Transport in graphene:

The search for a replacement to Silicon based digital electronics prompted interest in graphene, a 2-dimensional allotrope of carbon, also a group IV element. Various 1D derivatives of graphene such as carbon nanotubes and nanoribbons have also become topics of extensive research in the past decade and a half. It is now painfully clear that the very metallicity of graphene arising from its Dirac cone bandstructure that endows it with ultrahigh electron mobility (~million cm2/Vs on hBN at low temperature) also makes it very hard to switch it off. In fact, as we show below, there is a mobility-bandgap tradeoff, which translates ultimately to an energy-delay tradeoff as well. Opening a bandgap - e.g. by selective doping in graphene, quantization in nanowires and nanotubes, and vertical fields in bilayer graphene inevitably hurts its mobility by robbing graphene of its 'graphene-ness'. 



Electron optics, metamaterials and Klein tunneling:

However, electrons in graphene have nontrivial topological properties arising from their underlying pseudospin structure. They provide added symmetry rules for transmission, much like spin conservation poses added constraints in GMR devices while orbital selection rules control symmetry filering across CoFeB/MgO magnetic tunnel junctions. As seen below, the Bloch part of the graphene electron eigenstates are created by a phase coherent superposition of its dimer pz orbitals, so that the forward and backward moving branches are orthogonal (in fact, this orthogonality is responsible for the gaplessness of graphene, ultimately protected by 2-D spatial and 1-D temporal inversion symmetry). Due to the orthogonality of these superposed states or pseudospins, normally incident electrons cannot reflect at a P-N junction, giving rise to perfect transmission or Klein tunneling (this ultimately relates to the fact that 1D reflection is set by velocity discontinuity, which does not exist for Dirac cone bands). Since transverse modes can still reflect, this means that the presence of a P-N junction barrier, as in a split gated structure below, leads to a collimation of the electrons, making their transmission strongly anisotropic, focused perpendicular to the junction. 

In addition, the conversion of electrons into holes across the PN junction makes electrons bend the wrong way, much like light bends the wrong way in a material with negative refractive index. For optoelectronics, this is hard to engineer and needs artificial metamaterials with negative real part of the frequency-dependent dielectric constant (near an absorption edge), and negative permability with split rings. For graphene though, an electrostatic split gate suffices. Experiments by Cory Dean's group in Columbia U showing negative index behavior agree well with our predictions. In a four probe set-up, a magnetic field can swivel the electrons to allow them to reach a PN junction at or near normal incidence, and the measured magnetoconductance plots (peak vs valley, red vs blue) tell us which right side contact the electrons ended up at. The demonstration of negative index was voted one of the top-10 discoveries in physics by Physics World Editors in 2016.


So how can we use this?


A graphene pn junction offers a platform to draw an apt analogy between optics and electronics. As a result, we can design many electrical devices using optical principles such as Ray Optics. We saw two figures ago how the electron trajectories at an electrostatically doped graphene pn junction bend at the interface, in order to conserve group velocity and the transverse momentum. This produces an electron lens similar to Veselago lens in optics. The figure below shows a conceptual device with a tunnel barrier to eliminate the normally incident electrons to suppress Klein tunneling. It is theoretically predicted that this will induce a voltage dependent transmission gap, which can also be collapsed with gate voltage. The colorplot shows NEGF simulation of the predicted current densities.

But we can do one better, than relying on our ability to engineer holes at the PN junction and then create point sources for collimated injection. Recall that the pseudospin structure of graphene's dimer orbitals automatically causes electrons to collimate across a P-N junction, transmitting along a narrow lobe perpendicular to the junction. Rotating the junction causes the transmission lobe to rotate as well. In fact, we demonstrated the electronic equivalent of Malus' law for a polarizer-analyzer pair. 


