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We show that Hall-like current can be induced by acoustic phonons in a nondegenerate, semiconductor fluorine-doped single-walled carbon nanotube (FSWCNT) using a tractable analytical approach in the hypersound regime
*q* is the modulus of the acoustic wavevector and
*H*, the acoustic wave frequency,
*T*, the overlapping integral,
*q*. Qualitatively, the Hall-like current exists even if the relaxation time
*q* and

Transport of electrons in semiconductor and multi-quantum well structures is governed by diffusive, drift and tunneling current flow [

In the presence of a weak magnetic field, Ω τ ≪ 1 (where Ω is the cyclotron frequency and τ is the relaxation time), an acoustic wave propagating through a conductor also induces another type of effect called acoustomagnetoelectric effect (AME) [

AME was first theorized by Grinberg et al. [

Modification of single-walled carbon nanotube (SWCNT) skeletal structure with fluorine is one of the emerging and efficient processes for chemical activation and functionalization of carbon nanotubes [_{2}F without destroying the tube’s skeletal structure. The functionalization process is a fast exothermic reaction and the repulsive interactions of the fluorine atoms on the surface debundles the nanotube, thus, enhancing their electron dispersion [

Doping a SWCNT with fluorine atoms and thus, creating a double periodic band forms an FSWCNT, which modifies the metallic nature to semiconducting nature [

Consider a fluorine-doped SWCNT, where the fluorine atoms form a one-dimensional chain along the axial direction, the energy band is deduced as [

ε ( p z ) = ε o + Ξ n Δ cos 2 N − 1 ( a p z ) (1)

where a = 3 b / 2 ℏ , Ξ n is a constant, N is an integer and ε o is the minimum energy of the π-electrons within the first Brillouin zone. For N = 2 , the energy band for FSWCNT at the edge of the Brillouin zone is expressed as [

ε ( p z ) = ε o + Δ 1 cos ( 3 a p z ) + Δ 2 cos ( a p z ) (2)

where, p o is the quasi-momentum of an electron in the first Brillouin zone i.e. − π / a ≤ p o ≤ π / a , with Δ 1 = 2 Δ , and Δ 2 = 6 Δ .

As we previously described [

j = 2 e ( 2 π ℏ ) 2 ∑ n , n ′ ∫ U n , n ′ a c Ψ i ( p , H ) d 2 p z (3)

where Ψ i ( p z ) is the solution to the Boltzmann kinetic equation in the absence of a magnetic field, and is expressed as:

v ∂ Ψ i ∂ p + W p { Ψ } = v i . (4)

p z is the electron momentum along the axial direction of the FSWCNT and U n , n ′ a c in Equation (3) is the electron-phonon interaction transition rate which is obtained by using the Fermi golden rule as in refs. [

U n , n ′ a c = 2 π Φ ω q v s ∑ n , n ′ { | G p z − ℏ q , p z | 2 [ f ( ε n ( p z − ℏ q ) ) − f ( ε n ( p z ) ) ] δ ( ε n ( p z − ℏ q ) − ε n ( p z ) + ℏ ω q ) + | G p z + ℏ q , p z | 2 [ f ( ε n ′ ( p z + ℏ q ) ) − f ( ε n ′ ( p z ) ) ] δ ( ε n ′ ( p z + ℏ q ) − ε n ′ ( p z ) − ℏ ω q ) } (5)

Here f ( p z ) = f ( ε n , n ′ ( p z ) ) is the unperturbed electron distribution function, Φ is the sound flux density, ε n , n ′ ( p z ) is the electron energy band, n and n’ denote quantization of the electron energy band, and G p z ± ℏ q , p z is the matrix element of the electron-phonon interaction. Ψ i is the root of the kinetic equation given as [

e c ( v × H ) ∂ Ψ i ∂ p + W p { Ψ } = v i , (6)

where v i is the electron velocity and W p { ... } = ( ∂ f / ∂ ε ... ) − 1 W p ( ∂ f / ∂ ε ) . The operator W p is a Hermitian operator and it is the collision operator describing the relaxation of the nonequilibrium distribution of the electron [

Ψ i = Ψ i 0 + Ψ i 1 + Ψ i 2 + ⋯ (7)

Following the approach in ref. [

Ψ i 0 = v i τ (8)

Similarly, in the first approximation, we obtain

Ψ i 1 = − τ 2 e m c ( v × H ) i (9)

and i = x , y , z .

