Quantum transport of Dirac fermions in selected graphene nanosystems away from the charge-neutrality point

First, let’s look for a concise answer to the question: Where does electrical resistance come from at absolute zero?

In the familiar Drude model of electrical conduction Kit05 electrons are assumed to constantly bounce between heavier, stationary lattice ions, allowing one to express the material specific resistivity as a function of the electron’s effective mass, velocity, and the mean free path. In quantum-mechanical decription of solids, Drude model provides a reasonable approximation as long as the Fermi wavelength remains much shorter than electron’s mean free path and the conductor size.

The picture sketched above changes substantially when electic charge flows through a nanoscopic system, such as quantum point contact in semiconducting heterostructure Wee88 , a carbon nanotube Whi98 , or monoatomic quantum wire Agr03 (see Fig. 2, top part). Assuming for simplicity that such a system has no internal degrees of freedom leading to the degeneracy of quantum states, in other words — that in a sufficiently small energy range $\Delta{}E$ we have at most one quantum state (level) — we note that the time of flight of an electron through the system is limited from below by the time-energy uncertainty relation

$$\Delta{}t\geqslant{}\frac{\hbar}{\Delta{}E}.$$

(1)

Next, by linking the energy range $\Delta{}E$ with the electrochemical potential difference in macroscopic electrodes (reservoirs) connected to the nanoscopic system (see Fig. 2, bottom part), we can write

$$\Delta{}E=\mu_{L}-\mu_{R}=eU,$$

(2)

where $U$ denotes the difference in electrostatic potential on both sides of the system, and is the elementary charge (without sign). Combining the above equations we obtain the limit for the electric current flowing through the system,

$$I=\frac{e}{\Delta{}t}\leqslant{}\frac{e^{2}}{\hbar}U,$$

(3)

which means that the electrical conductivity

$$G=\frac{I}{U}\leqslant{}\frac{e^{2}}{\hbar}.$$

(4)

We thus see that the uncertainty principle of energy and time leads to a finite value of the conductivity, and therefore to a nonzero value of the electrical resistance, of a nanoscopic system. By rigorous derivation, the upper bound in Eq. (4) is replaced by $e^{2}/h$, introducing the Landauer-Sharvin resistance in noninteracting electron systems Sha65 ; Lan57 ; But85 . Obviously, many-body effects may alter this conclussion substantially. For instance, the resistivity of graphene sample may drop below the Landauer-Sharvin bound due to hydrodynamic effects Kum22 . In twisted bilayer graphene, both the interaction-driven insulating and superconducting (i.e., resistance-free) phases were observed Cao18a ; Cao18b ; Fid18 . These issues are, however, beyond the scope of the present wotk.

At a temperature close to absolute zero ($T\rightarrow{}0$) and in the limit of linear response, i.e., the situation in which the electrochemical potential difference also tends to zero ($\mu_{L}-\mu_{R}=eU\rightarrow{}0$), it can be shown that the electrical conductivity of a nanoscopic system is proportional to the sum of transition probabilities for the so-called normal modes in the leads Naz09 ,

$$G=g_{0}\sum_{n}{}T_{n}(E_{F}),$$

(5)

where $g_{0}$ denotes the conductance quantum; namely, $g_{0}=2e^{2}/h$ for systems exhibiting spin degeneracy (for graphene, we have $g_{0}=4e^{2}/h$ due to the additional degeneracy — called valley degeneracy — related to the presence of two nonequivalent Dirac points in the dispersion relation). The probabilities ($T_{n}$) are calculated by solving (exactly or approximately) the corresponding wave equation (Schrödinger or Dirac) for a fixed energy, which, given the assumptions made, can be identified with the Fermi energy $E_{F}$. Importantly, we perform the calculations under the additional assumption that there are so-called waveguides between the macroscopic reservoirs and the nanoscopic system, for which we can provide (for a fixed value of $E_{F}$) solutions in the form of propagating waves, the number of which is $N_{L}$ or $N_{R}$, for the left or right waveguide, respectively (see Fig. 2). We also assume that a wave that reaches the waveguide-reservoir interface is never reflected. It is worth noting that the sum appearing in formula (5) is the trace of the transmission matrix, the value of which does not depend on the choice of the basis; therefore, it can be expected that the result does not depend on how precisely we construct the aforementioned waveguides, which, it is worth emphasising, are an auxiliary construction that usually has no direct physical interpretation. (For the same reason, the result will be the same whether we consider scattering from left to right or in the opposite direction.)

As mentioned above, the details of the calculations (or computer simulations) leading to the determination of the probability values ($T_{n}$) will depend on the geometry of the system under consideration. If waveguides are modelled as strips of fixed width ($W$), at the edges of which the wave function disappears, the normal modes have the form of plane waves planewfoo , for which the longitudinal component of the wave vector ($k_{x}$) is continuous and the normal component ($k_{y}$) is quantized according to the rule

$$k_{y}^{(n)}=\frac{\pi{}n}{W},\ \ \ \ n=1,2,\dots.$$

(6)

The calculations are particularly simple in cases where the central region (marked with a dark square in Fig. 2) differs from the leads only in that it contains an electrostatic potential that depends on the $x$ coordinate (oriented along the main axis of the system), for example in the form of a rectangular barrier. Then the transmission matrix has a diagonal form (no scattering between normal modes occurs), and in special cases, such as the rectangular barrier mentioned above, but also e.g. the parabolic potential considered by Kemble in 1935 Kem35 , it is possible to provide compact analytical formulas.

