Tunneling in Fermi systems with quadratic band crossing points

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Annals of Physics

journal homepage:www.elsevier.com/locate/aop

Tunneling in Fermi systems with quadratic band crossing points

Ipsita Mandal

^∗

Faculty of Science and Technology, University of Stavanger, 4036 Stavanger, Norway Nordita, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden

a r t i c l e i n f o

Article history:

Received 21 April 2020 Accepted 20 June 2020 Available online 29 June 2020

a b s t r a c t

We investigate the tunneling of quasiparticles through a rect- angular potential barrier of finite height and width, in 2d and 3d semimetals with band structures consisting of a quadratic band crossing point. We compute the transmission coefficient at various incident angles, and also the conductivity and the Fano factor. We discuss the distinguishing signatures of these transport properties in comparison with other semimetals, as well as electrons in normal metals.

1. Introduction

Multiband fermionic systems may exhibit a band crossing point in the Brillouin zone where two or more bands cross. If the chemical potential is adjusted to lie exactly at that point, the Fermi surface shrinks to a Fermi node. The most famous example of such a Fermi node is the case of a linear band crossing, whose low energy properties are described by Dirac fermions, and are conspicuous in systems like nodal superconductors and graphene. In this paper, we consider systems with a quadratic band crossing point (QBCP) somewhere in their two-dimensional (2d) [1–3] or three-dimensional (3d) [4–6] Brillouin zones. 2d QBCPs can be realized in checkerboard [1]

(at 1

/

2 filling), Kagome [1] (at 1

/

3 filling), and Lieb [2] lattices. On the other hand, pyrochlore iridates A₂Ir₂O₇ (A is a lanthanide element [7,8]) have been shown to host a 3d QBCP. Such bandstructures have also been realized that in 3d gapless semiconductors in the presence of a sufficiently strong spin–orbit coupling [9], such that the resulting model of a semimetal is indeed relevant for materials like gray tin (

α

-Sn) and mercury telluride (HgTe). These systems are also known as ‘‘Luttinger semimetals’’ [10] due to the fact that the low-energy fermionic degrees of freedom are captured by the Luttinger Hamiltonian of inverted band-gap semiconductors [11,12].

∗ Correspondence to: Faculty of Science and Technology, University of Stavanger, 4036 Stavanger, Norway.

E-mail address: ipsita.mandal@gmail.com.

https://doi.org/10.1016/j.aop.2020.168235

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2 I. Mandal / Annals of Physics 419 (2020) 168235

Fig. 1. Tunneling through a potential barrier in a QBCP material. The upper panel shows the schematic diagrams of the spectrum of quasiparticles about a QBCP, with respect to a potential barrier in thex-direction. The lower panel represents the schematic diagram of the transport across the potential barrier. The Fermi level (indicated by dotted lines) lies in the conduction band outside the barrier, and in the valence band inside it. The blue fillings indicate occupied states. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Our aim is to compute the tunneling coefficients and other transport characteristics when the quasiparticles of the QBCP semimetals are subjected to a potential barrier of finite strength and width along one direction, which is chosen to be thex-axis. This scenario is represented in the cartoon inFig. 1. Our results will show how these transport characteristics are significantly different from those in normal metals, due to the presence of multiple bands. We will also compare their features with those of other semimetals like graphene, bilayer graphene and three-band pseudospin-1 systems.

The paper is organized as follows. In Section2, we study the 2d QBCPs, while Section3deals with the 3d QBCP case. We compare our findings with the results for some other known bandstructures in Section4. Finally, we end with a summary and outlook in Section5.

2. 2D model

For a 2d system, the particle–hole symmetric QBCP withC₆rotational symmetry is described by the Hamiltonian [1]:

H^kin_2d(k_x

,

^ky)

=

h

¯

² 2m

[

2k_xk_y

σ

x

+

( k²_y

−

k²_x)

σ

z

]

(1) in the momentum space, with eigenvalues

ε

^±_2d^(kx

,

^ky)

= ± ¯

h²( k²_x

+

k²_y)

2m

,

⁽²⁾

(3)

where the ‘‘

+

’’ and ‘‘

−

’’ signs, as usual, refer to the conduction and valence bands respectively. The corresponding eigenvectors are given by:

Ψ⁺

=

¹

√ k²_x

+

k²_y

{

k_y

,

^kx

} ,

^andΨ⁻

=

¹

√ k²_x

+

k²_y

{−

k_x

,

^ky

} ,

⁽³⁾

respectively.

