Terahertz Conductivity of Nanoscale Materials and Systems

Abstract

The history of RF technology can provide human beings a powerful lesson that the infrastructure of modern-day wireless communication depends on the complexity and configurability of silicon-based solid-state devices and integrated circuits. The field of THz technology is undergoing a developmental revolution which is at an inflection point and will bridge the ‘technology’ and ‘application’ gap in meaningful ways. This quantitative progress is a result of continuous and concerted efforts in a wide range of areas including solid-state devices, 2D materials, heterogeneous integration, nanofabrication and system packaging. In this chapter, the innovative theoretical approaches that have enabled significant advancement in the field of system-level THz technology are discussed. The focus is kept on the formulation of terahertz conductivity which plays a critical role in the modeling of devices that integrate technologies across electronics and photonics. Further, the findings build on coupling a probe pulse of terahertz illumination into the photoexcited region of amorphous silicon are presented and discussed in detail. Terahertz light has a higher penetration depth for opaque semiconductor materials which provides an accurate method to measure the conductivity of novel materials for the construction of efficient solar cells. This paves the way for the possibility to develop energy systems can address the need for reconfigurability, adaptability and scalability beyond the classical metrics.

1. Introduction

Terahertz radiation generally accounts for the photons of energy ranging from 414 μeV to 41.4 eV. This energy spectrum corresponds to the frequencies of 0.1–10 THz in the electromagnetic spectrum. In the past, it has been referred to as the ‘terahertz gap’ primarily due to the inefficient methods for generation and detection of waves in the frequency range [1, 2]. The major reason for the complexity in the designing of efficient THz systems was the presence of background noise sources in incoherent light (room temperature is 25 meV, or 6 THz) [2, 3, 4]. Over the past few decades, due to the advancement in the nanoscale fabrication, there has been a significant surge of progress in enabling integrated, compact and efficient chip-scale solid-state semiconductor technology that can operate at room temperature and can be manufactured at a low cost, exploiting economies of scale. Considerable work has been directed towards miniaturized technologies demonstrated with quantum-cascade lasers [5], microbolometers [6], nanowires [7], novel plasmonic nanostructures [8], metamaterials [9] and ultrafast conductive semiconductor materials [10]. This progress has resulted a major step towards a more holistic approach for realizing the development of THz systems for applications in communication, spectroscopy and hyperspectral imaging [11].

Interaction of the terahertz radiation with matter provides a vital low-energy probe to the electronic nature of doped semiconductors by generating carriers optically with a fraction of laser beam [12, 13], this technique is known as Optical Pump Terahertz Probe Spectroscopy (OPTPS). This technique has the benefit of allowing the carrier density to be readily controllable, and also allows the photoconductivity to be measured on picosecond time scales after photoexcitation. A detailed description of the OPTPS is presented by the authors of paper; [14], and further, the authors discussed the terahertz spectroscopy method to measure transient variation in the photoconductivity of bulk and nanostructured semiconductors. There are a number of terahertz spectroscopic techniques and applications which have been presented and discussed from various scientists around the world. The studies based on the terahertz spectroscopy of polymer-fullerene based hetrojunctions are reviewed by the authors of [15] and the utilization of near-field terahertz for ultrafast imaging is explained in depth by the authors of [16].

On the electronics front for the development of terahertz-based devices and integrated circuits (ICs), researchers across various disciplines have converged to address the technological challenges in the field. Scientists have made numerous efforts in engineering the fabrication of novel nanomaterials, and most importantly, in the development of theoretical framework to optimize the design parameters of the devices and ICs. The main result of this effort is reflected in the ability of various technologies to generate terahertz signals with sufficient power levels. Particularly in frequencies beyond 1 THz, the semiconductor technologies based on III–V group elements such as InP heterojunction bipolar transistors (HBTs), high electron mobility transistors (HEMTs) and GaAs-based Schottky diodes are now capable of generating power levels almost two to five times higher than a decade ago [17, 18]. Silicon-based integrated technology, providing a platform for massive integration has demonstrated complex phased array, imaging and communication systems with output power reaching up to the 100 μW at the terahertz frequency range [19, 20].

Semiconductor device modeling represents a physics-based analytical modeling approach to predict device operations at specific conditions such as applied bias (voltages and currents), environment (temperature and noise) and physical characteristics (geometry and doping levels). The fundamental understanding of the charge carrier dynamics plays a crucial role in the formulation of numerical models by implementing mathematically fitted conductivity equations. Numerical models are primarily used as virtual environments for device optimization to directly estimate the conductivity providing insight into the nature of charge carriers, their mobility and its dependence on time. In this chapter, the focus is on the conductivity at terahertz frequencies of bulk and nano-structured solid state materials. A variety of theoretical formalisms that describe the terahertz conductivity of bulk, mesoscopic and nanoscale materials are explained in the chapter (Section 2). The validity and limitations of the respective formulations are discussed to highlight the boundary of implementation for accurate calculations. The chapter starts with the definition of the complex conductivity (Section 3) and then the experimental method (Section 4) to obtain complex conductivity is described to develop the understanding of practical implementation. Finally, the effectiveness of surface passivation using hydrogenated amorphous silicon is quantified through time resolved terahertz spectroscopy and the results of the measurements are presented in the form of change in photoconductivity upon excitation from terahertz electric field (Section 6). Further, in the section, the results of Fourier transform infrared spectroscopy are discussed to comment upon the nature of bond formation in the amorphous silicon passivation layer.

