Colloquium: Andreev reflection and Klein tunneling in graphene

A colloquium-style introduction to two electronic processes in a carbon monolayer (graphene) is presented, each having an analog in relativistic quantum mechanics. Both processes couple electronlike and holelike states, through the action of either a superconducting pair potential or an electrostatic potential. The first process, Andreev reflection, is the electron-to-hole conversion at the interface with a superconductor. The second process, Klein tunneling, is the tunneling through a $p\text{−}n$ junction. The absence of backscattering, characteristic of massless Dirac fermions, implies that both processes happen with unit efficiency at normal incidence. Away from normal incidence, retro-reflection in the first process corresponds to negative refraction in the second process. In the quantum Hall effect, both Andreev reflection and Klein tunneling induce the same dependence of the two-terminal conductance plateau on the valley isospin of the carriers. Existing and proposed experiments on Josephson junctions and bipolar junctions in graphene are discussed from a unified perspective.

Figure 1

Albert Einstein, Paul Ehrenfest, Paul Langevin, Heike Kamerlingh Onnes, and Pierre Weiss at a workshop in Leiden (October 1920). The blackboard discussion, on the Hall effect in superconductors, has been reconstructed by 93. See also 111 for the historical context of this meeting.

Figure 2

(Color online) Atomic force microscope image (false color) of a carbon monolayer covered by two superconducting Al electrodes. From 54.

Figure 3

(Color online) Band structure $E(k_x,k_y)$ of a carbon monolayer. The hexagonal first Brillouin zone is indicated. The conduction band $(E>0)$ and the valence band $(E<0)$ form conically shaped valleys that touch at the six corners of the Brillouin zone (called conical points, Dirac points, or $K$ points). The three corners marked by a white dot are connected by reciprocal-lattice vectors, so they are equivalent. Likewise, the three corners marked by a black dot are equivalent. In undoped grapheme, the Fermi level passes through the Dirac points. Illustration by C. Jozsa and B. J. van Wees.

Figure 4

Honeycomb lattice of a carbon monolayer. The unit cell contains two atoms, labeled $A$ and $B$, each of which generates a triangular sublattice (open and closed circles). The lattice constant $a$ is $\sqrt{3}$ times larger than the carbon-carbon separation of $0.142\mathrm{nm}$. The reciprocal-lattice vector $\mathit{K}$ has length $4\pi ∕3a$. The edge of the lattice may have the armchair configuration (containing an equal number of atoms from each sublattice), or the zigzag configuration (containing atoms from one sublattice only). Dashed circles and bonds indicate missing atoms and dangling bonds, respectively. The separation $W$ of opposite edges is measured from one row of missing atoms to the opposite row, as indicated.

Figure 5

(Color online) Location of the valley isospin $\mathbf{\nu }$ on the Bloch sphere for a zigzag edge (arrows along the $z$axis) and for an armchair edge (arrows in the $x\text{−}y$ plane). The solid and dashed arrows correspond to opposite edges.

Figure 6

Two graphene flakes having the same zigzag boundary condition: $\Psi =\pm {\tau }_z\otimes {\sigma }_z\Psi$. The sign switches between + and − at the armchair orientation (when the tangent to the boundary has an angle with the $y$ axis that is a multiple of 60°) From 5.

Figure 7

(Color online) Electrostatic potential profile (solid line) producing two heavily doped graphene regions at the left and right and a weakly doped region (length $L$) at the center. The central region is undoped when the Fermi energy (dashed line) coincides with the energy of the Dirac point. Electrical conduction then proceeds via evanescent (exponentially decaying) modes.

Figure 8

(Color online) Electron and hole excitations in the conical band structure of graphene (filled and empty circles at energies $E_F\pm \epsilon$), converted into each other by Andreev reflection at a superconductor.

Figure 9

(Color online) Andreev retroreflection (left panel) at the interface between a normal metal and a superconductor. Arrows indicate the direction of the velocity, and solid or dashed lines distinguish whether the particle is a negatively charged electron $\left(e\right)$ or a positively charged hole $\left(h\right)$. Specular Andreev reflection (right panel) at the interface between undoped graphene and a superconductor. From 14.