The 2d system is modulated by a square electric potential barrier of heightV₀and widthL, giving rise to anx-dependent potential energy function:

V(x)

=

{V₀ for 0

<

^x

<

^L

0 otherwise

.

⁽⁴⁾

Hence, we need to consider the total Hamiltonian:

H^tot_2d

=

H^kin_2d(

−

i

∂

x

, −

i

∂

y)

+

V(x) (5) in position space. We choose the x-axis along the transport direction, and place the chemical potential at an energyE

>

0 in the region outside the potential barrier. The Fermi energyEcan in general be tuned by chemical doping or a gate voltage.

2.1. Formalism

For a material of a sufficiently large transverse dimensionW, the boundary conditions should be irrelevant for the bulk response, and we use this freedom to simplify the calculation. Here, on a physical wavefuncitonΨ^totwe impose periodic boundary conditions:

Ψ^tot^(x

,

^W⁾

=

_Ψ^tot(x

,

⁰⁾

.

⁽⁶⁾

The transverse momentum k_y is conserved, and it is quantized due to the periodicity in the transverse widthW, and hence takes the form:

k_y

=

²

π

ⁿ

W

≡

q_n

,

⁽⁷⁾

wheren

∈

_Z. For the longitudinal direction, we seek plane wave solutions of the forme^ik^x^x. Then the full wavefunction is given by:

Ψ^tot^(x

,

^y

,

ⁿ⁾

=

const.

×

_Ψ_n(x)e^iqⁿ^y

,

⁽⁸⁾

For any mode of given transverse momentum componentk_y, we can determine thex-component of the wavevectors of the incoming, reflected, and transmitted waves (denoted byk_ℓ), by solving

ε

^±_2d^(kx

,

ⁿ⁾

= ±

^¯^h

2( k²ℓ+_q²_n

)

2m . In the regionsx

<

^{0 and}^x

>

L, we have only propagating modes (k_ℓis real), while thex-components in the scattering region (denoted byk), are given by

˜

_k

˜

2

=

^2m^|^E⁻^V⁰^|

h¯²

−

q²_n, and may be propagating (imaginary part of_k

˜

is zero) or evanescent (imaginary part of_k

˜

is nonzero).

We will follow the procedure outlined in Refs. [13] and [14] to compute the transport coefficients. We consider the transport of positive energy states (Ψ⁺) corresponding to electron-like particles. The transport of hole-like excitations (Ψ⁻) will be similar. Hence, the Fermi level outside the potential barrier is adjusted to a valueE

= ε

_2d⁺^(kx

,

^ky). Such a scattering stateΨn,+, in the mode labeled byn, is constructed from the states:

Ψn(x)

=

⎧

⎨

⎩

φ

L forx

<

⁰

, φ

M for 0

<

^x

<

^L

, φ

R forx

>

^L

,

φ

L

=

^Ψ⁺^(k^ℓ

,

^qn)e^ik^ℓ^x

+

r_nΨ⁺⁽

−

k_ℓ

,

^qn)e⁻^ik^ℓ^x

√

V(k_ℓ

,

ⁿ⁾

, φ

M

=

[

α

nΨ⁺⁽_k

˜ ,

^qn)eⁱ^˜^{k x}

+ β

nΨ⁺⁽

−˜

k

,

^qn)e⁻ⁱ^{k x}^˜

]Θ

(

^E

−

V₀

)

(4)

+

[

α

nΨ⁻⁽_k

˜ ,

^qn)eⁱ^{k x}^˜

+ β

nΨ⁻⁽

−˜

k

,

^qn)e⁻ⁱ^˜^{k x}

]Θ

(

^V0

−

E

) , φ

R

=

t_nΨ⁺^(kℓ

,

^qn) e^ik^ℓ⁽^x⁻^L⁾

√

V(k_ℓ

,

ⁿ⁾

,

V(k_ℓ

,

ⁿ⁾

≡ | ∂

kℓ

ε

⁺^(kℓ

,

ⁿ⁾

| = ¯

h²k_ℓ

m

,

^kℓ

=

√ 2m E

h

¯

²

−

q²_n

,

_k

˜ =

√

2m

|

E

−

V₀

|

¯

h²

−

q²_n

,

⁽⁹⁾ where we have used the velocityV(k_ℓ

,

n) to normalize the incident, reflected and transmitted plane waves. Note that forV₀

>

E, the Fermi level within the potential barrier lies within the valence band, and we must use the valence band wavefunctions in that region.

