies? – Andrew Jul 17 '12 at 20:43
• Graphene, like semiconductors, has gate-dependent conductivity. Metals like copper do not. The Novoselov paper you cited was measuring "few-layer graphene", maybe 1 or 2 or 3 or 4 atoms thick (they didn't measure). In more recent papers they always measure the exact thickness. "Graphene" most often these days means "monolayer graphene", i.e. one atom thick, maybe 3 or 4 angstroms. Is 3-angstrom-thick-graphene more conductive than 3-angstrom-thick copper? Well, the latter doesn't exist, so I guess the answer is "yes"... – Steve Byrnes Jul 18 '12 at 3:00
• If I assume the few-layer graphene in the Novoselov paper is actually 3 layer graphene (just a guess), then the graphene is 1nm thick, and with a strong gate its sheet resistance is 200 ohm/square. If we pretended that you could make 1nm thick copper without changing the resistivity, its sheet resistance would be 20 ohm/square. So I guess you could say that copper is 10X more conductive than graphene, at least in this particular paper. But it seems to me that this is comparing apples and oranges. [Note: This comment replaces a previous version I just deleted where I misread the graph scale.] – Steve Byrnes Nov 13 '12 at 20:27

$\text{m}\Omega ^{-1}$, means milli-S, that means the resistivity is in the range of kilo-Ohm. What's the problem?

Apparently, the curve in your post shows very low conductivity compared to Cu.