We will not present the exact results here, but only point out that for solutions obtained by the mode-matching method for the Schrödinger equation, one can write approximately

$$T_{n}=T(k_{y}^{(n)})\approx\begin{cases}1\ \ \text{if }\ k_{y}\leq{}k_{F},\\ 0\ \ \text{if }\ k_{y}>k_{F},\end{cases}$$

(7)

which we write more briefly as $T_{n}\approx\Theta(k_{F}-k_{y}^{(n)})$, with $\Theta(x)$ denoting the Heaviside step function. In Eq. (7) we introduce the wave vector $k_{F}$ corresponding to the Fermi energy $E_{F}$ (assuming for simplicity that the dispersion relation is isotropic) calculated with respect to the top of the potential barrier in the central region. Furthermore, assuming that there are many modes for which $k_{y}^{(n)}<k_{F}$ (which occurs if $k_{F}W\gg{}1$), and therefore the summation in Eq. (5) can be replaced to a good approximation by integration, we obtain — via Eqs. (6) and (7) — the result known in the literature as the Sharvin conductance Sha65

$$G_{\rm Sharvin}\approx{}g_{0}\frac{k_{F}W}{\pi}.$$

(8)

It is worth noting that the reasoning leading to Eq. (8) can be relatively easily applied to the case where the electrostatic potential in the central region is approximately constant and the width of the conducting region is a function of the position along the longitudinal axis ($x$), changing slowly enough that the scattering between normal modes can be neglected. The above-mentioned case is the so-called quantum point contact (QPC), shown schematically in Fig. 2 (top part), which can be realized in semiconductor heterostructures hosting a two-dimensional electron gas (2DEG) Naz09 .

The second quantity, besides electrical conductivity, that characterizes nanoscopic systems at temperatures close to absolute zero is the shot-noise power. For the sake of brevity, let us point out the basic facts: First, the electric charge $Q$ flowing through the system shown schematically in Fig. 2 (lower part) in a short time interval $\Delta{}t$ is a random variable. Second, the expectation value of such a variable is closely related to the electrical conductivity $G$ in the linear-response limit,

$$\langle{}Q\rangle=GU\Delta{}t\ \ \ \ (U\rightarrow{}0).$$

(9)

The reason the measured value of $Q$ fluctuates at successive time intervals is due to the discrete (granular) nature of the electric charge.

Assuming (for the moment) that electrons jump from one reservoir to another completely independently, we conclude that the charge flow is a Poisson process, or more precisely, that the quantity $Q/e$ follows the Poisson distribution; the variance is therefore proportional to the expectation value given by Eq. (9),

$$\left\langle{}Q^{2}-\langle{}Q\rangle^{2}\right\rangle_{\rm Poisson}=e\langle{}Q\rangle=eU\Delta{}tg_{0}\sum_{n}{}T_{n}.$$

(10)

More generally, $m$-th central moment can be written as

$$\langle\langle{}Q^{m}\rangle\rangle_{\rm Poisson}\equiv\left\langle{}\left(Q-\langle{}Q\rangle\right)^{m}\right\rangle_{\rm Poisson}=e^{m-1}\langle{}Q\rangle,$$

(11)

with the integer $m\geqslant{}1$.

The Fano factor, quantifying the shot-noise power, is defined as the ratio of the actual measured variance of the charge flowing through the system to the variance given by Eq. (10), or more precisely

$$F=\frac{\left\langle{}Q^{2}-\langle{}Q\rangle^{2}\right\rangle}{\left\langle{}Q^{2}-\langle{}Q\rangle^{2}\right\rangle_{\rm Poisson}}=1-\frac{\sum_{n}{}T_{n}^{2}}{\sum_{n}{}T_{n}}.$$

(12)

(For compact derivation, see e.g. Ref. Naz09 .) In the following, we have limited our considerations to long time intervals such that $eU\Delta{}t\gg{}\hbar$; hence $F$ characterizes the zero-frequency noise, not to be confused with the celebrated $1/f$ noise in electronic systems Tak93 . A generalization of Eq. (12) for finite times (and nonzero temperatures) is also possible Sch07 .

In particular, it follows from Eq. (12) that the Poisson limit ($F\rightarrow{}1$) is realized in the case of a tunnel junction, for which we have $T_{n}\ll{}1$ for each $n$. This is a completely different case than the ballistic system considered above, which exhibits Sharvin conductance; then, replacing the summation with integration as before and using the approximation given by Eq. (7), we obtain

$$F_{\rm Sharvin}\approx{}1-\frac{\int{}dk_{y}\left({\Theta}(k_{F}-k_{y})\right)^{2}}{\int{}dk_{y}{}\,{\Theta}(k_{F}-k_{y})}\approx{}0.$$

(13)

In general, for fermionic systems we always have $0<F<1$; the factor $1-T_{n}$ appearing in the numerator in Eq. (12) is a consequence of the Pauli exclusion principle. In the case of the idealized ballistic system we have $F=0$, see Eq. (13), which means that the electron count ($Q/e$) does not fluctuate with time. One could say that the electrons avoid each other so much that they "march" at equal intervals. (Of course, this is only possible at absolute zero temperature, otherwise additional thermal noise appears, i.e. the Nyquist-Johnson noise proportional to the conductivity value, whose influence we have ignored here; see Ref. Naz09 .)