The boundary conditions can be obtained by integrating the equationH^tot_2dΨ^tot

=

EΨ^tot ^{over a} small interval in thex-direction around the pointsx

=

0 andx

=

L. The results are that the two components of the wavefunction be continuous at the boundaries. These conditions are sufficient to guarantee the continuity of the current flux along thex-direction.¹In particular, the reflection and transmission amplitudesr_n

,

^tn, and the two coefficients

(α

n

, β

n

)

, are determined from these boundary conditions. This mode-matching procedure gives us:

r_n(E

,

^V0)

=

⎧

⎪⎪

⎪⎨

⎪⎪

⎪⎩

(_k˜2k²ℓ−_q⁴_n )

sin(_kL)˜ (˜_k2k²ℓ+_q⁴_n

)

sin(˜_kL)−_{2 i}_{k k}˜ _ℓ_q2

ncos(˜_kL) forE

<

^V0 (

k²ℓ−˜_k² )

sin (˜_kL) (˜_k2+_k²

ℓ )

sin (˜_kL)

+_{2 i}_{k k}˜ ℓcos

(˜_kL) forE

>

^V0

.

(10)

and

t_n(E

,

^V0)

=

⎧

⎪⎪

⎨

⎪⎪

⎩

−

( ^{2 i}^˜^{k k}^ℓ^q²ⁿ k˜²k²ℓ+_q⁴_n

) sin

(˜_kL)

−_{2 i}˜_{k k}_ℓ_q2 ncos

(_kL˜) forE

<

^V0 2 i˜_{k k}

( ℓ k˜²+_k²_ℓ

) sin

(_kL˜)

+_{2 i}_{k k}˜ _ℓ_cos( kL˜

) forE

>

^V0

.

⁽¹¹⁾

The reflection and transmission coefficients at an energyEare given by

R(E

,

^V0

, φ

⁾

= |

r_n(E

,

^V0)

|

²andT(E

,

^V0

, φ

⁾

= |

t_n(E

,

^V0)

|

²

,

⁽¹²⁾ respectively, where

φ =

tan⁻¹

(qn kℓ

)

is the incident angle of the incoming wave.

2.2. Transmission coefficient, conductivity and Fano factor

Let us assumeW to be very large such thatq_ncan effectively be treated as a continuous variable.

We then numerically computeT(E

, φ

^).

Usingk_ℓ

=

√2mE

¯h² cos

φ ,

ⁿ

=

^W

√ 2mE

h sin

φ ,

^dn

=

²^W

√ 2mE

h cos

φ

^d

φ

, in the zero-temperature limit and for a small applied voltage, the conductance is given by [15]:

G(E

,

^V0)

=

^e

2

h

∑

n

|

t_n

|

²

→

^e

2

h

∫

|

t_n(E)

|

²dn

=

^e

2W

√

2m E h²

∫ ^π₂

−^π₂

T(E

,

^V0

, φ

^{) cos}

φ

^d

φ .

⁽¹³⁾ Therefore, the conductivity is given by:

σ

^(E

,

^V0)

=

^L W

G(E

,

^V0) e²

/

^h

=

2

π

√ E

h

¯

²

/

( 2mL²)

∫ ^π₂

−^π₂

T(E

,

^V0

, φ

^{) cos}

φ

^d

φ .

⁽¹⁴⁾

1 From wavefunction matching, we have two equations from the two boundaries. For 2d QPCB, each of these equations has two components as each wavevector has two components. Therefore we have four equations for four undetermined coefficients. We do not need to match the first derivatives of the wavefunction as those will be redundant equations.