Substituting Equation (8) and Equation (9) into Equation (3) and using the principle of detailed balance, i.e. | G p ′ , p | 2 = | G p , p ′ | 2 , we obtain the net current density as:

j i = 2 e ( 2 π ℏ ) 2 2 π Φ ω q v s ∑ n , n ′ ∫ | G p z + ℏ q , p z | 2 [ f ( ε n ′ ( p z ) ) − f ( ε n ′ ( p z + ℏ q ) ) ] × [ Ψ i ( p z + ℏ q ) − Ψ i ( p z ) ] δ ( ε n ′ ( p z + ℏ q ) − ε n ′ ( p z ) − ℏ ω q ) d 2 p z − 2 e ( 2 π ℏ ) 2 2 π Φ ω q v s e τ 2 m c ∑ n , n ′ ∫ | G p z + ℏ q , p z | 2 [ f ( ε n ′ ( p z ) ) − f ( ε n ′ ( p z + ℏ q ) ) ] × [ Ψ i ( p z + ℏ q ) − Ψ i ( p z ) ] δ ( ε n ′ ( p z + ℏ q ) − ε n ′ ( p z ) − ℏ ω q ) d 2 p z (10)

The matrix element of the electron-phonon interaction is given as:

| G p ′ , p | = 4 π e K 2 ρ ω q ϵ (11)

where K is the piezoelectric modulus, ϵ is the lattice dielectric constant, and ρ is the density of FSWCNT. Ψ i ( p z ) = l z , is the electron mean free path given as

l z = τ v z (12)

where

v z = ∂ ε ( p z ) ∂ p z (13)

Substituting Equations (11)-(13) into Equation (10) yields:

j = − 2 K 2 π Φ e 3 τ Θ ( 1 − α 2 ) ℏ 3 ω q 2 ϵ ρ a 1 − α 2 ∑ n , n ′ ∫ [ f ( ε n ′ ( p z ) ) − f ( ε n ′ ( p z + ℏ q ) ) ] × [ v z ( p z + ℏ q ) − v z ( p z ) ] δ ( ε n ′ ( p z + ℏ q ) − ε n ′ ( p z ) − ℏ ω q ) d p z − 2 K 2 π Φ e 4 τ 2 Θ ( 1 − α 2 ) ℏ 3 ω q 2 ϵ ρ a 1 − α 2 m c ∑ n , n ′ ∫ [ f ( ε n ′ ( p z ) ) − f ( ε n ′ ( p z + ℏ q ) ) ] × [ ( v ( p z + ℏ q ) × H ) − ( v ( p z ) × H ) ] δ ( ε n ′ ( p z + ℏ q ) − ε n ′ ( p z ) − ℏ ω q ) d 2 p z (14)

Under the orientation considered, the axial AE current ( j z A E ) is given by the first term and its solution if found in [

∂ f ( r , p , t ) ∂ t + v ( p ) ⋅ ∇ r f ( r , p , t ) + e E ∇ p f ( r , p , t ) = − f ( r , p , t ) − f o ( p ) τ (15)

which has a solution of the form:

f ( p z ) = ∫ 0 ∞ d t ′ τ exp ( − t / τ ) f o ( p z − e a E t ′ ) (16)

where f o ( p z ) is the shifted Fermi-Dirac distribution given as:

f o ( p z ) = 1 [ exp ( − ( ε ( p z ) − μ ) / k T ) + 1 ] . (17)

μ is the chemical potential which ensures the conservation of electrons, k is the Boltzmann’s constant, T is the absolute temperature in energy units. Substituting Equation (16) and Equation (17) into Equation (14), we obtained the equations for j z A E and j y A M E which contain the Fermi-Dirac integral F 1 / 2 of the order 1/2 as:

j = − 2 K 2 π Φ e 3 τ Θ ( 1 − α 2 ) ℏ 3 ω q 2 ϵ ρ a 1 − α 2 ∑ n , n ′ ∫ [ F 1 / 2 ( ε n ′ ( p z ) ) − F 1 / 2 ( ε n ′ ( p z + ℏ q ) ) ] × [ v z ( p z + ℏ q ) − v z ( p z ) ] δ ( ε n ′ ( p z + ℏ q ) − ε n ′ ( p z ) − ℏ ω q ) d p z − 2 K 2 π Φ e 4 τ 2 Θ ( 1 − α 2 ) ℏ 3 ω q 2 ϵ ρ a 1 − α 2 m c ∑ n , n ′ ∫ [ F 1 / 2 ( ε n ′ ( p z ) ) − F 1 / 2 ( ε n ′ ( p z + ℏ q ) ) ] × [ ( v ( p z + ℏ q ) × H ) − ( v ( p z ) × H ) ] δ ( ε n ′ ( p z + ℏ q ) − ε n ′ ( p z ) − ℏ ω q ) d 2 p z (18)

where

F 1 / 2 ( η f ) = 1 Γ ( 1 / 2 ) ∫ 0 ∞ η f 1 / 2 d η 1 + exp ( η − η f ) . (19)

Here, ( μ − ε c ) / k T ≡ η f and Γ ( 1 / 2 ) is the gamma function. For nondegenerate electron gas, where the Fermi level is several kT below the energy of the conduction band edge ε c ,( k T ≪ ε c ), the integral in Equation (19) approaches π / 2 exp ( η f ) . The unperturbed distribution function from Equation (19) can be expressed as:

f o ( p z ) = A † exp ( − [ ε ( p z ) ] / k T ) (20)

where A † is the normalization constant to be determined from the normalization condition ∫ f ( p ) d p = n o as:

A † = 3 n o a 2 2 I o ( Δ 1 * ) I o ( Δ 2 * ) exp ( ε o − μ k T ) (21)

n o is the electron concentration, and I o ( Δ x * ) is the modified bessel function of zero order, where x = 1 , 2 . Assuming the electrons are confined to the lowest conduction band, that is n = n ′ = 1 , then the quasi-velocity of the electrons in the system is also given as:

v z ( p z ) = − [ 3 a Δ 1 sin ( 3 a p z ) + a Δ 2 sin ( a p z ) ] (22)

In the absence of an external magnetic field, the axial AE current density in the first term of Equation (14) is deduced as:

j z A E = j z A E ( 0 ) { 1 − 4 ( Δ z * sin ( χ ( 1 − v d v s ) ) cos B sin ( a 2 ℏ q ) + Δ s * cos A sin ( 3 χ ( 1 − v d v s ) ) sin ( 3 2 a ℏ q ) ) × coth [ Δ s * cos ( 3 χ ( 1 − v d v s ) ) cos A cos ( 3 2 a ℏ q ) + Δ z * cos ( χ ( 1 − v d v s ) ) cos B cos ( a 2 ℏ q ) ] × coth [ Δ s * cos ( 3 χ ( 1 − v d v s ) ) sin A sin ( 3 2 a ℏ q ) + Δ z * cos ( χ ( 1 − v d v s ) ) sin B sin ( a 2 ℏ q ) ] } (23)

where

j z A E ( 0 ) = j o [ sinh { Δ s * sin ( 3 2 a ℏ q ) sin A + Δ z * sin ( a 2 ℏ q ) sin B } × sinh { Δ s * cos ( 3 2 a ℏ q ) cos A + Δ z * cos ( a 2 ℏ q ) cos B } ] (24)

and

j o = 4 A † π Φ e 3 K 2 τ Θ ( 1 − α 2 ) ℏ 3 ω q 2 ϵ 2 σ a q 1 − α 2 , χ = ℏ ω q a / v s , α = ω q / 12 Δ a q

Switching off the external electric field, Equation (23) then reduces to Equation (24). Similarly, the Hall-like current density in the second part of Equation (14) is obtained after some cumbersome calculations as:

j y A M E = − 2 A † K 2 π Φ e 3 τ 2 Θ ( 1 − α 2 ) Ω ℏ 3 ω q 2 ϵ ρ a 1 − α 2 ∫ 0 ∞ exp ( − d t ′ τ ) × { sinh [ Δ 1 * cos ( 3 e a E t ′ ) sin A sin ( 3 2 a ℏ q ) + Δ 2 * cos ( e a E t ′ ) sin B sin ( a 2 ℏ q ) ] × sinh [ Δ 1 * cos ( 3 e a E t ′ ) cos A cos ( 3 2 a ℏ q ) + Δ 2 * cos ( e a E t ′ ) cos B cos ( a 2 ℏ q ) ] − 4 ( Δ 2 * sin ( e a E t ′ ) cos B sin ( a 2 ℏ q ) + Δ 1 * cos A sin ( 3 e a E t ′ ) sin ( 3 2 a ℏ q )