In an attempt to determine higher charge cumulants, it is convenient to introduce characteristic function

$$\Lambda(\chi)=\left\langle\,\exp(i\chi{Q}/e)\,\right\rangle,$$

(14)

such that

$$\langle\langle{}Q^{m}\rangle\rangle\equiv\left\langle{}\left(Q-\langle{}Q\rangle\right)^{m}\right\rangle=e^{m}\left.\frac{\partial^{m}{}\ln\Lambda(\chi)}{\partial(i\chi)^{m}}\right|_{\chi=0}.$$

(15)

Assuming $U>0$ for simplicity, we arrive at the Levitov-Lesovik formula Naz09 ; Sch07

$$\ln\Lambda(\chi)=\frac{g_{0}U\Delta{t}}{e}\sum_{n}\ln\left[1+T_{n}\left(e^{i\chi}-1\right)\right],$$

(16)

expressing the full counting for noninteracting fermions.

Substitution of the above into Eq. (15) with $m=1$ and $m=2$ reproduces (respectively) Eqs. (5) and (12). Analogously, for $m=3$ and $m=4$, we get

	$\displaystyle R_{3}$	$\displaystyle\equiv\frac{\langle\langle{Q^{3}}\rangle\rangle}{\langle\langle{Q^{3}}\rangle\rangle_{\rm Poisson}}=\left(\sum_{n}{}T_{n}-3\,\sum_{n}{}T_{n}^{2}+2\,\sum_{n}{}T_{n}^{3}\right)\Big{/}\sum_{n}{}T_{n},$		(17)
	$\displaystyle R_{4}$	$\displaystyle\equiv\frac{\langle\langle{Q^{4}}\rangle\rangle}{\langle\langle{Q^{4}}\rangle\rangle_{\rm Poisson}}=\left(\sum_{n}{}T_{n}-7\,\sum_{n}{}T_{n}^{2}+12\,\sum_{n}{}T_{n}^{3}-6\,\sum_{n}{}T_{n}^{4}\right)\Big{/}\sum_{n}{}T_{n}.$		(18)

For the Sharvin regime, see Eq. (7),

$$\left(R_{3}\right)_{\rm Sharvin}\approx\left(R_{4}\right)_{\rm Sharvin}\approx 0.$$

(19)

Using the introductory information gathered above, we will now calculate — with some additional simplifying assumptions — the electrical conductivity as well as the higher charge cimulants of a graphene strip. The effective wave equation for itinerant electrons in this two-dimensional crystal is the Dirac-Weyl equation, the detailed derivation of which can be found, e.g., in Katsnelson’s textbook Kat20 , and which can be written in the form

$$\left[v_{F}\,{\boldsymbol{p}}\cdot{\boldsymbol{\sigma}}+V(x)\right]\Psi=E\Psi,$$

(20)

where the energy-independent Fermi velocity is given by $v_{F}=\sqrt{3}t_{0}a/(2\hbar)$, where $t_{0}\approx{}2.7\,{\rm eV}$ denotes the nearest-neighbor hopping integral in the graphene plane and $a=0.246\,{\rm nm}$ is the lattice constant (as a result, the approximate value of $v_{F}$ is about $10^{6}\,{\rm m/s}$, which is several times lower than typical Fermi velocities in metals). The remaining symbols in Eq. (20) are the quantum mechanical momentum operator ${\boldsymbol{p}}=-i\hbar\left(\partial_{x},\partial_{y}\right)$ (the notation $\partial_{j}$ here means differentiation with respect to the selected coordinate, $j=x,y$), ${\boldsymbol{\sigma}}=\left(\sigma_{x},\sigma_{y}\right)$ is a vector composed of Pauli matrices valleyfoo , and the electrostatic potential energy $V(x)$ is assumed to depend only on the position along the principal axis of the system.

The above assumptions imply that we can look for solutions to Eq. (20) in the form of a two-component (i.e., spinor) wave function

$$\Psi=\begin{pmatrix}\phi_{a}\\ \phi_{b}\end{pmatrix}e^{ik_{y}{}y},$$

(21)

where $\phi_{a}$ and $\phi_{b}$ are functions of $x$. By substituting the above ansatz into Eq. (20) we obtain a system of ordinary differential equations

	$\displaystyle\phi_{a}^{\prime}$	$\displaystyle=k_{y}\phi_{a}+i\frac{E-V(x)}{\hbar{}v_{F}}\phi_{b},$		(22)
	$\displaystyle\phi_{b}^{\prime}$	$\displaystyle=i\frac{E-V(x)}{\hbar{}v_{F}}\phi_{a}-k_{y}\phi_{b},$		(23)

where the primes on the left-hand side denote derivatives with respect to $x$. We see that in the system of Eqs. (22), (23) the quantities $k_{y}$ and $E$ play the role of parameters on which the solutions depend (in the following, when calculating, among others, the electrical conductivity, we will identify the electron’s energy with the Fermi energy by setting $E=E_{F}$).

