Klein Tunneling Effect in Graphene

Chapter 8 Experimental Considerations of 2D Graphene

8.11 Klein Tunneling Effect in Graphene

180 Graphene for Defense and Security

The transmission T and reflection R coefficients are derived from the probability of amplitude currents. The probability current connected with the Dirac equation:

J it i

x i

1=ψ σ ψ, =1 2, (8.60) In our case:

J1 A A

2 2

=2[ − ` ], J2 B 2 2

= ; (8.61)

The reflection and transmission coefficient are:

R A

= A`²

2; T B

= A

2

2 ; (8.62)

At x = 0, the continuity of the wave function:

A²= B²; A²=0; (8.63)

The explanation to the Klein tunneling comes from the fact that a potential step cannot change the direction of the group velocity of a relativistic particle without a mass.

V0

y

x Positive velocity

FIGURE 8.33 The dispersion relation for the incoming particles: the momentum increases in x-direction and the energy grows in y-direction.

A series of experiments using the Klein tunneling effect has been conducted⁴⁶. The diffusive scattering remains in graphene and the characteristic effects are not possible to distinguish from the bulk resistance that depends on the total trans- parency of the pn-junction. An applied magnetic field in y-direction causes a pro- nounced curve that corresponds to a charge motion in x-direction. The structure maintains the device function. The metal gate’s width is approximately 20 nm.

The thickness and permittivity of the dielectric layer determine the gate voltage.

The induced carrier density versus x coordinate and the Klein barrier located at x = 0 is:

n x( )=

{

V CTG TG/ [¹+( / ) ]x ^{ω 2 5.} +V CBG BG

}

/ ;e ^(8.64) where CTG = 1490 aF/µm² and CBG = 116 aF/µm², VTG = -10 V and VBG = 50 V, the exponent 2.5 is chosen to match the results of the conducted simulations.

The geometry of the structure for which the above calculations were perform is shown in Fig. 8.34. Transmission through the Klein barrier structure depends on multiple internal reflection similar to those in a Fabry-Perot etalon. This pro- cess is shown in Fig. 8.35. The applied magnetic field is equal to zero or larger than zero.

V_TG

GL

VBG SiO₂ HSQ/HfO₂

FIGURE 8.34 Geometry of a Klein barrier in a Fabry-Perot etalon device⁴⁷. The top gate is 20 nm wide crosses the graphene layer GL. The gate induces carrier concentration n₂. HfO2 and poly-hydroxysilane form the dielectric for the top gate. The conductance G = dI/dV (between the source and the drain) is measured.

182 Graphene for Defense and Security

The angle of incidence is α and L = x2 – x1. The back-reflection coefficient changes at α = 0. At B = 0 the phase changes when back-reflection amplitudes cancel resulting in a half-period shift in the fringes caused by Fabry-Perot interference.

In a Fabry-Perot etalon, the conductance G of the Klein barrier is calculated:

G e h t t L LGR r r i L LGR

ky

=^{4 2/}

∑

^{[1 2exp(}− ^{/2λ )] / [1}− ^{1 2exp( )exp(θ} − ^{/λ )]2;; (8.65)} where the summation gives the transverse wavevector, L, t₁, t₂ are the transmission coef- ficients, L = the width of the barrier, λLGR = mean free path in the vicinity of the gate, r₁, r₂ are reflection coefficients, θ= ∆θ is the phase difference caused by the particle moving between the two junctions of the transmission and reflection coefficients.

WKB approximation is a technique in mathematical physics for finding solutions to linear differential equation that have coefficients varying in space. The abbreviation stands for the first letters of Wentzel-Kramers-Brillouin. The WKB approximation is used to find an analytical solution for simple tunneling-barrier models. In prac- tice, exact solutions often do not exist and the WKB approximation can give an approximate solution.

In Fig. 8.35, one trajectory is transmitted through both barriers and the second trajectory is reflected at the second barrier. There are two traversals more of the length L that intersect the first trajectory (Fig. 8.35 geometry).