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Fig. 2. 2d QBCP: The polar plots show the reflection coefficient R(E,^V0, φ⁾⏐

⏐E≤V0 and the transmission coefficient T(E,^V0, φ⁾⏐

⏐_E_≤_V

0 as functions of the incident angleφ for the parametersE=0.³^V0(red),E=0.⁵^V0(green),E=0.⁸^V0

(magenta) andE=1.^0V0(blue). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Shot noise is the measure of the fluctuations of the current away from their average value. The zero-temperature shot noise is given by [15]:

S(E

,

^V0)

=

²^e

2Φ h

∑

n

|

t_n

|

²

|

r_n

|

²

→

e³Φ^W√ _E

¯h²/

(

^2mL²

) π

^{h L}

∫ ^π₂

−^π 2

T(E

,

^V0

, φ

^{) [1}

−

T(E

,

^V0

, φ

^)]^d

φ ,

(15) whereΦis the applied voltage, and is characterized by the Fano factor:

F(E

,

^V0)

=

∫^π₂

−π 2

T(E

,

^V0

, φ

⁾^d

φ

∫^π₂

−^π 2

T(E

,

^V0

, φ

^{) [1}

−

T(E

,

^V0

, φ

^)]^d

φ .

⁽¹⁶⁾ We express E and V₀ in units of ^h^¯²

2mL², and study the behavior of T(E

,

^V0

, φ

^),

σ

^(E

,

^V0) and F(E

,

^V0).Figs. 2and3show the polar plots ofR(E

,

^V0

, φ

^{) and}^T^(E

,

^V0

, φ

) as functions of the incident

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Fig. 3. 2d QBCP: The polar plots show the reflection coefficient R(E,^V0, φ⁾⏐

⏐E>V0 and the transmission coefficient T(E,^V0, φ⁾⏐

⏐_E_>_V

0 as functions of the incident angleφfor the parametersE=1.^1V0(red),E=1.^5V0(green),E=1.^8V0

(magenta) andE=2.⁵^V0(blue). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

angle

φ

, for the cases E

≤

V₀ and E

>

^V0 respectively. They also serve to demonstrate that R(E

,

^V0

, φ

⁾

+

T(E

,

^V0

, φ

⁾

=

1. From the expression of transmission coefficient in Eq.(34), it is clear that the transmission is zero at normal incidence (

φ =

0), as long asE

<

^V0. Hence, we do not have a Klein tunneling analogue in the 2d QBCP, unlike graphene [16] or three-band pseudospin-1 Dirac–

Weyl systems [17,18]. However, we still have the resonance conditions_{k L}

˜ = π

^N

,

^N

∈

_Z, under which the barrier becomes transparent (T

=

1). InFig. 4, we illustrate the conductivity

σ

^(E

,

^V0) and the Fano factorF(E

,

^V0), as functions ofE

/

^V0, for some values ofV₀.

3. 3D model

We consider a model for 3d QBCP semimetals, where the low energy bands form a four- dimensional representation of the lattice symmetry group [6]. Then the standard

(

^k

·

p

)

Hamiltonian for the particle–hole symmetric system can be written by using the five 4

×

4 Euclidean Dirac

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Fig. 4. 2d QBCP: Plots of the (a) conductivity (σin units of 2π), and (b) Fano factor (F), as functions ofE/^V0, for various values ofV0.

matricesΓaas [12,19]:

H^kin_3d(k_x

,

^ky

,

^kz)

=

h

¯

² 2m

5

∑

a=₁

d_a(k)Γa

.

⁽¹⁷⁾

The Γa’s form one of the (two possible) irreducible, four-dimensional Hermitian representations of the five-component Clifford algebra defined by the anticommutator

{

_Γ_a

,

Γb

} =

2

δ

ab. The five anticommuting gamma-matrices can always be chosen such that three are real and two are imaginary [12,20]. In the representation used here, (Γ1

,

Γ2

,

Γ3) are real and (Γ1

,

Γ3) are imaginary [12]:

Γ1

= σ

3

⊗ σ

2

,

Γ2

= σ

3

⊗ σ

1

,

Γ3

= σ

2

⊗ σ

0

,

Γ4

= σ

1

⊗ σ

0

,

Γ5

= σ

3

⊗ σ

3

.

⁽¹⁸⁾ The five functionsd_a(k) are the real

ℓ =

2 spherical harmonics, with the following structure:

d₁(k)

= −

√

3k_yk_z

,

^d2(k)

= −

√

3k_xk_z

,

^d3(k)

= −

√

3k_xk_y

,

d₄(k)

= − √

3

2 (k²_x

−

k²_y)

,

^d5(k)

= −

1 2

(2k²_z

−

k²_x

−

k²_y)

.