Figure 10

(Color online) Dispersion relation (14) in graphene for two values of the Fermi energy $E_F=\hslash v{k}_F$, for the case of normal incidence ($\delta k_y=0$, $\delta k_x\equiv \delta k$). Electron excitations (filled states above the Fermi level, from one valley) and hole excitations (empty states below the Fermi level, from the other valley) are both indicated. Solid and dotted lines distinguish the conduction and valence bands, respectively. The electron-hole conversion upon reflection at a superconductor is indicated by the arrows. Specular Andreev reflection (right panel) happens if an electron in the conduction band is converted into a hole in the valence band. In the usual case (left panel), the electron and hole both lie in the conduction band. From 14.

Figure 11

(Color online) Trajectories of an incident electron and the Andreev reflected hole, for different excitation energies $\epsilon$ relative to the Fermi energy $E_F$, at fixed angle of incidence. For $\epsilon <E_F$, the hole is in the conduction band (solid lines), while for $\epsilon >E_F$ the hole is in the valence band (dashed lines). The reflected trajectories rotate clockwise with increasing $\epsilon$, jumping by 180° when $\epsilon =E_F$.

Figure 12

(Color online) The transition from retroreflection to specular Andreev reflection in a graphene channel with superconducting boundaries induces a transition from a localized level (left) to a propagating mode (right). The latter state contributes to thermal transport along the channel, but not to electrical transport. From 105.

Figure 13

(Color online) Josephson junction, formed by a graphene layer (N) with two superconducting electrodes (S) a distance $L$ apart, having a phase difference $\varphi ={\Phi }_1-{\Phi }_2$. From 104.

Figure 14

Critical current $I_c$ and $I_c{R}_N$ product of a ballistic Josephson junction (length $L$ short compared to the width $W$ and superconducting coherence length $\xi$), as a function of the Fermi energy $E_F$ in the normal region. Small and large $E_F$ asymptotes are indicated by dashed lines. From 104.

Figure 15

Product of the critical current $I_c$ and the normal state resistance $R_N$ vs gate voltage $V_{\text{gate}}$, measured at $T=30\mathrm{mK}$ in the Josephson junction of Fig. 2. The carrier density in the graphene layer is linearly proportional to $V_{\text{gate}}$, while the Fermi energy $E_F\propto {\sqrt{V}}_{\text{gate}}$. The resistance $R_N$ is measured in the presence of a small magnetic field to drive the electrodes in the normal state. From 54.

Figure 16

(Color online) Classical trajectories of an electron in the presence of a uniform electric field in the $x$ direction. All three trajectories are at the same energy; only the component $p_y$ of the momentum transverse to the field lines is varied. Two trajectories are for $p_y>0$, while the other trajectory is for $p_y=0$. The electron is in the conduction band of graphene for $x<0$ (solid trajectories, velocity parallel to momentum) and in the valence band for $x>0$ (dashed trajectories, velocity antiparallel to momentum). Solid and dashed trajectories are coupled by Klein tunneling.

Figure 17

(Color online) Band structure of a single valley at two sides of a potential step (height $U_0$, width $d$). The equilibrium Fermi energy $E_F$ is the same at both sides, so that for $U_0>E_F$ an electron just above the Fermi level is in the conduction band at one side and in the valence band at the other side. Arrows indicate the electron velocity, which is parallel to the wave vector (or momentum) in the conduction band (left) and antiparallel in the valence band (right).

Figure 18

(Color online) $n\text{−}p\text{−}n$ junction in graphene: (a) Cross-sectional view of the device. (b) Electrostatic potential profile $U\left(x\right)$ along the cross section of (a). The combination of a positive voltage on the back gate and a negative voltage on the top gate produces a central $p$-doped region flanked by two $n$-doped regions. (c) Optical image of the device. The barely visible graphene flake is outlined with a dashed line and the dielectric layer of PMMA appears as a shadow. From 55.

Figure 19

Two trajectories in a $p\text{−}n\text{−}p$ junction, the lower one (transmitted) in zero magnetic field and the upper one (reflected) in a small but nonzero field. Because only trajectories with an angle of order $1∕\sqrt{k_{F}d}⪡1$ around normal incidence are transmitted through the $p\text{−}n$ and $n\text{−}p$ interfaces, a relatively weak magnetic field suppresses the series conductance of the interfaces by bending the trajectories away from normal incidence. From 32.

Figure 20

(Color online) Electron trajectories along a $p\text{−}n$ interface in a magnetic field $B>B_{*}$ (when there is no transmission through the interface). The electron rotates in opposite directions in the conduction band (solid trajectories) and in the valence band (dashed). The trajectory centered at the interface represents an “ambipolar snake state.”