The complex conductivity ($\sigma$) can be defined as a mathematical term that relates the current density vector ($\overrightarrow{J}$) to the electric field vector ($\overrightarrow{E}$). This relation is similar to the way that current ($I$) and voltage ($V$) in an ac circuit are related by the complex conductance ($G$), the stated analogy can be understood by the following set of equations,

$$I=G\times V$$E1

$$G=\sigma \times \frac{A}{L}$$E2

$$V=E\times L$$E3

$$J=\frac{I}{A}=\frac{\sigma \times \frac{A}{L}\times E\times L}{A}$$E4

where $L$ is the length of the conductor and $A$ is the cross-sectional area of the conductor. Hence in vector form,

$$\overrightarrow{J}=\sigma \overrightarrow{E}.$$E5

The response of a material to an electromagnetic wave is generally described by the dielectric constant of the material. Fundamentally, the refractive index of a material quantifies the combined response of a material to a electric and magnetic field which is given by the following equation,

$$\tilde{n}=n+\mathit{i\kappa }=\sqrt{\mathit{\epsilon \mu }}$$E6

where the dielectric function or electrical permittivity ($\epsilon$) describes the ability of an electric field to penetrate the material medium and the magnetic permeability ($\mu$) quantifies magnetic response of the medium. The dielectric function is generally complex with the significantly higher imaginary part (${\epsilon }_2$) as compared to the real part (${\epsilon }_1$) in spectral regions with dominant absorption.

$$\epsilon ={\epsilon }_1\left(\omega \right)+i{\epsilon }_2\left(\omega \right)=\mid \epsilon \mid e^{-\mathit{\iota \delta }}=\mid \epsilon \mid \left(cos\delta -\iota sin\delta \right)$$E7

where ${\epsilon }_1$ and ${\epsilon }_2$ is the real and imaginary part of the permittivity and $\delta$ is the loss angle. It is often convenient to disentangle the contribution to $\epsilon$ from bound modes (lattice vibrations and core electrons) and the component arising from free charges. In the THz and mid-IR frequency range, lattice vibrations contribute via transverse optical phonon absorption (in polar materials), while at optical frequencies interband transitions alter $\epsilon$. The lattice component can be derived from the Maxwell’s equations and can be mathematically written in the following way,

$$\epsilon \left(\omega \right)={\epsilon }_L\left(\omega \right)+\frac{\mathit{i\sigma }\left(\omega \right)}{{\mathit{\omega \epsilon }}_0}$$E8

where $\omega$ is the angular frequency of the electromagnetic wave, $\sigma$ is the electrical conductivity, ${\epsilon }_0$ is the electrical permittivity of free space and ${\epsilon }_L\left(\omega \right)$ is the component of $\epsilon$ from bound modes (lattice vibrations and core electrons). The electrical conductivity ($\sigma$) of mobile charges is also defined complex in nature and hence can be written as,

$$\sigma \left(\omega \right)={\sigma }_1\left(\omega \right)+i{\sigma }_2\left(\omega \right)$$E9

Generally, a valid assumption for the equations derived from Maxwell’s equation is the excitation from a conventional monochromatic plane wave propagating in the $\widehat{k}$ direction, with a time dependent electrical field, which is defined as,

$$E\left(t\right)=E_0{e}^{i\left(k.r-\mathit{\omega t}\right)}$$E10

where $E_0$ represents the maximum electric field strength, $r$ is the three-dimensional position vector of a point, and $k\cdot r$ denotes the projection of the direction vector $\widehat{k}$ along the position vector $r$.

2. Models for conductance at terahertz frequency

In this chapter, various models for the terahertz conductivity are compared and outlined in sequential manner, initially the basic fundamental physics of charge transport is described to develop the understanding of terahertz conductivity in a homogeneous media. The chapter starts with the description of Drude-Lorentz model which is applicable in classical domain of charge transport and further discussed about the limitation in the applicability of the model. Further, the chapter provides the discussion of the models that functions effectively at the quantum mechanical domain and elucidates the mathematical expressions for the terahertz conductivity. The relationship between the length scale and the applicability of various conductivity models is demonstrated with the help of a schematics shown in Figure 1, it clearly separates the quantum and classical conductivity models based on the length scale of electrons mean free path in the semiconductor. The schematics also shows the conductivity models namely Drude-Smith and Plasmon model which consider the interfacial scattering of electrons into account for the sub-micrometer length scale of the semiconductor material.

Figure 1.

The relevant terahertz conductivity models of blends and micro/nano-materials at different length scales.

3. Complex conductivity

A classical circuit theory forms a simple analog to examine the complex conductivity $\sigma ={\sigma }_1+i{\sigma }_2$. A voltage wave of the mathematical form $V=V_0{e}^{-\mathit{i\omega t}}$ generates a complex impedance $Z=Z_R+i\mid \left(Z_L+Z_C\right)=R+i\left(-\mathit{\omega L}+1/\mathit{\omega C}\right)$ where R, L and C are resistance, inductance and capacitance of the circuit respectively. A non-polarisable material provides a good model system to theoretically understand the functioning of a bulk semiconductor with a single charge carrier, in which the current lags behind the applied voltage (a resistor $R$ in series with an inductor $L$). It is known that conductance is defined as an inverse of resistance, hence it can be written that the complex conductivity ($\sigma$) is proportional to the inverse of complex impedance ($Z$) and therefore, mathematically it can be expressed as, $\sigma \propto 1/Z\propto R+\mathit{i\omega L}$. Similarly, the charge transport in a polarisable material can be examined by including a capacitor $C$ in the equivalent circuit. A circuit consisting of resistor and capacitor in series configuration, the complex impedance is given by $Z=R+i/\mathit{\omega C}$ and the complex conductivity is proportional to $\sigma \propto 1/Z\propto R-i/\mathit{\omega C}$. It should be noted that in the case of non-polarisable material the imaginary part of complex conductivity is positive but for polarisable material it has a negative sign. This property plays a very important role in the mechanics of charge transportation for polarisable medium.

3.1 Drude-Lorentz model

Drude-Lorentz model provides the simplest approach to mathematically interpret the frequency-dependent conductivity of metals and semiconductors. This model considers the charge carriers i.e. electrons and holes transportation as a non-interacting gas plasma. It is assumed in the model that the charge carriers undergo random collisions at a rate $\mathrm{\Gamma }=1/\tau$ where $\tau$ denotes the relaxation time between collisions and $\mathrm{\Gamma }$ is assumed to be independent of energy in the system. According to the Drude-Lorentz model, the displacement $x$ of the charge carrier with effective mass $m^{\ast }$ and charge $q$ from its equilibrium position is given by,

$$\frac{d^{2}x}{{\mathit{dt}}^2}+\mathrm{\Gamma }\frac{\mathit{dx}}{\mathit{dt}}=\frac{{\mathit{qE}}_{\text{local}}}{m^{\ast }}$$E11