At this point it is worth to comment on the problem of quantizing the value of the transverse momentum ($k_{y}$) in Eqs. (22), (23). Assuming that the component of the current density perpendicular to the axis of the graphene strip disappears at its edges (i.e., for $y=0$ and $y=W$, see Fig. 3(a)), what is known as the so-called mass confinement Ber87 , we get a slightly different quantization than in the case of the Schrödinger system, see Eq. (6), namely

$$k_{y}^{(n)}=\frac{\pi{}(n+1/2)}{W},\ \ \ \ n=0,1,2,\dots.$$

(24)

In practice, however, the assumptions made in the following part mean that when calculating measurable quantities ($G$, $F$, etc.) we will approximate the sums appearing in Eqs. (5), (12), (17), (18), with integrals with respect to $dk_{y}$; the quantization change described above is therefore insignificant for further considerations.

The solution of the system of Eqs. (22), (23) is particularly simple in the case if the electrostatic potential energy, i.e. the function $V(x)$, is piecewise constant. Then, the solutions in individual sections (i.e., areas where $V(x)$ is constant) have the form of plane waves. For instance, for $E>V(x)$ waves traveling in the positive $(+)$ and negative $(-)$ directions along the $x$ axis, look as follows

$$\phi^{(+)}=\begin{pmatrix}1\\ e^{i\theta}\end{pmatrix}e^{ik_{x}{}x},\ \ \ \ \phi^{(-)}=\begin{pmatrix}1\\ -e^{-i\theta}\end{pmatrix}e^{-ik_{x}{}x},$$

(25)

where we have defined

	$\displaystyle e^{i\theta}$	$\displaystyle=(k_{x}+ik_{y})/k_{F},$
	$\displaystyle k_{F}$	$\displaystyle=\left(E-V(x)\right)/\hbar{}v_{F},$		(26)
	$\displaystyle\text{and }\ \ \ k_{x}$	$\displaystyle=\sqrt{k_{F}^{2}-k_{y}^{2}}.$

For $E<V(x)$, propagating-wave solutions also exist (this is, by the way, the main difference between the solutions of the massless Dirac equation and the Schrödinger equation, which leads in particular to the phenomenon known as Klein tunneling Rob12 ; Gut16 ) and differ from those given in Eq. 25 only in some signs. We leave the straightforward derivation to the reader.

At the interface of regions differing in the (locally constant) value of $V(x)$, we perform a matching of wave functions, which for the two-dimensional Dirac equation reduces to solving the continuity conditions for both spinor components matchfoo . For instance, if we consider the scattering from the right side of the discontinuity to the left side, we write

$$t\phi^{(L,-)}=\phi^{(R,-)}+r\phi^{(R,+)},$$

(27)

where the spinor functions with indices $L$ and $R$ differ in the values of $k_{F}$ and $k_{x}$ [see Eqs. (25), (26)], but are characterized by the same value of $k_{y}$. Since the considerations concern the interface between the graphene sample and the graphene region covered with a metal electrode (see Fig. 3(a)), the calculations can be simplified by adopting the model of a heavily doped electrode, in which we set $V(x)=-V_{\infty}$, where $V_{\infty}\rightarrow\infty$; we can then write the wave functions on the left in asymptotic form

$$\phi^{(L,\pm)}\simeq\begin{pmatrix}1\\ \pm{}1\end{pmatrix},$$

(28)

where we have omitted the phase factor, which is not important for further considerations. After substituting the above into Eq. (27), the calculations are straightforward; we now present the results for the transition and reflection probabilities

$$T_{1}=|t|^{2}=\frac{2\cos\theta}{1+\cos\theta},\ \ \ \ R_{1}=|t|^{2}=\frac{1-\cos\theta}{1+\cos\theta},$$

(29)

which turn out to depend only on the angle of incidence $\theta$ of the plane wave, or — more precisely — on the value of $\cos\theta=\sqrt{1-(k_{y}/k_{F})^{2}}$. In particular, we see that for $\theta=0$ we have $T_{1}=1$ (and $R_{1}=0$), which is a manifestation of the Klein tunneling mentioned above (let us emphasize that the potential barrier considered here has an infinite height).