⁽¹⁹⁾

The energy eigenvalues are

ε

^±_3d^(kx

,

^ky

,

^kz)

= ± ¯

h²(

k²_x

+

k²_y

+

k²_z)

2m

,

⁽²⁰⁾

where the ‘‘

+

’’ and ‘‘

−

’’ signs, as usual, refer to the conduction and valence bands. Each of these bands is doubly degenerate.

A set of orthogonal eigenvectors are given by:

Ψ+^T,¹

=

¹

N+,1

{

−

^(k^x

+

ik_y)

(

^k

+

k_z

)

(k_x

−

ik_y)²

,

ⁱ

(

^k

+

3k_z

)

√

3 (k_x

−

ik_y)

, −

ⁱ

(

−

2k_z

(

^k

+

k_z

) +

k²_x

+

k²_y)

√

3 (k_x

−

ik_y)²

,

¹ }

,

Ψ+^T,2

=

¹

N+,²

{(k_x

+

ik_y)

(

^k

−

k_z

)

(k_x

−

ik_y)²

, −

ⁱ

(

^k

−

3k_z

)

√

3 (k_x

−

ik_y)

, −

ⁱ

(2k_z

(

^k

−

k_z

) +

k²_x

+

k²_y)

√

3 (k_x

−

ik_y)²

,

¹ }

,

Ψ−^T,¹

=

¹

N−,1

{

−

ⁱ

(

^k

+

k_z

)

√

3 (k_x

−

ik_y)

,

^k

−

k_z k_x

+

ik_y

,

¹

, −

ⁱ

(2k_z

(

^kz

−

k

) +

k²_x

+

k²_y)

√

3 (k_x

+

ik_y)² }

,

Ψ−^T,2

=

¹

N−,²

{ i

(

^k

−

k_z

)

√

3 (k_x

−

ik_y)

, −

^k

+

k_z k_x

+

ik_y

,

¹

, −

ⁱ

(2k_z

(

^k

+

k_z

) +

k²_x

+

k²_y)

√

3 (k_x

+

ik_y)² }

,

⁽²¹⁾

(8)

wherek

=

√

k²_x

+

k²_y

+

k²_z, and the ‘‘

+

’’ (‘‘

−

’’) indicates an eigenvector corresponding to the positive (negative) eigenvalue. The symbols _N¹

±,¹ and _N¹

±,² denote the corresponding normalization factors.

The 3d system is modulated by a square electric potential barrier of heightV₀and widthL, as described in Eq.(4). Here, we need to consider the total Hamiltonian:

H^tot_3d

=

H^kin_3d(

−

i

∂

x

, −

i

∂

y

, −

i

∂

z)

+

V(x) (22) in position space. As before, we choose the x-axis along the transport direction, and place the chemical potential at an energyE

>

0 in the region outside the potential barrier.

3.1. Formalism

We consider the tunneling in a slab of height and widthW. Again, we assume that the material has a sufficiently large widthWalong each of the two transverse directions, such that the boundary conditions are irrelevant for the bulk response, and impose the periodic boundary conditions:

Ψ

˜

^tot^(x

,

⁰

,

^z)

= ˜

_Ψ^tot(x

,

^W

,

^z)

,

_Ψ

˜

^tot_(x

,

^y

,

⁰⁾

= ˜

_Ψ^tot(x

,

^y

,

^W)

.

⁽²³⁾ The transverse momentumk⊥

=

(k_y

,

^kz) is conserved, and its components are quantized. Due to periodicity, we conclude that:

k_y

=

²

π

ⁿy

W

≡

q_n_y

,

^kz

=

²

π

ⁿz

W

≡

q_n_z

,

⁽²⁴⁾

where (n_x

,

ⁿy)

∈

_Z. For the longitudinal direction (along thex-axis), we seek plane wave solutions of the forme^ik^x^x. Then the full wavefunction is given by:

Ψ

˜

^tot^(x

,

^y

,

^z

,

ⁿ⁾

=

const.

× ˜

_Ψ_n(x)eⁱ (

q_nyy+_q_nz_z )

withn

=

(ny

,

ⁿz)

.

