Modulation‐doped semiconductor nanostructures exhibit extraordinary electrical and optical properties that are quantum mechanical in nature. The heart of such structures lies in the heterojunction of two epitaxially grown semiconductors with different band gaps. Quantum confinement in this heterojunction is a phenomenon that leads to the quantization of the conduction and the valence band into discrete subbands. The spacing between these quantized bands is a very important parameter that has been perfected over the years into device applications. Most of these devices form low‐dimensional charge carriers that potentially allow optical transitions between the subbands in such nanostructures. The transition energy differences between the quantized bands/levels typically lie in the infrared or the terahertz region of the electromagnetic spectrum and can be designed according to the application in demand. Thus, a proper understanding and a suitable external control of such intersubband transitions (ISTs) are not only important aspects of fundamental research but also a necessity for optoelectronic device applications specifically towards closing the terahertz gap.
- intersubband transition
Low‐dimensional semiconductor heterostructures, otherwise known as semiconductor nanostructures, have tremendously revolutionized both the technical and the fundamental aspects of semiconductor industry in terms of device applications. With the ability to grow clean and high‐quality samples, device implementations have become a huge success [1–3]. When the dimensions of a region with free carriers (electrons) are reduced as compared to the bulk and approach the deBroglie wavelength, the electronic motion is quantized, thus resulting in carrier confinement that is quantum mechanical in origin. The phenomenon has been widely used for carrier confinement in one, two and three dimensions that consequently gives rise to nanostructures such as quantum wells, quantum wires and quantum dots, respectively. Due to the quantum confinement, the energy bands (i.e. the conduction and valence bands) are quantized into discrete energy levels/bands and are no longer continuous as in the bulk systems. These quantized energy states are known as subbands for 2D or 1D systems and sublevels for 0D systems. The energetic spacings between these quantized subbands and the sublevels are very important parameters that define the device applications both from an optical and from an electrical point of view.
The intersubband spacings in GaAs‐based 2D systems are typically in the order of 10–30 meV [4, 5], as seen in the case of two‐dimensional electron gas (2DEGs) with a triangular confinement potential formed across a GaAs/Al
2. High electron mobility transistor design
The low‐dimensional charge carriers, trapped in the heterojunction of the HEMT design, form the core of such field‐effect transistors. This transistor design also goes by the name of modulation‐doped field‐effect transistors (MODFET). These designs are used in various high‐power  and high‐speed  electronics, high‐resolution imaging  and various gas, chemical and biomedical applications . We begin with the design concept of this semiconductor heterostructure along with an overview of its band structure (see Figure 1(a)) that is obtained by solving the
These gates circumvent all the abovementioned disadvantages of Schottky gates and are typically composed of a 25‐nm‐thick bulk carbon‐doped GaAs layer (with an acceptor density of
3. Characteristics of HEMTs
After being introduced in 1980s, these transistors based on high‐mobility modulation‐doped heterostructures have revolutionized the semiconductor industry in terms of being the most high‐performance compound semiconductor FETs.
Due to spatial separation of the electrons from the ionized impurities, the scattering between them is highly reduced as compared to the bulk semiconductors, enhancing the electron mobility especially at low temperatures where the abovementioned scattering mechanism is dominant. The spacer layer further increases the electron‐to‐donor separation. While the larger separation reduces the scattering mechanism, as a negative contribution, the carrier concentration is also reduced which reduces the performance. Hence, the spacer thickness should be optimized. Typical values range from 1 to 30 nm. In order to explain how the high mobility of the electrons in HEMT makes them fast transistors, we use the
where is the total impedance, is the dimension of the gate and is the threshold voltage. The HEMTs have a clear advantage of lower access resistance particularly in terms of channel resistance due to the high mobility electrons in the channel in comparison to standard FETs. To summarize, the HEMT design principles allow:
High carrier concentration of 2DEG in the channel
High mobility by optimization of the spacer‐layer thickness
Low access resistance by using buried/recessed gates
Better confinement of carriers in the channel due to high barriers
Reduced interface and alloy scattering mechanisms, thus enhancing mobility
Typical transistor characteristic curves of GaAs/AlGaAs HEMT structures under dark and after 1 s of illumination with a near‐infrared light emitting diode (NIR LED) are shown in Figure 2(a) and (b), respectively. Let us now briefly discuss the transistor operation. Even under zero bias or for a small positive voltage applied to the gate, an inversion layer is formed at the semiconductor surface, the
3.1. Quantum confinement and intersubband transitions in HEMTs
One of the most popular terms in nanoscience is the