where at normal incidence the local electric field $E_{\text{local}}$ of the terahertz pulse is related to the incident electric field $E_{\mathrm{THz}}$ by the following equation, $E_{\text{local}}=2E_{\mathrm{THz}}/\left(n+1\right)$. The term ${\mathit{qE}}_{\text{local}}\tau /m^{\ast }$ represents the velocity by which the charge carriers drift in the presence of constant applied field. It can be assumed from the linear nature of the differential equation that if the incident electric field oscillates at angular frequency $\omega$ and has the $E_{\mathrm{THz}}=E_0{e}^{-\mathit{i\omega t}}$ mathematical form, then the displacement $x$ has the solution of an identical time variation i.e. $x=x_0{e}^{-\mathit{i\omega t}}$. The value of $x_0$ can be yielded by substituting the terms of $E$ and $x$ in Eq. (11) which represents the mean oscillating amplitude of the charge carrier. The dielectric function $\epsilon$ of the medium can be determined from the definition of the polarization from $P={\mathit{\chi \epsilon }}_{0}E=\mathit{Nqx}$, and then the conductivity as a function of frequency can be expressed from the following equation,

$$\sigma \left(\omega \right)=\frac{{\mathit{Nq}}^2}{m^{\ast }}\frac{\tau }{1-\mathit{i\omega \tau }}.$$E12

It should be noted form the equation that the real part (${\sigma }_1$) of the complex conductivity has a maximum at the zero frequency and similarly the imaginary part has a maximum at the frequency $\omega =1/\tau$. Intraband conductivity of some metals and semiconductors are adequately modeled by the Drude-Lorentz conductivity, however there are some variables which are considered constants for mathematical simplicity, such as electron scattering rate which is dependent on energy, has been considered constant in the Drude-Lorentz model. Most importantly, it is assumed in the model that the material is uniform over the length scale that is traversed by electrons during their motion which often significantly varies in the experimental observations [21]. These limitations are the fundamental reason for the Drude-Lorentz model to be invalid for two-dimensional materials, having examined these limitations it is important to consider alternative terahertz conductivity models to establish more accurate simulation environment.

3.2 Plasmon model

The plasmon model is a straightforward extension to the Drude-Lorentz model, in this model a restoring force with an external electromagnetic wave derives the motion of electrons which is governed by the equation of motion of a damped, driven simple harmonic oscillator [22]. The restoring force can be provided by electromagnetic or for instance electrostatic, by a surface depletion layer or accumulation layer, which is often the case when calculating for the charge transportation at the interface of semiconductors. Hence, the plasmon model is mostly applicable to model the terahertz conductivity of semiconductor nanomaterials. A mathematical term ${\omega }_0^{2}x$ is added to the left-hand side of Eq. (11) which represents the applied restoring force and where ${\omega }_0$ is the angular frequency of the oscillatory response.

$$\frac{d^{2}x}{{\mathit{dt}}^2}+\mathrm{\Gamma }\frac{\mathit{dx}}{\mathit{dt}}+{\omega }_0^{2}x=\frac{{\mathit{qE}}_{\text{local}}}{m^{\ast }}$$E13

A larger restoring force requires a greater value of ${\omega }_0$. The complex conductivity in the model is given by the equation,

$$\sigma \left(\omega \right)=\frac{N\times e^2}{m^{\ast }}\frac{\tau }{(1-(\mathit{i\tau }\left(\omega -{\left({\omega }_0\right)}^2/\omega \right)}$$E14

The oscillation frequency (${\omega }_0$) can be mathematically linked to plasmon frequency (${\omega }_p$) for some particular geometries by the following equations,

$${\omega }_p=\sqrt{\frac{N\times e^2}{m^{\ast }{\epsilon }_{\infty }{\epsilon }_0}}$$E15

and

$${\omega }_0^2=f\times {\omega }_p^2$$E16

The equations are valid for the assumption that the charges are located on the surface of a small spherical particle which has the geometric factor $f=1/3$ [22]. while for cylindrical wires $f=0$ when the electric field $E_{\mathrm{THz}}$ is axial and the wave vector $q_{\mathrm{THz}}$ is radial, $f=1/2$ when $E_{\mathrm{THz}}$ and $q_{\mathrm{THz}}$ are radial, and $f=1/3$ for radial $E_{\mathrm{THz}}$ and axial $q_{\mathrm{THz}}$ [23, 24]. There has been many practical implementation of the classical conductivity expression such as it has been applied in temperature-dependent terahertz time domain spectroscopy studies of doped silicon, however there are several reasons why the classical relaxation-effect model is not sufficient for a quantitative account of experimental observation, even when interband transitions can be safely ignored or calculated separately:

1. Conduction bands with normal metals can be significantly different from the ideal spherical energy band (that assumes an effective electron mass, $m$).

2. Scattering relaxation time is assumed to be independent of energy $E\left(k\right)$, where $k$ = wave vector of the electron, even though it depends on both energy and position. For this reason, $\tau$ is considered to be a semi-empirical parameter.

3. A local-response regime is assumed.

The above reasons highlight the importance to achieve even better modeling accuracy and therefore it is necessary to move towards the more complicated semiclassical treatment of terahertz conductivity of materials.

3.3 Phenomenological conductivity model

In the Drude-Lorentz model, it is assumed that the charge carriers are propagating in the bulk of the material which means that the carrier displacement is less than the largest dimension $L$ of the material. Thus, it can be argued that when an electron or hole’s displacement under the electric field $E_{\mathrm{THz}}$ becomes comparable to the dimension $L$ of the material then the conductive response of the material will differ significantly from that of the bulk material. Mathematical, this can be understand by the criterion of mean free path such that the electron or hole mean free path $l$ should be greater than $L$, i.e. $l=\mathit{v\tau }>L$, where the carrier velocity is often equated with the thermal velocity which is given by, $v=\sqrt{\frac{3k_{B}T}{m^{\ast }}}$. The carrier velocity defines the This relation can be demonstrated by some examples; firstly, for room temperature electrons close to the $\mathrm{\Gamma }$-valley minimum of GaAs with the values of $v\sim 5\times {10}^5$ m/s and $\tau \sim 100$ fs, this yields the mean free path ($l$) of 50 nm and thus, the dimension $L$ of the material should be less than 50 nm. Secondly, for silicon $m^{\ast }=0.26$ me, $v\sim 2\times {10}^5$ m/s, and $\tau \sim 100$ fs, the estimate for $l$ is 20 nm. The materials which satisfy this dimensional constraint, the scattering from the interface boundary must play an important role in the calculation of charge-carrier transportation. This argument is based on three important underlying assumptions which are stated as follows, firstly, the carrier velocity should be equated to the Fermi velocity ($v_F$) rather than be equal to the thermal velocity. The electrons which occupies energy states within $k_{B}T$ of the Fermi surface will be able to perform excitation from an occupied state to an unoccupied state. When the material is excited through photons of energy very high as compared to the bandgap, then the mean carrier velocity will be a function of time which considers the fact the charge carriers will decay to an equilibrium energy state after the excitation source is removed. Secondly, the terahertz electric field can provide electrons significant amount of energy when $E_{\mathrm{THz}}$ is larger compared to the electron affinity. Thirdly, at the scale of tens of nanometers, quantum confinement effects can become significant and could play an important role in the conductivity. For instance, in a GaAs quantum well with infinite energy barriers the spacing between the lowest two electron sub-bands when $L=63$ nm corresponds to 1 THz. Therefore, it can be concluded that the phenomenological conductivity models are applicable when the average electron displacement is approximately equals to the dimension of device but when the quantum confinement effects are not of significant magnitude.