The probability of passing through the entire graphene sample, i.e., through two electrostatic potential steps occurring at the sample-lead interface (see Fig. 3(a)), is most easily calculated using the double-barrier formula, the clear derivation of which can be found, e.g., in the Datta’s handbook Dat97

$$T_{12}=\frac{T_{1}{}T_{2}}{1+R_{1}R_{2}-2\sqrt{R_{1}R_{2}}\cos\phi},$$

(30)

where a phase shift

$$\phi=k_{x}L=L\sqrt{k_{F}^{2}-k_{y}^{2}},$$

(31)

related to the propagation of a plane wave along the main axis $x$, is introduced. (Note here that the phase shift introduced in this manner also implies the assumption that any reflections from the side edges of the system do not change the value of $k_{y}$; in practice, this implies that we restrict our considerations to systems for which $W\gg{}L$.) Assuming barrier symmetry, $T_{2}=T_{1}$, $R_{2}=R_{1}$, and substituting the formulas given in Eq. (29), we can now write $T_{12}$ explicitly as a function of $k_{y}$ and $E$,

$$T_{12}=T_{k_{y}}(E)=\left[1+\left(\dfrac{k_{y}}{\varkappa}\right)^{2}\sin^{2}\left(\varkappa{}\,L\right)\right]^{-1},$$

(32)

where

$$\varkappa=\begin{cases}\sqrt{k_{F}^{2}-k_{y}^{2}},&\text{for }\ |k_{y}|\leqslant{}k_{F},\\ i\sqrt{k_{y}^{2}-k_{F}^{2}},&\text{for }\ |k_{y}|>k_{F},\\ \end{cases}$$

(33)

and the Fermi wave vector, assuming $V(x)=0$ for the sample region, is equal to $k_{F}=|E|/(\hbar{}v_{F})$. The absolute value in the last expression arises from the fact that formulas in Eq. (29) and the following results are identical for $E<0$; we leave the verification of this property to the reader.

The physical consequences of the above expression for the transition probabilities $T_{k_{y}}(E)$, see Eqs. (32) and (33), are now discussed for two physical situations: a charge-neutral sample ($k_{F}=0$) and the Sharvin limit ($k_{F}W\gg{}1$). (Unless otherwise stated, we also assume geometry with long, parallel sample-lead interfaces and $W\gg{}L$.)

In the first case ($k_{F}=0$) we obtain $\varkappa=i|k_{y}|$ and can use the identity $\sin(ix)=i\sinh x$, resulting in a surprisingly simple expression

$$T_{k_{y}}(0)=\frac{1}{\cosh^{2}(k_{y}L)},$$

(34)

visualized in Fig. 3(b). In the wide-sample limit, $W\gg{}L$, the sums appearing in the formulas for Landauer conductance $G$ [see Eq. (5)], Fano factor $F$ [Eq. (12)], and higher cumulants $R_{3}$, $R_{4}$ [Eqs. (17) and (18)] can be approximated with integrals [see also Eq. (24)], leading to

	$\displaystyle\sum_{n}{}T_{n}(0)$	$\displaystyle\approx{}\frac{W}{\pi}\int_{0}^{\infty}dk_{y}\,T_{k_{y}}(0)=\frac{W}{\pi{}L},$		(35)
	$\displaystyle\sum_{n}{}\left[T_{n}(0)\right]^{2}$	$\displaystyle\approx{}\frac{W}{\pi}\int_{0}^{\infty}dk_{y}\,\left(T_{k_{y}}(0)\right)^{2}=\frac{2}{3}\frac{W}{\pi{}L},$		(36)

or, more generally,

$$\sum_{n}{}\left[T_{n}(0)\right]^{m}\approx{}\frac{W}{2\sqrt{\pi}{}\,L}\,\frac{\Gamma(m)}{\Gamma(m+1/2)}\ \ \ \ \text{for }\ m>0,$$

(37)

where $\Gamma(x)$ is the Euler gamma function. To facilitate future comparisons with other transport regimes, we will additionally define

$$\langle{}T^{m}\rangle_{k_{F}=0}=L\int_{0}^{\infty}dk_{y}\,\left(T_{k_{y}}(0)\right)^{m}=\frac{\sqrt{\pi}\,\Gamma(m)}{2\Gamma(m+\frac{1}{2})},$$

(38)

such that $\langle{}T\rangle_{k_{F}=0}=1$. Taking into account the graphene-specific fourfold degeneracy of states due to the presence of spin and valley degrees of freedom (the conductance quantum is therefore $g_{0}=4e^{2}/h$), we obtain

	$$\displaystyle G\approx\frac{4e^{2}}{\pi{}h}\frac{W}{L},\ \ \ \ F\approx 1-\frac{2}{3}=\frac{1}{3},$$		(39)
	$$\displaystyle R_{3}\approx\frac{1}{15},\ \ \ \ R_{4}\approx{}-\frac{5}{512}.$$		(40)

The value of $G\propto{}W/L$ (instead of $G\propto{}W$, as in a typical ballistic system) means that charge-neutral graphene exhibits universal specific conductivity, $\sigma_{0}=4e^{2}/(\pi{}h)$, the value of which is additionally determined only by the universal constants of nature. The value of the Fano factor $F=1/3$ is also not accidental, as it is a value characteristic for ohmic (disordered) conductors. (The same applies to higher cumulants.) In the context of graphene, the term pseudodiffusive conductivity is often used to emphasize that this ballistic system perfectly emulates an ohmic conductor within the appropriate parameter range. It should be emphasized that the first two theoretical values, given in Eq. (39) and originally derived in Refs. Kat06 (conductance) and Two06 (Fano factor), have been experimentally confirmed with satisfactory accuracy in 2008 Dan08 . (For the comprehensive theoretical discussion of full counting statistics for graphene at the Dirac point, see Ref. Sch10 .)

