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Acoustoelectric effect (AE) in a non
-degenerate fluorinated single walled carbon nanotube (FSWCNT) semiconductor was carried out using a tractable analytical approach in the hypersound regime
*q* is the acoustic wavenumber and
*q*, temperature
*T* and the electron
-phonon interactions parameter,
*T*=300K. However, FSWCNT will offer the potential for room temperature application as an acoustic switch or transistor and also as a material for ultrasound current source density imaging (UCSDI) and AE hydrophone devices in biomedical engineering. Moreover, our results prove the feasibility of implementing chip
-scale non
-reciprocal acoustic devices in an FSWCNT platform through acoustoelectric amplification.

Drift, diffusive and tunnelling current flow are the dominant carrier transport mechanisms in semiconductor structures [

Other interesting effects are observed during the process of energy and momentum exchange. These effects occur not only during the scattering of quasi-momentum carriers by lattice vibrations, but also occur when acoustic waves are propagating through these structures. Among the mechanisms witnessed include: absorption (amplification) of acoustic phonons [

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

Fluorination plays a significant role in the doping process, as it provides a high surface concentration of functional groups, up to C_{2}F without destruction of the tube’s physical structure. Doping is an easy, fast, exothermic reaction. The repulsive interactions of the fluorine atoms on the surface debundles the nanotube, thus enhancing their electron dispersion. Consider a fluorine modified SWCNT (n, n) with the fluorine atoms forming a one-dimensional chain. A nanotube of this nature is equivalent to a band with unit cell as shown in

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. Choosing N = 2 , the energy dispersion for FSWCNT at the edges of the Fermi surface is expressed as:

ε ( p ) = ε o + 8 Δ cos 3 ( a p z ) (2)

Expanding Equation (2) yields:

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

where Δ 1 = 2 Δ , Δ 2 = 6 Δ , and Δ is the overlapping integral for jump. ε o is the minimum electron energy in the first Brillouin zone with momentum p o , i.e. − π / a ≤ p o ≤ π / a .

Following the model developed in refs. [

j z A E = − e ∑ n , n ′ ∫ U n , n ′ a c Ψ i ( p z ) d 2 p z (4)

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

v ∂ Ψ i ∂ p + W p { Ψ } = v i (5)

p z is the electron momentum along the axial direction of the FSWCNT and U n , n ′ a c in Equation (4) is the electron-phonon interaction term expressed as:

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 ) } (6)

here f ( p z ) = f ( ε n , n ′ ( p z ) ) is the unperturbed electron distribution function, Φ is the sound flux density, v s is the sound velocity in the medium, ε n , n ′ ( p z ) is the energy band, n and n’ denotes the quantization of the energy band, and G ( p z ± ℏ q , p z ) is the matrix element of the electron-phonon interaction. Denoting p ′ z = p z ± ℏ q and substituting into Equation (6) and employing the principle of detailed balance, we obtain:

| G p ′ , p | 2 = | G p , p ′ | 2 (7)

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

| G p ′ , p | = 4 π e K 2 ρ ω q ϵ (8)

where K is the piezoelectric modulus, ϵ is the lattice dielectric constant, ρ is the density of FSWCNT and the net AE current density is given as:

j z A E = − 2 e ( 2 π ℏ ) 2 2 π Φ ω q v s ∑ n , n ′ | G p ′ z , 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 ) (9)

where Ψ i ( p z ) = l i ( p z ) is the electron mean free path defined as:

l z = τ v z (10)

and

v z = ∂ ε ( p z ) ∂ p z (11)

Substituting Equation (10) and Equation (11) into Equation (9) yields

j z A E = − 2 e ( 2 π ℏ ) 2 2 π Φ ω q v s 16 π 2 e 2 K 2 2 ρ ω q ϵ ∑ n , n ′ [ f ( ε n ′ ( p z ) ) − f ( ε n ′ ( p z + ℏ q ) ) ] × [ l z ( p z + ℏ q ) − l z ( p z ) ] δ ( ε n ′ ( p z + ℏ q ) − ε n ′ ( p z ) − ℏ ω q ) (12)

The electron distribution function in the presence of the applied electric field, E ( t ) is obtained by solving the Boltzmann transport equation in the τ-approximation. That is

∂ 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 (13)

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 ′ ) (14)

and f o ( p z ) is the Fermi-Dirac distribution given as

f o ( p z ) = 1 [ exp ( − ( ε ( p z ) − μ ) / k T ) + 1 ] (15)

where μ is the quasi Fermi-level which ensures the conservation of electrons, k is the Boltzmann’s constant, T is the absolute temperature in energy units. Substituting Equation (14) and Equation (15) into Equation (10), we obtain an equation for j z A E as:

j z A E = − 2 e ( 2 π ℏ ) 2 2 π Φ ω q v s 16 π 2 e 2 K 2 2 ρ ω q ϵ ∑ n , n ′ [ F 1 / 2 ( ε n ′ ( p z ) ) − F 1 / 2 ( ε n ′ ( p z + ℏ q ) ) ] × [ l z ( p z + ℏ q ) − l z ( p z ) ] δ ( ε n ′ ( p z + ℏ q ) − ε n ′ ( p z ) − ℏ ω q ) (16)

which contains the Fermi-Dirac integral ( F 1 / 2 ) of the order 1/2 as

F 1 / 2 ( η f ) = 1 Γ ( 1 / 2 ) ∫ 0 ∞ η f 1 / 2 d η 1 + exp ( η − η f ) (17)

where ( μ − ε c ) / k T ≡ η f . For nondegenerate electron gas, where the Fermi level is several kT below the conduction band edge ε c , (i.e. k T ≪ ε c ), the integral in Equation (17) approaches 2 / π exp ( η f ) . The unperturbed distribution function can be expressed as:

f o ( p z ) = A † exp ( − [ ε ( p z ) ] / k T ) (18)

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 ) (19)

here n o is the electron concentration, and I o ( x ) is the modified bessel function of zero order. Assume the electrons are confined to the lowest mini-band, then n = n ′ = 1 . The velocity of 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 ) ] (20)

Making use of the transformation

∑ p → 2 e ( 2 π ℏ ) 2 ∬ d 2 p z (21)

and substituting Equation (14)-(20) into Equation (12), with a little bit of algebra, we obtain the AE current density as:

j z A E = 4 A † π Φ e 3 K 2 Θ ( 1 − α 2 ) ℏ 3 ω q 2 ϵ 2 ρ a q 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 ( p ′ a ) 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 ) ] } (22)

where

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

Simplifying Equation (22) yields

j z A E = j z A E ( 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 ) ] } (23)

where j z A E ( 0 ) is the acoustoelectric current density in the absence of an external electric and is given in [

j z A E ( 0 ) = j o [ 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 } ] (24)

and

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

The AE current density obtained in Equation (23) shows a strong nonlinear dependence on the acoustic wavenumber (q), frequency ( ω q ) temperature (T) and the dimensionless electric field (

Equation (23) can be solved explicitly under two conditions: 1) in the absence of an electric field when

The AE current density

The dependence of

The AE current as shown in ^{16} - 10^{19} cm^{−3} without introducing strong electron-electron interactions. Higher

However, in

AE was studied in a non-degenerate FSWCNT semiconductor using a tractable analytical approach in the hypersound regime

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. and Jebuni-Adanu, C. (2020) Semiconductor Fluorinated Carbon Nanotube as a Low Voltage Current Amplifier Acoustic Device. World Journal of Condensed Matter Physics, 10, 12-25. https://doi.org/10.4236/wjcmp.2020.101002