The conductivity of charge carrier undergoing restricted motion is derived from an extension of the Drude-Lorentz model, in which the charge carrier conserves a part of its initial velocity upon scattering from a boundary layer energy barrier [25]. It is assumed that the collisions are randomly distributed in time and $c_n$ denotes the fraction of initial velocity of charge carrier which is conserved after $p$ scattering events in time. With these numerical assumptions, the Drude-Lorentz model can be generalized into a different mathematical form which can be written as,

$$\sigma \left(\omega \right)={\sigma }_D\left(\omega \right)\left[1+\sum _{p=1}^{\infty }\frac{c_p}{{\left(1-\mathit{i\omega \tau }\right)}^p}\right]$$E17

${\sigma }_D\left(\omega \right)$ denotes the conductivity of Drude-Lorentz model, and $c_p$ is the expectation value $⟨cos\theta ⟩$ for scattering angle $\theta$. Statistically, if the scattering is a total random process then the carrier’s momentum will be randomized which gives $c_p=0$, on the other hand if the carrier is completely backscattered then $c_p=-1$. The infinite series of Eq. (17) is generally truncated at $p=1$, which corresponds to the assumption that the charge carrier conserves a part of its initial momentum after the first scattering event, but in every subsequent scattering events the velocity is randomized. This model is called as Drude-Smith model. The Drude-Smith model can be overlapped with surface plasmon model when $c_1=-1$ and $c_{p>1}=0$, and with the following mathematical substitution,

$${\omega }_0^2=\frac{1}{{\tau }_{\mathit{DS}}^2}$$E18

$$\frac{1}{{\tau }_{\mathit{SP}}}=\frac{2}{{\tau }_{\mathit{DS}}}.$$E19

The subscripts DS and SP are used to distinguish the scattering times denoted in the Drude-Smith model and surface plasmon model. There is an alternate approach to simulate the conductivity of materials with length scales comparable to the mean free path which includes the effect of non-uniformity on the electronic states. The non-uniformity of the electronic states is the result of disordered materials for which long-range delocalized eigen states such as Bloch functions cannot be used to describe the electronic wavefunctions. Tight-binding approach more appropriately describes the localized states of disordered material [26]. The first order correction to the Drude-Lorentz model take the following form in the limit of weak disorder [26, 27, 28, 29],

$${\sigma }_1\left(\omega \right)=\mathit{Re}\left[{\sigma }_D\left(\omega \right)\right]\left(1-A{\tau }^{-2}+{\left(3\omega \right)}^{1/2}A{\tau }^{-3/2}\right)$$E20

where $A$ is the fit parameter ($A=C/{\left(k_F{v}_F\right)}^2$) and it is dependent on the Fermi wavevector ($k_F$), Fermi velocity ($v_F$) and a constant $C$ of order unity. This conductivity expression is referred to as the localized-modified Drude-Lorentz model and can be derived from the quantum conductivity model which is discussed later in the chapter.

Drude-Lorentz model is attempted to provide another extension which include the influence of an energy dependent $\tau$. The Cole-Cole and Cole-Davidson models add exponents to the denominator of Eq. (12) producing the following conductivity equation,

$$\sigma \left(\omega \right)=\frac{N\times e^2}{m^{\ast }}\times \frac{\tau }{{\left(1-{\left(\mathit{i\omega \tau }\right)}^{1-\alpha }\right)}^{\beta }}$$E21

where the Cole-Cole ($\alpha$) and Cole-Davidson ($\beta$) parameter are in the range $0<\alpha <1$ and $0<\beta <1$. The Drude-Lorentz model is the special case of the model of Eq. (21) where $\alpha \to 1$ and $\beta \to 1$. This expression can model small deviations from the Drude-Lorentz conductivity but does not provide elaborate insight into the dominant scattering mechanisms. Most importantly, it cannot reproduce the negative imaginary conductivities which are reported for semiconductor nanomaterials, and this leads to further development of conductivity models.

3.4 Boltzmann conductivity model

A classical approach has been used to model the terahertz conductivity where quantum mechanical effects such as a finite density of energy states, or an energy dependent $\tau$, has not been accounted for in the mathematical equation. Boltzmann transport equation can be used to derive a generalized expression of conductivity in which both the contributions are addressed quantitatively. Under an external perturbation such as an electric field, it is assumed that the rate of change of the electron distribution function is inversely proportional to the scattering time $\tau$, this is called as relaxation time approximation. The frequency dependent conductivity of an isotropic three-dimensional material with parabolic energy bands is derived from the Boltzmann transport equation which is given by,

$$\sigma \left(\omega \right)=\frac{2\sqrt{2m^{\ast }}{e}^2}{3{\pi }^2{\mathrm{\hslash }}^3}{\int }_E\frac{\tau \left(E\right)E^{3/2}}{1-\mathit{i\tau }\left(E\right)\omega }\left(-\frac{\partial f_0}{\partial E}\right)\mathit{dE}.$$E22