This means if we create two split gated structures at an angle, then for homogeneous electrical doping (+ voltage on all gates, giving an n-n-n structure), we will continue to benefit from the high mobility of bulk graphene. However, switchiing polarity on the middle gate to create an n-p-n junction will force each junction to Klein tunnel and transmit collimated electrons perpendicular to itself. The lack of overlap between the transmission nodes will cut off the current, creating an ON-OFF in the gate transfer characteristic, and saturationi in the drain output characteristic. This is basically a Klein tunnel FET where we are engineering a transmission gap for the OFF but not for the ON state. Simulations below. 

Experiments have now realized this Klein tunnel transistor, giving us a decent ON-OFF ratio (~10) at high mean free path (> 1 micron). See data from Philip Kim below (similar data exist from an European group).  While this ratio is 'good but no cigar',  it also predicts a highly saturating output characteristic, which seems consistent  with measurements from Columbia with ultra-high output resistances ~600 Kilo-Ohm Microns, >> contact resistances.

Remember what made graphene exciting was its high electron mobility! But the lack of a gap made its RF characteristics really weak - the linear I-V made its transconductance and output resistance low, giving poor current gain dominated by contact resistances (~1 kiloOhm-micron), at least an order of magnitude larger than III-Vs or Silicon. But a Klein TFET builds such an intrinsic resistance while preserving mobility, meaning that it's RF characteristic is quite impressive. Preliminary experiments show an intrinsic resistance of ~600 KiloOhm-micron, with a strongly saturating current (with bulk graphene!) 




AntiKlein tunneling of electrons in bilayer graphene.

Interestingly, for bilayer graphene, you expect zero transmission at normal incidence, rather than perfect transmission, which is called anti-Klein tunneling (you can see this by plotting the pseudospin rotation, ie, without any serious maths). Transmission maximizes along specific lobes related to the inverse tangent of the gate voltage ratio for a PN junction. 

It is important to appreciate how nontrivial this behavior is. For optics, we all know that reflection is minimum (transmission max) at normal incidence. The max is set by the refractive index ratio. We see plots like this in textbooks.


For monolayer graphene, the plots are similar, except it is not just minimum, but zero at normal incidence! This is Klein tunneling (KT). And for bilayer graphene, it is actually the opposite - reflection is 100% at normal incidence!  This is anti-Klein tunneling (AKT). See plot below.

In fact, if you keep going to trilayer, quadrilayer, pentalayer etc, the curves each oscillate (roughly as sine and cosine) - which would be super-cool to see experimentally!! I suspect nature isn't as kind with graphene because with more layers you get trigonal warping, gap opening with stray vertical fields, and stronger electron correlation effects. However, with 'artificial graphene' (e.g. hexagonal array of cold atoms) one might actually be able to see  them. 


Key Publications.

1. Proposed idea of the Graphene Klein Tunnel FET (APL and ACS Nano)
A.  "High efficiency switching using graphene based electron-optics" Applied Physics Letters, vol. 99 , pp. 123101, 2011
B. "Manipulating Chiral transmission by Gate Geometry: Switching in Graphene with transmission gaps" ACS Nano, vol. 7 :11 , pp.  9808-9813, 2013

2. Demo of Malus' law in graphene PN junctions with Albany (PRB)
"Manifestation of chiral tunneling at a tilted graphene p-n junction", Physical Review B, vol. 86 , pp. 155412, 2012

3. Demo of negative index in graphene with Columbia (Science)
 "Electron optics with p-n junctions in ballistic graphene", Science 353, 1522 (2016) 

4. Use for superior RF design with current saturation (Nature Scientific Reports)
  "Graphene Klein tunnel transistors for high speed analog RF applications",

5. Impact of non-idealities on Klein tunneling with Columbia (ACS Nano Articles ASAP)
    "Atomic scale characterization of Graphene p-n junctions for Electron-Optical Applications", ACS Nano 13, 2558 (2019) 

6. Demo of Klein tunnel transistor with Phil Kim group at Harvard (PNAS)
    "Graphene Transistor Based on Tunable Dirac-Fermion-Optics," PNAS 116, 6575 (2019)