⁽²⁵⁾

For any mode of given transverse momentum componentk⊥, we can determine thex-component of the wavevectors of the incoming, reflected, and transmitted waves (denoted byk_ℓ), by solving

ε

_3d^±^(kx

,

ⁿ⁾

= ±

¯h²( k²ℓ+k⊥²)

2m . In the regionsx

<

^{0 and} ^x

>

L, we have only propagating modes (k_ℓ is real), while thex-components in the scattering region (denoted by k), are given by

˜ ˜

_k2

=

2m|_E−_V₀|

¯h²

−

k⊥², and may be propagating (_k

˜

is real) or evanescent (_k

˜

is imaginary).

We will follow the same procedure as described for the 2d QBCP. Again, without any loss of generality, we consider the transport of one of the degenerate positive energy states (Ψ⁺,¹) corresponding to electron-like particles, with the Fermi level outside the potential barrier being adjusted to a valueE

= ε

⁺_3d^(kx

,

^ky

,

^kz). In this case, a scattering state_Ψ

˜

_n, in the mode labeled byn, is constructed from the states:

Ψ˜n(x)=

⎧

⎪⎨

⎪⎩

φ˜L forx<⁰, φ˜M for 0<^x<^L, φ˜R forx>^L, φ˜L=^Ψ⁺^,¹^(k^ℓ,^qny,^qnz)eⁱ^k^ℓ^x+∑

s=1,²rn,sΨ⁺,s(−k_ℓ,^qny,^qnz)e⁻ⁱ^k^ℓ^x

√_V˜_(k_ℓ,n) , φ˜M =[ ∑

s=1,2

αn,sΨ⁺,s(_k˜,^qny,^qnz)eⁱ^{k x}^˜ +∑

s=1,2

βn,sΨ⁺,s(−˜k,^qny,^qnz)e⁻ⁱ^{k x}^˜

]Θ(^E−V₀)

+[ ∑

s=1,2

αn,^sΨ⁻,^s(_k˜,^qny,^qnz)eⁱ^{k x}^˜ +∑

s=1,2

βn,^sΨ⁻,^s(−˜k,^qny,^qnz)e⁻ⁱ^{k x}^˜

]Θ(^V0−E) , φ˜R=

∑

s=1,2t_n_,_sΨ⁺,s(k_ℓ,^qny,^qnz)

√_V˜_(k_ℓ,n)

eⁱ^k^ℓ⁽^x⁻^L⁾,

V˜(k_ℓ,ⁿ⁾≡ |∂k_ℓε⁺3d(k_ℓ,ⁿ⁾| =h¯²k_ℓ m , ^kℓ=

√ 2m E

h¯²

−q²_n

y−q²_n

z, _k˜=

√

2m|E−V₀|

¯h²

−q²_n

y−q²_n

z, (26)

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where we have used the velocity_V

˜

_(k_ℓ

,

n) to normalize the incident, reflected and transmitted plane waves. Note that forV0

>

E, the Fermi level within the potential barrier lies within the valence band, and we must use the valence band wavefunctions in that region.

The usual mode-matching procedure atx

=

0 andx

=

Lgives us²:

rn,¹(E,^V0)=

⎧

⎪⎪

⎪⎨

⎪⎪

⎪⎩

sin(

˜kL)

(ⁱ^k^y⁺^kℓ)^[^kℓkz+iky

√ k²ℓ+k²_y+k²_z

) (˜k²(

8k²ℓ−k²_y−k²_z) +(

k²_y+k²_z)(

5k²ℓ−4( k²_y+k²_z))]

2(^kℓ−iky)²^√^k²ℓ+k²_y+k²_z[ sin(

˜kL){

k˜²( 4k²ℓ+k²_y+k²_z)

+ (

k²_y+k²_z)(

k²ℓ+4( k²_y+k²_z))}

−6 ikk˜ℓcos(

˜kL)(

k²_y+k²_z)] forE<^V⁰

( k²ℓ−˜k²

) sin

(_kL˜) (kℓ+iky)

( kℓkz+iky

√ k²ℓ+k²y+k²z

)

(kℓ−iky)²

[(˜_k2+k²ℓ )

sin (˜_kL)

+2 i_kk˜_ℓ_cos(

˜kL

)] forE>^V0

,

rn,²(E,^V0)=

⎧

⎪⎪

⎨

⎪⎪

⎩

−

k˜²[ 4k²ℓ−5(

k²_y+k²_z)]

+( k²_y+k²_z)[

k²ℓ−8( k²_y+k²_z)]

sin(

˜kL) sin(

˜kL)[

˜k²( 4k²ℓ+k²_y+k²_z)