The electron distribution function in equilibrium at temperature $T$ is given by the Fermi-Dirac distribution,

$$f_0\left(E\right)=\frac{1}{e^{\left(E-\mu \right)/k_{B}T}+1}$$E23

where the chemical potential $\mu$ can be determined mathematically from the given numerical equation,

$$N={\int }_0^{\infty }{f}_0\left(E\right)g\left(E\right)\mathit{dE}$$E24

where $g\left(E\right)$ denotes the electronic density of states. Fermi’s golden rule can be applied to calculate the energy dependent scattering times $\tau \left(E\right)$ for various scattering mechanisms such as with impurities, acoustic or optical phonons. This approach has the benefit of converting the scattering time which is a fit parameter into a numerically determined quantity obtained from the carrier density of semiconductor. This model provides a physical insight into the underlying microscopic physics because the calculated scattering time $\tau \left(E\right)$ will highlight the dependence on the inclusion of relevant scattering mechanisms. In addition, it can be inferred from the model that the conductivity is dominated by the contribution of electrons with energies close to the chemical potential ($\mu$), this can be understand from the fact that the peak value of the function $\partial f_0/\partial E$ comes at the energy $E=\mu$. The function $\partial f_0/\partial E$ is given by the equation,

$$\frac{\partial f_0\left(E\right)}{\partial E}=\frac{e^{\left(E-\mu \right)/k_{B}T}}{k_{B}T{\left(e^{\left(E-\mu \right)/k_{B}T}+1\right)}^2}.$$E25

1. Skin depth is much less than linear dimensions of metal, thus regarding it as planar and infinite in extent.

2. Normal incidence of propagation, thus simplifying to a one-dimensional problem.

3. Conduction electrons are quasi-free, having a kinetic energy $E\left(k\right)=\mathrm{\hslash }{\left|k\right|}^2/2m$ with a parabolic band approximation, where $\mathrm{\hslash }$ = modified Planck’s constant.

4. Collision mechanism is always described in terms of $l_m$.

5. A fraction p of electrons arriving at the surface is scattered specularly, while the rest are scattered diffusely.

6. This one-band free-electron model does not apply to multivalent metals, in which the electrons occupy more than one energy band (e.g., aluminum or tin).

7. $K\left(s\right)$ is derived for transverse conductivity, and holds for the whole $\left(q,\omega \right)$ plane, for a spherical energy band.

8. While the semiclassical treatment takes into account partially-filled conduction band energy dispersion, $E\left(k\right)$, it ignores electron wavefunctions.

These assumptions laid the foundation of a more accurate theoretical framework which considers the quantum behavior of electrons for the purpose of calculating the charge carrier dynamics inside the semiconductor materials.

3.5 Quantum conductivity model

The conductivity associated with interband and intraband transitions can be determined accurately from the linear response theory or Kubo-Greenwood theory. The transition rate of electrons denoted by $W_{\mathit{ij}}$ between an initial state $i$ and a final state $j$ is calculated by the Fermi’s golden rule, and $\mathrm{\hslash }\omega W_{\mathit{ij}}$ is equated to the power lost in the form of electromagnetic field due to absorption (mathematical formulation of this statement yields the absorption coefficient $\alpha$). Further, the real part of the refractive index can be determined from the absorption coefficient ($\alpha$) by the Kramers-Kronig relations. The resulting expression for the conductivity is given by the equation,

$$\sigma \left(\omega \right)=\frac{2e^2\mathrm{\hslash }}{{\left(m^{\ast }\right)}^{2}V}\sum _{\mathit{ij}}\frac{{\left|⟨{\psi }_j|\widehat{e}\cdot p|{\psi }_i⟩\right|}^2}{E_j-E_i}\times \frac{i\left[f\left(E_j\right)-f\left(E_i\right)\right]}{E_j-E_i-\mathrm{\hslash }\omega -i\mathrm{\hslash }/\tau }$$E26

where $\widehat{e}$ denotes the unit vector in the direction of the electric field ($\overrightarrow{E}$), $p$ is the momentum operator and $V$ represents the volume in which the summation of the dipole matrix element ${\left|⟨{\psi }_j|\widehat{e}\cdot p|{\psi }_i⟩\right|}^2$ is applicable. The matrix element creates selection rules for observable absorption peaks, for instance in semiconductor quantum wells intersubband absorption is only observed if the applied electric field is parallel to the direction of quantum confinement, and if the initial and final states have parity. It is important that the wavefunctions ($\psi$) of initial and final states are already determined, to calculate the quantum expression of conductivity. Additionally, a common formalism should be designed to include the interband, intersubband and intraband transitions. Boltzmann expression of Eq. (22) is the result of solving Eq. (26) for intraband transitions in a parabolic band. The conductivity expression of Eq. (26) can be simplified for intersubband transitions in quantum wells, such as absorption from lowest state 1 to the next highest state $1$, this yields the real part of the conductivity which is given as,

$${\sigma }_1\left(\omega \right)=\frac{e^2{f}_{10}}{2m^{\ast }}\int g\left(E\right)f\left(E\right)\frac{\mathrm{\hslash \Gamma }\left(E\right)}{{\left(\mathrm{\hslash }\omega -E_{10}\right)}^2+{\mathrm{\Gamma }}^2\left(E\right)}\mathit{dE},$$E27

where $f_{10}$ is the oscillator strength and the linewidth in units of energy is denoted by $\Gamma =\mathrm{\hslash }/\mathrm{tau}$. The expression can further be solved in the limit of an energy-independent $\mathrm{\Gamma }$, which yields the following equation,

$${\sigma }_1\left(\omega \right)\propto \frac{\mathrm{\Gamma }}{{\left(\mathrm{\hslash }\omega -E_{10}\right)}^2+{\mathrm{\Gamma }}^2}.$$E28

The most prominent systems to form excitons from the binding of electrons and holes together, are the ones which exhibits quantum confinement and where the excitonic binding energy is enhanced by the quantum effects. The model of hydrogen atom can be used to derive a comparison for the eigen states of an exciton, where 1s, 2s, 2p, 3s,… are in spectroscopic notation. It is important to highlight the difference between the classical approach and the current approach, for the hydrogen atom, the lowest order electric dipole allowed transition (1s-2p) falls in the ultraviolet range while for GaAs quantum wells due to the lower effective masses of electrons and holes, and high dielectric constant, 1s-2p transition results in the excitons of terahertz frequency range. The expression of frequency dependent complex conductivity is yielded from the Fermi’s golden rule,

$$\sigma \left(\omega \right)=\frac{N_X{e}^2}{m^{\ast }}\sum _k\frac{f_{j,k}\omega }{{\omega }_{j,k}^2-{\omega }^2-\mathit{i\omega }/{\tau }_X}$$E29

where $N_X$ and $T_X$ denote the excitonic density and linewidth, $\mathrm{\hslash }{\omega }_{j,k}$ denote the energy difference between the initial state $j$ and final state $k$, and $f_{j,k}$ is the electric dipole matrix element for the transition.