In the Sharvin limit ($k_{F}W\gg{}1$) the situation looks a bit different. We can then assume that the contribution of modes for which $k_{y}>k_{F}$ (i.e., evanescent modes) is negligible and limit the considerations to $k_{y}\leqslant{}k_{F}$. Next, we notice that as the values of $T_{k_{y}}(E)$ (or their powers) are summed, the $\sin^{2}\left(L\sqrt{k_{F}^{2}-k_{y}^{2}}\right)$ term of Eq. (32) oscillates rapidly, especially as $k_{y}$ approaches $k_{F}$. Therefore, it seems reasonable to replace the sine argument with a random phase and average the result, which leads to the following approximation

	$\displaystyle T_{k_{y}}(E)\approx\{T_{k_{y}}\}_{\rm incoh}$	$\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}\frac{d\varphi}{1+\left(k_{y}^{2}/\varkappa^{2}\right)\sin^{2}\varphi}$
		$\displaystyle=\sqrt{1-\left(k_{y}/k_{F}\right)^{2}},$		(41)

where we used the table integral Ryz-2-553-3

$$I(a,b)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{du}{a+b\cos{}u}=\frac{1}{\sqrt{a^{2}-b^{2}}},\ \ \ \ \text{for}\ a>|b|,$$

(42)

substituting

	$\displaystyle u$	$\displaystyle=2\varphi,$
	$\displaystyle a$	$\displaystyle=\frac{1-\frac{1}{2}\eta^{2}}{1-\eta^{2}},\ \ \ \ b=a-1=\frac{\frac{1}{2}\eta^{2}}{1-\eta^{2}},$		(43)
	$\displaystyle\text{with }\ \eta$	$\displaystyle=k_{y}/k_{F}.$

The comparison between the approximation given in Eq. (41) and the actual $T_{k_{y}}(E)$, see Eqs. (32), (33), for $k_{F}L=25$, is presented in Fig. 3(d).

Eq. (41) is essentially the Dirac version of Eq. (7) describing a standard ballistic system; when calculating the Landauer conductance, we can again approximate the summation by integration and obtain

$$G\approx{}\frac{g_{0}W}{\pi}\int_{0}^{k_{F}}dk_{y}\,\{T_{k_{y}}\}_{\rm incoh}=\frac{\pi}{4}G_{\rm Sharvin},$$

(44)

where we recall the value of Sharvin conductance given in Eq. (8). The prefactor $\pi/4$ is a consequence of the fact that in the last expression in Eq. (41), where previously there was a step function $\Theta(k_{F}-k_{y})$, a term describing an arc of a circle has appeared; the conductance of graphene beyond the charge-neutrality point is therefore reduced compared to a typical ballistic system.

Interestingly, in deriving Eq. (44) we did not explicitly assume, as in Eq. (39), that the width to length ratio of the sample is $W/L\gg{}1$; hypothetically, the result given in Eq. (44) can therefore be applied whenever the condition $k_{F}W\gg{}1$ is satisfied, regardless of the value of $W/L$. In practice, however, it is difficult to imagine that the double-barrier transmission formula, Eq. (30), which is the basis of the entire reasoning, could be applied to samples that do not satisfy the $W/L\gg{}1$ condition. It seems that in the case of graphene samples with $L\gtrsim{}W$, hard-to-control edge effects can significantly alter the conductivity Mia07 . We will address this issue later in this paper, but first let us calculate the approximate values of the Fano factor and higher cumulants in the Sharvin limit.

In order to construct the approximation analogous to this in Eq. (41), but for $T_{k_{y}}^{2}$, it is sufficient to calculate

	$\displaystyle\{T_{k_{y}}^{2}\}_{\rm incoh}$	$\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}\frac{d\varphi}{\left[1+\left(k_{y}^{2}/\varkappa^{2}\right)\sin^{2}\varphi\right]^{2}}$
		$\displaystyle=\sqrt{1-\eta^{2}}\left[1-\frac{1}{2}\eta^{2}\right],$		(45)

where we used the first derivative of $I(a,b)$, see Eqs. (42) and (43), with respect to $a$. More generally, the $m$-th power of $T_{k_{y}}$ can be approximated by

	$\displaystyle\{T_{k_{y}}^{m}\}_{\rm incoh}$	$\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}\frac{d\varphi}{\left[1+\left(k_{y}^{2}/\varkappa^{2}\right)\sin^{2}\varphi\right]^{m}}=\frac{(-1)^{m-1}}{(m-1)!}\frac{\partial^{m-1}}{\partial{}a^{m-1}}\,I(a,b)$
		$\displaystyle=\frac{2^{m-1}\Gamma(m-\frac{1}{2})a^{m-1}}{\sqrt{\pi}\,\Gamma(m)(a^{2}-b^{2})^{m-1/2}}\,{}_{2}F_{1}\left(1-\frac{m}{2},\frac{1-m}{2};\frac{3}{2}-m;1-\frac{b^{2}}{a^{2}}\right),$		(46)
		$\displaystyle=\frac{2^{m-1}\Gamma(m-\frac{1}{2})}{\sqrt{\pi}\,\Gamma(m)}\sqrt{1-\eta^{2}}\,z^{-m+1}\,{}_{2}F_{1}\left(1-\frac{m}{2},\frac{1-m}{2};\frac{3}{2}-m;z\right),\ \ \ \ \text{with }\ z=\frac{1-\eta^{2}}{(1-\frac{1}{2}\eta^{2})^{2}},$