+ Δ 1 * Δ 2 * sin ( e a E t ′ ) sin ( 3 e a E t ′ ) cos A cos B sin ( a 2 ℏ q ) sin ( 3 2 a ℏ q ) ) × cosh [ Δ 1 * cos ( 3 e a E t ′ ) cos A cos ( 3 2 a ℏ q ) + Δ 2 * cos ( e a E t ′ ) cos B cos ( a 2 ℏ q ) ] × cosh [ Δ 1 * cos ( 3 e a E t ′ ) sin A sin ( 3 2 a ℏ q ) + Δ 2 * cos ( e a E t ′ ) sin B sin ( a 2 ℏ q ) ] } (25)

where Ω = μ H / ℏ c , Θ is the Heaviside step function, Δ 1 * = Δ 1 / k T , Δ 2 * = Δ 2 / k T . Simplifying further, we obtain

j y A M E = j y ( 0 ) { 1 − 4 ( Δ 2 * sin ( χ ( 1 − v d v s ) ) cos B sin ( a 2 ℏ q ) + Δ 1 * cos A sin ( 3 χ ( 1 − v d v s ) ) sin ( 3 2 a ℏ q ) ) × coth [ Δ 1 * cos ( 3 χ ( 1 − v d v s ) ) cos A cos ( 3 2 a ℏ q ) + Δ 2 * cos ( χ ( 1 − v d v s ) ) cos B cos ( a 2 ℏ q ) ] × coth [ Δ 1 * cos ( 3 χ ( 1 − v d v s ) ) sin A sin ( 3 2 a ℏ q ) + Δ 2 * cos ( χ ( 1 − v d v s ) ) sin B sin ( a 2 ℏ q ) ] } (26)

Switching off the external electric field from Equation (26) yields:

j y ( 0 ) = j o y [ sinh { Δ 1 * sin ( 3 2 a ℏ q ) sin A + Δ 2 * sin ( a 2 ℏ q ) sin B } × sinh { Δ 1 * cos ( 3 2 a ℏ q ) cos A + Δ 2 * cos ( a 2 ℏ q ) cos B } ] (27)

where

j o y = − 4 e 3 A † K 2 π Φ τ 2 Θ ( 1 − α 2 ) ℏ 4 ω q 2 v s ϵ ρ a q 1 − α 2 μ H c

and

A = 3 4 sin − 1 ( ω q 12 Δ a q ) , B = 1 4 sin − 1 ( ω q 12 Δ a q )

The dependence of the Hall-like current density on ω q , q and T as expressed above is highly nonlinear.

The acoustic wave considered in this study is treated as packets of coherent phonons (monochromatic phonons) with wavelength λ = 2 π / q , smaller than the mean free path of the FSWCNT electrons in the hypersound region q l ≫ 1 (q is the modulus of the acoustic wavevector and l is the electron mean free path).

Equations (23) and (26) and Equations (24) and (27) can be written in terms of the axial AE current as:

j y A M E = j z A E Ω τ . (28)

The Hall-like current is proportional to the axial acoustoelectric current but depends on the dimensionless quantity Ω τ which is a measure of the magnetic strength i.e. j A M E / j A E = Ω τ . It is therefore empirical that the existence of the Hall-like current in FSWCNT is due to the electron quantization and the non-parabolicity of the energy band, and not on the dependence of τ on ε ( p z ) [

It can be inferred from

increases the conductivity as shown in

Similarly, we display in

In

transfer their energy and momentum to the electrons to increase j z A E and j y A M E . However, this is not the case at high temperatures because the energy of the electron-phonon interactions increases due to increase in their kinetic energies and collisions with other excitations. This leads to a handful of electrons undergoing intraminiband transition, leading to a decrease in both j z A E and j y A M E . Therefore, there is a threshold temperature for which the j z A E and j y A M E turn on, suggesting that FSWCNT can be used as an acoustic switch or acoustic transistor. These occurs at T = 35 K for q = 4 × 10 6 cm − 1 , T = 56 K for q = 5 × 10 6 cm − 1 , T = 84 K for q = 6 × 10 6 cm − 1 and T = 112 K for q = 7 × 10 6 cm − 1 . As shown in