Further, for superconducting materials, Mattis-Bardeen theory provides the quantum mechanical expression for the conductivity. When the terahertz photon energy exceeds the Cooper pair binding energy $2\mathrm{\Delta }$, this results in strong absorption and production of a ${\sigma }_1$ that increases with frequency above $\mathrm{\hslash }\omega =2\mathrm{\Delta }$. Apart from the Mattis-Bardeen theory, there is another model called as two-fluid model to simulate the conductivity and is often utilized to approximate the conductivity in the terahertz range. The condition associated with the applicability of two-fluid model states that the terahertz photon energy should be very less than the Cooper pair binding energy ($E<<2\mathrm{\Delta }$). In this model, the conductivity is the summation of two terms, first is the Drude-Lorentz term which arises from the thermally excited carriers and the second is the superconducting term (${\sigma }_s$) which is purely imaginary. The fraction ($x_S$) of carrier in superconducting state is zero at the superconducting transition temperature while the the fraction ($x_n$) of thermally excited carrier is zero at zero temperature. The conductivity is given by the following equation,

$$\sigma \left(\omega \right)=\frac{{\mathit{Ne}}^2}{m^{\ast }}\left[\frac{x_n\tau }{1-\mathit{i\omega \tau }}+x_s\left(\frac{i}{\omega }+\mathit{\pi \delta }\left(\omega \right)\right)\right]$$E30

where $N$ denotes the sum of the densities of the superconducting Cooper pairs and the normal carriers, and $x_s+x_n=1$. By the Kramers-Kronig relations, it can be stated that the infinite DC conductivity in the superconducting state is inversely proportional to the frequency (${\sigma }_2\propto 1/\omega$).

The extended Drude model provides an additional accurate approach to simulate the terahertz conductivity in quantum materials [30, 31]. It has been extensively applied to the visible and infrared conductivity of metals and superconductors, and is applicable for Fermi liquids with strong electron–electron interactions, rather than the weaker interaction of a Fermi gas. In this model, the single-particle effective mass ($m^{\ast }$) in Eq. (12) is replaced by a frequency-dependent effective mass ($m^{\ast \ast }(\omega$) which is given by the equation,

$$m^{\ast \ast }\left(\omega \right)=m^{\ast }\left[1+\lambda \left(\omega \right)\right]$$E31

and the energy-independent scattering time $\tau$ is replaced by a photon frequency-dependent function ${\tau }^{\ast }\left(\omega \right)$. The effective mass renormalization factor $1+\lambda \left(\omega \right)$ can be experimentally determined from the electron density $N$ by using the equation,

$$m^{\ast \ast }\left(\omega \right)=-{\mathit{Ne}}^2\times \mathit{Im}\left[1/\sigma \right]/\omega .$$E32

4. Experimental determination of complex conductivity

Generally, the studies based on broadband spectroscopy are obtained from a method known as terahertz-time domain spectroscopy (THz-TDS) which measures the electric field of pulses of electromagnetic radiation directly in the real time after the interaction of radiation with the sample material. Short pulses ($<100$fs) of infrared radiation of centre wavelength $800$nm are used in THz-TDS, for example: mode-locked Ti: sapphire laser can be used to generate and detect single-cycle pulses of the radiation. It is known that the time and frequency resolution are inversely proportional to each other, and as the duration of pulses are in sub-picosecond range, this results in a broadband frequency spectrum with significant amount of power from tens of GHz to a few THz. There are other alternate methods to generate terahertz radiation, firstly the down-conversion of IR radiation into the THz range for which non-linear techniques such as difference frequency generation, optical rectification and gas plasma generation are the efficient methods for pulses of high fluences [21, 32]. Secondly, photoconductive emitters can be utilized to generate a pulse of lower fluences where a transient photocurrent generates a pulse of THz radiation [33, 34]. The photocurrent can be a result of various fields that cause charge separation such as an applied electric field (Auston switches), field due to the surface accumulation layer or depletion layer [35, 36], and photo-Dember fields which are created by the difference in the ballistic motion of electron and hole due to the effective mass [36, 37].

Now, lets discuss about the detection methods, there are generally two competing technologies that are being used for the coherent detection of THz pulses which are as follows,

1. The first detection method is based on the change in the polarization state of an IR gate pulse due to the interaction with the electric field of THz pulse, this is known as electro-optic sampling [38, 39]. The polarization state of the pulse is often analyzed by a Wollaston prism and balanced photodiodes, and the electric field of THz pulse can be directly calculated from the experimental signals using known parameters of the electro-optic crystal [40].

2. The second method is the photoconductive detection of THz radiation pulses which is basically a inverse process of emission [34]. A photocurrent created by a gate beam flows between two contacts under the influence of the THz electric field, producing a signal that is proportional to $E_{\mathrm{THz}}$ (when the photoconductive decay time of the material is very short) [41].

Commonly, a THz-TDS setup contains a photoconductive antenna which generates and detect THz radiation. The emitter chip generates a burst of terahertz radiation when excited by the photons of a subpicosecond laser pulse of average beam power in the range of few mW. A silicon lens is generally included in the setup which collimates the excited THz pulse for parallel free space propagation before reflecting from a parabolic mirror. A second mirror reflects the THz pulse onto the detector, and the time-dependence of the THz pulse is obtained by scanning the relative time delay between the emitter excitation pulse and the detection pulse. The conductive sample which is located between the parabolic mirrors, causes a time delay and a reduction in the amplitude of the THz pulse which can be compared with the reference THz pulse measured without the sample. This comparison gives the information about the magnitude and phase of the spectra and hence results in the numerical calculation of absorption coefficient and refractive index. Hence, the conductivity ($\sigma$) of the sample is determined experimentally as the it is defined by the quantity of absorption and refractive index. However, THz-TDS system has a number of advantages including the ability to extract complex optical properties of material, there are of course some disadvantages of the system, such as the high cost of the technology and a narrow frequency range as compared to the Fourier-transform infrared spectrometers. Most importantly, the reflection geometry needs to be precisely monitored to prevent errors entering into the calculated optical properties of the sample [42, 43].