where ${}_{2}F_{1}(\alpha,\beta;\gamma;z)$ is the hypergeometric function Abr65 . (Notice that for a positive integer $m$, $\alpha=1-\frac{m}{2}$ or $\beta=\frac{1-m}{2}$ is a non-positive integer, so the function reduces to a polynomial of $z$; after multiplying by $z^{-m+1}$, the resulting expression can be further simplified to a degree $2m-2$ polynomial of the variable $\eta$.) For $m=1$ and $m=2$, the above reproduces the results of Eqs. (41) and (45), respectively. The next two expressions are

	$\displaystyle\{T_{k_{y}}^{3}\}_{\rm incoh}$	$\displaystyle=\sqrt{1-\eta^{2}}\left(1-\eta^{2}+\frac{3}{8}\eta^{4}\right),$		(47)
	$\displaystyle\{T_{k_{y}}^{4}\}_{\rm incoh}$	$\displaystyle=\sqrt{1-\eta^{2}}\left(1-\frac{\eta^{2}}{2}\right)\left(1-\eta^{2}+\frac{5}{8}\eta^{4}\right).$		(48)

Setting $[T_{k_{y}}(E)]^{m}\approx{}\{T_{k_{y}}^{m}\}_{\rm incoh}$ for $m=1,\dots,4$, and approximating the summations occuring in Eqs. (12), (17), and (18) by integrations, namely,

	$\displaystyle\sum_{n}{}\left[T_{n}(E)\right]^{m}$	$\displaystyle\approx{}\frac{k_{F}W}{\pi}\left\langle\{T_{k_{y}}^{m}\}_{\rm incoh}\right\rangle$		(49)
	$\displaystyle\text{with }\ \left\langle\{T_{k_{y}}^{m}\}_{\rm incoh}\right\rangle$	$\displaystyle=\int_{0}^{1}d\eta\,\{T_{k_{y}}^{m}\}_{\rm incoh},$		(50)

we obtain

$$F\approx{}\frac{1}{8},\ \ \ \ R_{3}\approx{}-\frac{1}{23},\ \ \ \ R_{4}\approx{}-\frac{5}{512}.$$

(51)

(For the first four numerical values of $\langle\{T_{k_{y}}^{m}\}_{\rm incoh}\rangle$, see Table 1.)

The surprising (non-zero) value of the shot noise in Eq. (51) is close to the experimental results obtained by Danneau et. al. Dan08 , which are in the range of $F=0.10\div 0.15$. (The aspect ratio of the sample used in this experiment was $W/L=24$.) It should be emphasized that measuring the shot noise of nanoscopic devices containing components of different materials is rather challenging; there are also results in the literature that suggest that the dependence of the shot noise on the system filling is weak, with the value always close to the pseudodiffusive $F=1/3$ Dic08 . Measurements of higher charge cumulants are so far missing.

In Fig. 4, the approximations for $G$, $F$, $R_{3}$, and $R_{4}$, both for $k_{F}=0$ and for $k_{F}W\gg{}1$, are compared with and the actual values following from Eqs. (32), (33) for $T_{k_{y}}(E)$. Briefly speaking, the higher cumulant is considered, the larger value of $k_{F}W$ is necessary to observe the convergence to the sub-Sharvin limit; however, for $W/L=10$ and for the heavily-doped leads, $k_{F}W\gtrsim{}5$ is sufficient. The discussion of more realistic situations (finite doping in the leads, other sample shapes) is presented later in this paper.

Using the conformal mapping technique Ryc09 ; Gui08 , it can be shown that for charge-neutral graphene ($k_{F}=0$), the pseudodiffusive values given in Eqs. (37), (39), (40) are essentially valid for an arbitrary sample shape, provided that the prefector $W/L$ (if present) is replaced by an appropriate geometry-dependent factor defined by the conformal transformation. In particular, when mapping the rectangle onto the Corbino disk, one needs to substitute $W/L\rightarrow{}2\pi/\log(R_{\rm o}/R_{\rm i})$, where $R_{\rm o}$ and $R_{\rm i}$ are the outer and inner disk radii (respectively), see Fig. 5(a). An additional condition is that the system must be in the multimode regime, i.e. $\log(R_{\rm o}/R_{\rm i})\ll{}1$ (or $R_{\rm i}\approx{}R_{\rm i}$) in the disk case. Otherwise, if $R_{\rm i}\ll{}R_{\rm o}$, a nonstandard tunneling behavior is observed, with $G\propto{}R_{\rm i}/R_{\rm o}$ and $F\rightarrow{}1$ Ryc09 . (Since the transport is governed by a single mode, we also have $R_{3}\rightarrow{}1$ and $R_{4}\rightarrow{}1$ in such a case.)