L B = 0

α

-α

x = x1 x = x2 x

0

>

→

py

y 0 p

y B > 0

<

→

FIGURE 8.35 Electron transmission through semiconductor p-n-p structure at B = 0 (left) and B > 0 (right)⁴⁸.

∆θ=2θWKB+∆θ1+∆θ2; (8.66) where θWKB= h⁻¹

∫

p x dxx^{( )}' ¹ and ∆θ1 2( ) are the phase shifts for the interfaces (three shaded regions)⁴⁸.

Eq. (8.67) may be somewhat simplified by realizing that r₁ and r₂ are small, giving the oscillatory part of Eq. (8.67):

GOSC=⁸e h^2/

∑

kyt t^{12 22}r r^{1 2cos( )exp(θ} −²L^/λLGR^{); (8.67)}

The oscillating part extraction is possible because in the Fabry-Perot model has an oscillating dependence on the phase shift (Eq. 8.67).

θWKB= h⁻¹

∫

p x dxx^{( ) ;' ' (8.68)}

The shift comes from electron wave accumulation from reflection between the interfaces:

θWKB= h⁻¹

∫

p x dxx^{( ) ;' ' (8.69)}

where ∆θ1 2( ) are the back-reflection phases for the interfaces 1 and 2. In Fig. 8.34 the net phase contribution (∆θ1+∆θ2) is altered by magnetic field B. At zero magnetic field B the phase difference cancels but for the arc-like trajectories, the signs of angles of incidence can be equal. For a given transverse momen- tum p_y, there exists a magnetic field B^* for the condition –B^*L/2<p_y<B^*L/2.⁴⁸ And the phase sum ∆θ1+∆θ2=π. The modeling of the process assuming that the Klein barrier has a parabolic potential υ( )x =ax²−t creates pn interfaces at x= ±x x_ε( _ε = ε/ ). Based on these dependences and assuming a magnetic field a B with the WKB phase.

θWKB π y

L L

n x k eBx h dx

=  − −











−

∫

Re [ ( ) ( / ) ]^/ ;

/ /

2 1 2 2

2

(8.70)

where p h/ is the unit of the integrand expression.

The reflection phases mentioned earlier may be expressed through the Heaviside function. H is zero except for x > 0 H(x) = 1:

∆θ1=π[ (H k− +y eBL/2h)]; (8.71)

∆θ2= −π[ (H k− −y eBL/2h)]; (8.72)

184 Graphene for Defense and Security The transmission coefficient t_1,2 and the reflection coefficient r_1,2 for junction at

±L / 2 are:

t1 2, =exp[−π(hvF/2eE k)( y±eBL/2h) ];2 (8.73) r1 2, =exp[i H kπ (− yeBL/2h)][1−t1 2, 2 1 2]/ (8.74) where E is the electric field.

The transmission is maximum when ky =B= 0 or ky= −eBL/2h; In Eq. (8.73) eE≈2 1. hυF(dn dx/ )^{2 3/} (8.75) where dn/dx = the carrier velocity at the junction.

Relying on the approach considered above the observed conductance oscillators may be modelled⁴⁷. The induced carrier density n₂at the center of the region with the gate for dG/dn₂ conductance differential is expressed in general as:

n x( )=

{

V CTG TG/ [¹+( / ) ]x ^{ω 2 5.} +V CBG BG

}

/ ;e ^(8.76) where ω is a parameter in the range of 45 – 47 nm.

The prediction shows a fringe shift in the Fabry – Perot oscillation vs magnetic fields data range (Fig. 8.36).

e²/h

2

1 3 4

0 30

n2xx10¹², cm^-2 60

FIGURE 8.36 Conductance oscillating part that corresponds to magnetic field of 0, 200, 400, 600 and 800 mT (from top to bottom). The peaks of the curves correspond to fringe shift on the observed (or modelled) traces of 6o sc⁴⁷. The four arrows for each curve show half- phase shift for the Klein scattering specific half-phase shift.