Figure 21

(Color online) Comparison of two systems that can be mapped onto each other by the transformation (29). The upper graphs show the electrostatic potential profile (solid lines) of a $p\text{−}n$ junction (left) and the corresponding NS junction (right, with $U_{\infty }⩾U_0$). The upper right graph also shows the superconducting pair potential $\Delta$ (dashed line). The excitation spectrum of the two systems is the same for $\epsilon ⪡{\Delta }_0$. Classical trajectories in the two systems are related by reflection along the interface, as shown in the lower graphs for $\epsilon =0$ (solid lines indicate electronlike trajectories and dashed lines holelike trajectories).

Figure 22

Trajectories of an incident and refracted electron at a $p\text{−}n$ interface, for different excitation energies $\epsilon$ relative to the potential step height $U_0$, at fixed angle of incidence. For $\epsilon <U_0$, the refracted electron is in the valence band (dashed lines), while for $\epsilon >U_0$ it is in the conduction band (solid lines). The refracted trajectories rotate counterclockwise with increasing $\epsilon$, jumping by $180°$ when $\epsilon =U_0$. The transformation $x\mapsto -x$ maps this transition from negative to positive refraction onto the transition from retroreflection to specular reflection in the NS junction of Fig. 11.

Figure 23

(Color online) Classical trajectories (dotted lines) in an $n\text{−}p\text{−}n$ junction at an energy $\epsilon =0$ that is halfway the potential step across the $n\text{−}p$ and $p\text{−}n$ interfaces, so that the refraction precisely inverts the angle of incidence. A scatterer in the $n$-region (solid diagonal line) has an inverted image in the central $p$ region and a noninverted image in the other $n$-region. This is the principle of operation of the Veselago lens.

Figure 24

Schematic top view of a graphene nanoribbon containing an interface between a $p$-doped and $n$-doped region (left panel) and between a normal (N) and superconducting (S) region (right panel). Electronlike and holelike edge states in the lowest Landau level are indicated by solid and dashed lines, respectively, with arrows pointing in the direction of propagation. From 108.

Figure 25

(Color online) Conductance of an armchair nanoribbon containing the potential step $U\left(x\right)=\frac{1}{2}[\tanh (2x∕L)+1]U_{\infty }$, calculated numerically from a tight- binding model in a perpendicular magnetic field (magnetic length $l_m\equiv \sqrt{\hslash ∕eB}=5a$). The step height $U_{\infty }$ is varied from below $E_F$ (unipolar regime) to above $E_F$ (bipolar regime), at fixed $E_F=\hslash v∕l_m$ and $L=50a$. The solid curves are without disorder, while the dashed curves are for a random electrostatic potential landscape (correlation length $\xi =10a$). A different number $\mathcal{N}$ of hexagons across the ribbon are represented, and hence a different width $W=(\mathcal{N}+3∕2)a$: $\mathcal{N}=97$, 98, and 99. The dashed horizontal line marks the plateau at $G=\frac{1}{4}\times 2e^2∕h$. From 108.

Figure 26

(Color online) Experimental conductance of a gate-controlled $p\text{−}n$ junction in graphene. The conductance of the $n$-doped region at one side of the interface is fixed at $\mid f_1\mid e^2∕h$, with $f_1=2$, while the conductance $\mid f_2\mid e^2∕h$ at the other side of the interface is varied by the gate voltage (values of $f_2$ are indicated, with negative numbers corresponding to a $p$-doped region). In the unipolar regime $(f_1{f}_2>0)$, the conductance of the junction is $G=\min (\mid f_1\mid ,\mid f_2\mid )e^2∕h$, while in the bipolar regime $(f_1{f}_2<0)$ the conductance is the Ohmic series conductance $G\times h∕e^2=\mid f_1{f}_2\mid ∕(\mid f_1\mid +\mid f_2\mid )$. From 118.

Figure 27

(Color online) The density of states $\rho \left(\epsilon \right)$ for the $p\text{−}n$ junction shown in (a). The dotted line is the value in the isolated $p$ and $n$ regions, which is energy independent for $\mid \epsilon \mid ⪡U_0$. The density of states vanishes at the Fermi level $(\epsilon =0)$, according to Eq. (37). The NS junction shown in (b) has the same density of states. In both the NS and $p\text{−}n$ geometries, the suppression of the density of states is due to destructively interfering of the electronlike and holelike segments of periodic orbits at the Fermi level (indicated by solid blue and dashed green trajectories). From 15.

Figure 28

Persistent current through a ring containing an abrupt $p\text{−}n$ interface, as a function of the magnetic flux through the ring. From 15.