In

The relation between the attenuation coefficient, and the surface electric field due to the Hall-like current was proposed by Yamada [

Γ a b s Φ = n o e E S A M E μ H / ℏ c (29)

The absorption coefficient Γ a b s for FSWCNT is given in ref. [

Γ a b s = 2 A † π Φ 2 K 2 Θ ( 1 − α 2 ) 3 ℏ 2 ω q 2 ρ v s Δ ϵ a q 1 − α 2 × [ sinh { Δ 1 * sin ( 3 2 a ℏ q ) sin A + Δ 2 * sin ( a 2 ℏ q ) sin B } × cosh { Δ 1 * cos ( 3 2 a ℏ q ) cos A + Δ 2 * cos ( a 2 ℏ q ) cos B } ] (30)

Thus, the surface electric field due to the Hall-like current E S A M E is expressed as:

E S A M E = 2 A † π Φ 2 K 2 Θ ( 1 − α 2 ) 3 ℏ 2 ω q 2 ρ v s Δ ϵ a q 1 − α 2 n o e ( μ H ℏ c ) × [ sinh { Δ 1 * sin ( 3 2 a ℏ q ) sin A + Δ 2 * sin ( a 2 ℏ q ) sin B } × cosh { Δ 1 * cos ( 3 2 a ℏ q ) cos A + Δ 2 * cos ( a 2 ℏ q ) cos B } ] (31)

The drift velocity of the electrons is given as:

v d = μ E S A M E = 2 A † π Φ 2 K 2 Θ ( 1 − α 2 ) 3 ℏ 2 ω q 2 ρ v s Δ ϵ a q 1 − α 2 n o e ( μ 2 H ℏ c ) × [ sinh { Δ 1 * sin ( 3 2 a ℏ q ) sin A + Δ 2 * sin ( a 2 ℏ q ) sin B } × cosh { Δ 1 * cos ( 3 2 a ℏ q ) cos A + Δ 2 * cos ( a 2 ℏ q ) cos B } ] (32)

Thus, we display the dependence of the Hall-like field, E S A M E , on q in the absence of an electric field in

Hall-like field decreases but shifts toward higher q values (see

We display the dependence of the E S A M E on 1 − v d / v s when the external electric field is switched on as calculated in Equation (31) (see

To put the results in perspective, we show in

We have shown that Hall-like current and field can be induced by acoustic phonons in a nondegerate, semiconductor FSWCNT using a tractable analytical approach in the hypersound regime q l ≫ 1 . A strong nonlinear dependence of Hall-like current j y A M E and the field E S A M E on H, q, T and Δ are observed.

This is due to the attenuation of acoustic phonons by electric field driven electrons experiencing intraminiband transition. Qualitatively, Hall-like current and field exist even if τ does not depend on the carrier energy but has a strong spatial dispersion. This result is different from that obtained in bulk semiconductors. In the case when τ is constant, the effect is only present in non-degenerate electron gas but absent in degenerate electron gas. These results suggest that FSWCNT could offer a huge potential for room temperature applications, however, novel techniques are needed in reducing its high electrical resistance. The acoustic wavenumber and the overlapping integral can be used to tune the Hall-like current and field of the FSWCNT for room temperature applications such as acoustic switch or transistor and also as a material for ultrasound current source density imaging (UCSDI) and AE hydrophone devices in biomedical engineering.

The authors declare no conflicts of interest regarding the publication of this paper.

Sakyi-Arthur, D., Mensah, S.Y., Adu, K.W., Dompreh, K.A., Edziah, R., Mensah, N.G. and Jebuni-Adanu, C. (2020) Induced Hall-Like Current by Acoustic Phonons in Semiconductor Fluorinated Carbon Nanotube. World Journal of Condensed Matter Physics, 10, 71-87. https://doi.org/10.4236/wjcmp.2020.102005