+( k²_y+k²_z){

k²ℓ+4( k²_y+k²_z)}]

−6 ikk˜ℓcos(

˜kL)(

k²_y+k²_z)

×

kℓ(^kℓ+iky)^(√^k²ℓ+k²y+k²z+kz )√

( kz

( kz−

) +k²ℓ+k²y

)

(^k^ℓ⁻^ik^y)²^√^k²ℓ+k²y+k²z

√ kz

(√ k²ℓ+k²y+k²z+kz

) +k²ℓ+k²y

forE<^V0

kℓ (˜k²−k²ℓ

) sin(

˜kL) (kℓ+iky)

(√ k²ℓ+k²_y+k²_z+kz

)√ kz

( kz−

√ k²ℓ+k²_y+k²_z

) +k²ℓ+k²_y

(kℓ−iky)²

√ kz

(√ k²ℓ+k²y+k²z+kz

) +k²ℓ+k²y

[(_k˜2+k²ℓ )

sin (_kL˜)

+2i_kk˜_ℓ_cos(

˜kL )]

forE>^V⁰

, (27)

and

t_n_,₁(E

,

^V0)

=

⎧

⎪⎪

⎨

⎪⎪

⎩

−

^{6 i}

˜k kℓ (

k²_y+_k²_z )

sin (˜_kL)[

k˜² (

4k²ℓ+_k²_y+_k²_z )

+ (

k²y+_k²_z ){

k²ℓ+₄ (

k²y+_k²_z )}]

−_{6 i}˜_{k k} ℓcos

(_kL˜)(

k²y+_k²_z

) forE

<

^V0

−2 i_{k k}˜ ( ℓ

˜k²+_k² ℓ )

sin (_kL˜)

+_{2 i}˜_{k k} ℓcos

(_kL˜) forE

>

^V0

,

t_n_,₂(E

,

^V0)

=

0

.

⁽²⁸⁾

The reflection and transmission coefficients at an energyEare given by

R(E

,

^V0

, θ, φ

⁾

= |

r_n_,₁(E

,

^V0)

|

²

+ |

r_n_,₂(E

,

^V0)

|

²andT(E

,

^V0

, θ, φ

⁾

= |

t_n_,₁(E

,

^V0)

|

² (29) respectively, where

θ =

cos⁻¹

(√¯h q_nz 2m E

)

and

φ =

tan⁻¹ (_q

ny kℓ

)

define the incident angle (solid) of the incoming wave in 3d.

3.2. Transmission coefficient, conductivity and Fano factor Again, we assumeWto be very large such that(

q_n_y

,

^qnz

)can effectively be treated as continuous variables. Usingk_ℓ

=

√2mE

¯h² sin

θ

^cos

φ ,

ⁿy

=

^W

√ 2mE

h sin

θ

^sin

φ ,

ⁿz

=

^W

√ 2mE

h cos

θ ,

^dnydn_z

=

W²×_{2m E}

h² cos

φ

^sin²

θ

^d

φ

, in the zero-temperature limit and for a small applied voltage, the conductance is given by [15]:

G(E

,

^V0)

=

²^e

2

h

∑

n

|

t_n_,₁

|

²

→

²^e

2

h

∫

|

t_n_,₁

|

²dn_xdn_y

=

⁴

π

^e²^W² h

¯

(2m E

¯

h² ) ∫ _π

θ=₀

∫ ^π₂

φ=−π 2

T(E

,

^V0

, θ, φ

^{) cos}

φ

^sin²

θ

^d

φ ,

⁽³⁰⁾

2 From wavefunction matching, we have two equations from the two boundaries. For 3d QPCB, each of these equations has four components as each wavevector has four components. Therefore we have eight equations for eight undetermined coefficients. We do not need to match the first derivatives of the wavefunction as those will be redundant equations.

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Fig. 5.Contourplots of the reflection coefficient (R) and transmission coefficient (T) for 3dQBCP as functions of (θ, φ^), for various values ofV0andE.

leading to the conductivity expression:

σ

^(E

,

^V0)

=

(L

W )2

G(E

,

^V0) e²

/

^h

=

8

π

²

[ E

¯

h²

/

( 2mL²)

]∫ _π

θ=₀

∫ ^π₂

φ=−π 2

T(E

,

^V0

, θ, φ

^{) cos}

φ

^sin²

θ

^d

φ .

(31)