Compared with the THz-TDS method, Time-Resolved Terahertz Spectroscopy (TRTS) is an another important method to probe the conductivity at higher frequencies and is therefore sensitive to THz conductivity instead of static conductivity of materials. The sub-picosecond time resolution is suitable to study the ultrafast dynamics of carriers. The complex photoconductivity is calculated from the real time monitoring of amplitude and phase of emitted THz wave which is excited by the optical pump probe. For THz-TDS, electric field waveforms transmitted through the unexcited sample, $E_0\left(t\right)$, and reference, $E_i\left(t\right)$ are scanned and averaged which typically results in waveforms with a reproducible measured phase delays of femtoseconds range. However, for TRTS measurements, the pump delay is scanned while holding the delay between the gate and terahertz probe pulses fixed at the position that gives the maximum differential electro-optic response $\Delta E_{\mathrm{EXC}}$ to yield the TRTS decay dynamics. The optical pump-probe induced change in conductivity can be calculated by collecting and averaging the data from pump delay scans. Spectral changes to the photoconductivity is determined by measuring the differential electrical field waveforms, $\Delta E_{\mathrm{EXC}}\left(t\right)$, where the delay time between the THz probe and gate is scanned and the delay between the visible pump and THz probe is kept constant. Further, the conductivity of photoexcited samples is determined by analyzing the measured electric field waveforms of $\mathit{\Delta E}\left(\text{peak}\right)$ under the thin-film approximations. The electric field transmission through a sample consisting of a photoexcited layer smaller than the overall sample thickness relative to an non-photoexcited sample is given by, [44].

$$\frac{{\widehat{T}}_s^p\left(\omega \right)}{{\widehat{T}}_s^{\mathit{np}}\left(\omega \right)}=\frac{{\widehat{E}}_s^p\left(\omega \right)}{{\widehat{E}}_s^{\mathit{np}}\left(\omega \right)}=\frac{1+n_{\mathit{sub}}+Z_0{d}^{\mathit{np}}{\widehat{\sigma }}^{\mathit{np}}\left(\omega \right)}{1+n_{\mathit{sub}}+Z_0{d}^p{\widehat{\sigma }}^p\left(\omega \right)}$$E33

where the substrate is defined as the unexcited portion of the sample and the superscripts $\mathit{np}$ and $p$ indicate the non-photoexcited and photoexcited samples respectively. The rearrangement of the expression in Eq. (33) gives the explicit representation for the conductivity of the photoexcited sample. The real part of the conductivity determined by Eq. (33) is related to the direct current conductivity in the low frequency limit by $\sigma =\mathit{eN\mu }$, where $N$ and $\mu$ are the charge carrier density and mobility, respectively, and $e$ is the electron charge. In general, bulk semiconductors generate equal number of both electrons and holes with unit quantum yield following photoexcitation so the conductivity is given by a sum of the electron and hole contributions. This analysis method therefore provides the sum of the electron and hole conductivity in the photoexcited sample regardless of the intrinsic carrier type present and hence can be utilized directly to doped-semiconductor samples.

5. Charge carrier dynamics in hydrogenated amorphous silicon

Solar energy is one of the clean source of technology that has the potential to reduce the anthropogenic damage to the earth and its ecosystem. This puts the photovoltaic materials that make up solar cell as the subject of intensive study and scientists all around the world are analyzing the photovoltaic properties of novel materials to discover the true potential of solar energy [45, 46]. Time Resolved Terahertz Spectroscopy is one of the methods to analyze the conductivity of photovoltaic materials where spectroscopy, in general defines the study of interactions between matter and electromagnetic radiation. The sentence has been corrected. Generally, the field of spectroscopy plays an important role in the study of photovoltaic materials which is necessary for the efficient designing of solar cells. The investigation of early-time film dynamics offers unique opportunities to improve the photoconductive characteristics of the active materials by better understanding the carrier evolution properties.

Terahertz spectroscopy uses ultrashort pulse lasers to study the charge carrier dynamics of matter and electromagnetic radiation within extremely short time scales (nanoseconds to picoseconds). The lasers excite the specimen material in short pulses, which initiates the generation of charge carriers in the materials. Hence, provides a non-contact and accurate method to measure the photo conductivity of the material providing insight into the nature of charge carriers, respective mobility and time dependence of the conductivity. Hydrogenated amorphous silicon (a-Si:H) thin films are attracting increasing attention for their applications to silicon hetrojunction solar cells in surface passivation [47]. The state-of-art deposition method of intrinsic a-Si:H thin films is plasma enhanced chemical vapor deposition (PECVD), in this section we test the passivation capabilities of a-Si:H films on n-type silicon (Si) surface by measuring the change in conductivity ($\mathrm{\Delta }\sigma$) of the material after the optical excitation as expressed by the following equation,

$$\mathrm{\Delta }\sigma =\xi \times \left({\mu }_e+{\mu }_h\right)=-\frac{\mathrm{\Delta }{E}_{\mathit{exc}}\left(\omega \right)}{\mathrm{\Delta }{E}_{\mathit{gs}}\left(\omega \right)}\times \frac{{\epsilon }_{0}c}{{\mathit{Fe}}_0}\times \frac{1}{1-e^{-\mathit{\alpha L}}}$$E34

where $\xi$ is quantum yield of charge generation, ${\mu }_e$ and ${\mu }_h$ are the electron and hole mobility, respectively, $\Delta E_{\mathrm{exc}}$ is the terahertz electric field transmitted through the sample after photo excitation while $\Delta E_{\mathrm{gs}}$ is the ground state terahertz electric field, ${\epsilon }_0$ is permittivity of vacuum, $c$ is velocity of light, F is the fluence in number of photons/cm², $e_0$ is the elementary charge, $\alpha$ is the absorption coefficient, and $L$ the thickness of the sample. The quantity that is obtained from the Eq. (34) has the unit of mobility in cm²V⁻¹ s⁻¹ which is defined as the product of two quantities namely, quantum yield and mobility. The quantum yield is assumed to be unity ($\xi =1$) which means that all absorbed photons are converted to mobile charges. The interplay between the time-dependent change in charge population and mobility defines the shape of the terahertz photoconductivity kinetics. On one hand, the rise in the photo conductivity kinetics elucidates generation of charged species and/or increase in mobility of the charges. On the other hand, decay represents decrease of the mobility due to relaxation and/or disappearance of charge carriers by recombination mechanisms.