However, in the Sharvin limit ($k_{F}{}R_{\rm i}\gg{}1$ in the disk case), a different set of universal charge-transport characteristics is predicted Ryc22 . Regardless of the exact size or shape of the outer sample-lead interface, one can assume that the double-barrier formula, Eq. (30), is still applicable and that $T_{2}\approx{}1$ and $R_{2}=1-T_{2}\approx{}0$ due to the Klein tunneling. Therefore, $T_{12}\approx{}T_{1}$, the role of a phase shift $\phi$ is negligible, and one can write — for the wave leaving the inner lead with the total angular momentum $\hbar{}j$ (with $j=\pm{}1/2,\pm{}3/2,\dots$) — the transmission probability as

	$\displaystyle\left(T_{j}\right)_{R_{\rm i}\ll{}R_{\rm o}}$	$\displaystyle\approx{}T(u_{j})=\begin{cases}{\displaystyle\frac{2\sqrt{1-u_{j}^{2}}}{1+\sqrt{1-u_{j}^{2}}}}\ \ &\text{if }\ |u_{j}|\leq{}1,\\ 0\ \ &\text{if }\ |u_{j}|>1,\end{cases}$
	$\displaystyle\text{with }\ u_{j}$	$\displaystyle=\frac{j}{k_{F}{}R_{\rm i}}.$		(52)

Subsequently, the summation for the $m$-th power of $T_{j}$ can be approximated (notice that we have assumed $k_{F}{}R_{\rm i}\gg{}1$) by integration over $-1\leqslant{}u\leqslant{}1$, leading to

$$\sum_{j}(T_{j})^{m}\approx{}k_{F}{}R_{\rm i}\int_{-1}^{1}{}du\left[T(u)\right]^{m}=2k_{F}{}R_{\rm i}\langle{}T^{m}\rangle_{u},$$

(53)

where, also using the symmetry $T(-u)=T(u)$, we have defined

$$\langle{}T^{m}\rangle_{u}=\int_{0}^{1}du\left(\frac{2\sqrt{1-u^{2}}}{1+\sqrt{1-u^{2}}}\right)^{m}=\frac{\sqrt{\pi}\,\Gamma(m+2)}{4\Gamma(m+\frac{5}{2})}\left[2m+3-2m\,{}_{2}F_{1}\left(\frac{1}{2},1;m+\frac{5}{2};-1\right)\right].$$

(54)

The first four values of $\langle{}T^{m}\rangle_{u}$ are listed in Table 1. Substituting the above into Eqs. (5), (12), (17), and (18), we obtain

	$\displaystyle{G}$	$\displaystyle\approx(4-\pi)\,{G_{\rm Sharvin}},$
		$\displaystyle\ \text{with }\ {G_{\rm Sharvin}}=2g_{0}k_{F}R_{\rm i},$
	$\displaystyle F$	$\displaystyle\approx\frac{9\pi-28}{3(4-\pi)}\simeq{}0.1065,$	$\displaystyle(R_{\rm i}\ll{}R_{\rm o})$		(55)
	$\displaystyle R_{3}$	$\displaystyle\approx\frac{204-65\pi}{5(4-\pi)}\simeq{}-0.04742,$
	$\displaystyle R_{4}$	$\displaystyle\approx\frac{1575\pi-4948}{21(4-\pi)}\simeq{}0.0004674.$

For the Corbino disk, it is also possible to perform the analytical mode matching for the heavily-doped leads, but arbitrary disk doping $k_{F}$ and radii ratio $R_{\rm o}/R_{\rm i}$ Ryc09 ; Ryc20 . We skip the details of the derivation here and just give the transmission probabilities

$$T_{j}=\frac{16}{\pi^{2}{}k^{2}{}R_{\rm i}{}R_{\rm o}}\,\frac{1}{\left[\mathfrak{D}_{j}^{(+)}\right]^{2}+\left[\mathfrak{D}_{j}^{(-)}\right]^{2}},$$

(56)

where

	$\displaystyle\mathfrak{D}_{j}^{(\pm)}$	$\displaystyle=\mbox{Im}\left[H_{j-1/2}^{(1)}(k_{F}R_{\rm i})H_{j\mp{}1/2}^{(2)}(k_{F}R_{\rm o})\right.$
		$\displaystyle\ \ \ \ \ \ \ \ \ \ \pm\left.H_{j+1/2}^{(1)}(k_{F}R_{\rm i})H_{j\pm{}1/2}^{(2)}(k_{F}R_{\rm o})\right],$		(57)

and $H_{\nu}^{(1)}(\rho)$ [$H_{\nu}^{(2)}(\rho)$] is the Hankel function of the first [second] kind. In Fig. 5, we provide the comparison between the values of the conductance and the next three charge cumulants obtained by substituting the exact $T_{j}$-s given above into Eqs. (5), (12), (17), (18), and performing the numerical summation over $j$ with the approximate values given in Eq. (55). It is easy to see that the radii ratio of $R_{\rm o}/R_{\rm i}=5$ is sufficient to reproduce our predictions for the narrow-opening limit, with good accuracy, typically starting from $k_{F}{}R_{\rm i}=50-100$. (This time the higher cumulant is considered, the faster the convergence.)