Intrinsic a-Si:H films are deposited by an oxford instruments PECVD system using a source of silane gas (SiH₄), where the hydrogenation of films is achieved by providing hydrogen gas (H₂) for the generation of plasma in the deposition chamber. The total pressure inside the chamber is maintained around 400 mTorr with the partial pressure of SiH₄ and H₂ in the ratio of 5:4, which is determined experimentally for the optimum hydrogenation of amorphous silicon. Firstly, the samples are RCA cleaned and then, a dry oxidation of the sample is performed to grow a 100 nm double-sided layer of silicon dioxide (SiO₂) on the silicon sample. Secondly, a forming gas annealing at 300°C is performed followed by the conductivity measurement to determine the reference measurements for the comparison purposes. Thirdly, a wet etching procedure using HF is performed to remove the top-layer of SiO₂ and lastly, the sample is immediately placed in the chamber for the deposition of 30 nm of a-Si:H for surface passivation. The thickness of deposited amorphous silicon films are measured using a J.A. Woollam N-2000 spectroscopic ellipsometry system.

The schematic of the experimental setup is shown in Figure 2 and the results prepared for the measurements performed are shown in Figure 3. A pump probe delay time is varied in order to obtain the photoconductivity kinetics while the gating delay time is fixed at the peak of the terahertz electric field ($E_{\mathrm{THz}}$). The pump-probe delay time is scanned within an interval of 1500 s to record the transmitted terahertz electric field. It should be noted that the photoconductivity obtained with this process represents the lower limit of the mobility (Figure 3(c)). The transmitted terahertz electric field ($E_{\mathrm{GS}}$ and $E_{\mathrm{EXC}}$) responses can be fitted to Drude or Drude-Smith model to elucidate the change generation efficiency (quantum yield) and the mobility independently. The results of the photoconductivty point to a very important parameter that controls charge dynamics in photovoltaic materials: defects and dangling bonds, as evident by the difference in the photoconductivity kinetics of a-Si:H-passivated silicon sample and non-passivated silicon sample (Figure 3(c)). Through the use of time resolved terahertz spectroscopy, these early time processes and parameters of novel solar cell materials can be investigated in nanometer scale. The presence of hydrogen (H) in the passivation layer is by using Fourier Transform Infrared Spectroscopy (FTIR) and the many repetitive measurements were taken to check that no hydrogen loss takes place with time. The integration times were sufficient so as to reduce the statistical errors to a few percent of the average value. Infrared absorption measurements were made with a Perkin Elmer spectrometer on films deposited on crystalline silicon. The infrared spectrum of hydrogenated a-Si generally displays two groups of bands due to the stoichiometric ratio of silicon and hydrogen i.e. $\mathrm{Si}-{\mathrm{H}}_X$ bonds. The first group is designated to the bond wagging (rocking, rolling) band at $\approx 640$ cm⁻¹ which is directly proportional to the hydrogen concentrations and the second group corresponds to the band in the region of wavenumber 2000 cm⁻¹ where the vibrations are designated by the stretching of the Si-H bonds.

Figure 2.

Schematic diagram of the terahertz setup used in probing charge carrier dynamics in hydrogenated amorphous silicon passivation layer. The transmitted electric fields, $E_{\mathrm{GS}}$ (ground state) and $E_{\mathrm{EXC}}$ (excited state), are Fourier transformed and used the conductivity Eq. (34) to obtain the photoconductivity kinetics at the maximum of terahertz electric field ($E_{\mathrm{THz}}$).

Figure 3.

Surface passivation effectiveness of a layer of hydrogenated amorphous silicon (a-Si:H). (a) Schematic of the double-layered sample under observation, it consists of one-sided growth of 100 nm SiO₂ and on the other side 30 nm a = Si:H is deposited through PECVD process. (b) Scanning Electron microscopy (SEM) image of the top surface of the sample where an uniform layer of a-Si:H is clearly visible due to contrast difference between Si and a-Si:H. (c) the photoconductivity kinetics of the sample under study normalized to the excitation density where $N_{\mathrm{EXC}}$= $1\times {10}^{15}$ and $e_0$ is the elementary electronic charge. The effectiveness of surface passivation from a-Si:H can be stated from the difference in the terahertz conductivity between the passivated Si sample and non-passivated Si sample. (d) the results of Fourier transform infrared spectroscopy (FTIR) of the a-Si:H sample compared with the Si substrate to highlight the presence of $\mathit{Si}-H$ and $\mathit{Si}-H_2$ bonds in the passivation layer.

6. Conclusions

Terahertz radiation, in particular the theoretical framework, has become a requirement to design faster and energy efficient devices. Successful first-pass design of components of a system requires the ability to accurately predict the dissipative losses due to charge-carrier scattering in semi-conductor and metallic materials used in waveguides, filters, antennas and optics. A comprehensive description of different formalisms was presented that allows the complex conductivity of semiconductor materials to be calculated at terahertz frequencies by including the energy dependence of the scattering rate. The generalized conductivity reduces to the Drude model in the limit of zero temperature or an energy independent scattering rate. Terahertz time-domain spectroscopy was used to determine the conductivity and the photoconductivity of hydrogen-doped amorphous silicon, which was semi-quantitatively accounted for by the generalized conductivity model. The breadth and depth of the studies of conductive solid state materials using terahertz radiation continues to advance and as the field is very broad, it is very important to understand the fundamental models to better utilized materials for future. Terahertz based spectroscopy methods can be valuable to not only in revealing fundamental processes or micro/nano strategies but also in providing insights for possible optimization of manufacturing procedures for solar cell industry.

Acknowledgments

Rahul Goyal would like to thank Ministry of Human Resource and Development, Government of India, for providing financial support and Centre for Nanoscience and Engineering for the research facilities.

Conflict of interest

The authors declare no conflict of interest.