In low‐dimensional semiconductor nanostructures, the restriction of the electronic motion in one, two and three dimensions leads to the modification of the density of states (DOS) as compared to the bulk states. The electronic DOS is defined as the number of electronic states per unit volume per unit energy, the finiteness of which is a result of the
where is the quasi‐Fermi energy. The subbands can thus be split as:
where is the number of electrons in the subband with energy . The above classical Boltzmann distribution, , is given by :
where is the Boltzmann constant and is the temperature in Kelvin. Using the above equation, we obtain:
In the limit of low temperature, where electrons are degenerate, the 2D electron density is given by :
Figure 3(a) and (b) show a triangular potential well and a schematic of the filling of the subbands, respectively. Based on the position of the Fermi level, the corresponding subbands are occupied. Under triangular confinement potential (as in the HEMT design), the energy spacing decreases for higher subband energies and finally forms the continuum. When the Fermi energy is higher than but less than , only the ground subband is filled. Similarly, when the Fermi level is above , but less than , the lower two subbands are filled with electrons and so on. The position of the quasi‐Fermi level can be tuned by changing the band structure externally, that is, by applying either an electric or a magnetic field. With the external field, the conduction band can be raised or lowered with respect to the quasi‐Fermi level around the Fermi‐pinning point, hence depleting or filling the subbands with electrons. Figure 3(c) shows a schematic of the intersubband transition from the filled ground subband to an empty excited subband. In quantum mechanics, Fermi’s golden rule is used to calculate the transition rate (i.e. the probability of a transition to occur per unit time), from one state with a given eigenenergy to another state of higher eigenenergy or to the continuum of energy eigenstates, subjected to some kind of perturbation. According to Fermi’s golden rule, this rate of transition, from an initial state to a final state under the In optical experiments with wavelengths λ in the order of micrometers, the width of the quantum well, which practically sets the scale of the electronic wavefunctions, is much shorter than λ. To a good approximation, the momentum of the photon can be neglected and the electric field can be assumed as constant across the electronic states (with or at ). This approximation is also known as the
In optical experiments with wavelengths λ in the order of micrometers, the width of the quantum well, which practically sets the scale of the electronic wavefunctions, is much shorter than λ. To a good approximation, the momentum of the photon can be neglected and the electric field can be assumed as constant across the electronic states (with or at ). This approximation is also known as the
where is the amplitude of the electric field and and are the charge and the effective mass of electrons. and are the energies of the initial and the final state. Now, the absorption coefficient, , is defined as the ratio of the absorbed electromagnetic energy per unit time and area (considering a 2D system) and the intensity of the incident radiation, summed over all the filled initial and empty final states. In order to ensure that the initial state is filled and the final state is empty, a condition necessary for the transition to occur, we introduce the Fermi factors: for the initial state and for the final state. The absorption coefficient is thus given by:
where is the absolute permittivity, is the velocity of light and is the refractive index of the material. The intersubband absorption takes place within the quantized levels of the conduction or the valence band, schematically shown in Figure 3(c). The total wavefunction can be written as the product of the lattice‐periodic Bloch wave (for electrons in a crystal), , and a slowly varying envelope function . According to the
where is the normalization constant, indicates the index for the bands and represents the subband indices. The complete matrix element in can be split as follows:
where the first term indicates the interband transition and the second term is the intersubband transition. The first term has the dipole matrix element of the Bloch functions that explains the selection rule for the interband transition and an overlap integral of the envelope functions. In case of transitions within the subbands of the conduction or the valence band, the first term vanishes and the second term becomes more relevant in the
It can be observed that only the third term, , in the curly bracket survives, giving a contribution at a finite frequency. Except for and (i.e. the initial and the final states are equal), all the other terms vanish, implying the free‐carrier absorption at zero frequency when no scattering processes are involved . Hence, only the following matrix element determines the intersubband absorption in the one‐band model:
The above equation states that the electric field of the incident radiation must have a component perpendicular to the semiconductor layers or parallel to the growth direction (which is a necessary condition) in order to couple to the ISTs. This is known as the
The above quantity is used to understand and compare the strength of the transitions between initial and final states in different physical systems and obeys the
4. Intersubband‐Landau coupling under tilted magnetic fields
When a magnetic field is applied in a plane perpendicular to the semiconductor surface, the free electrons that carry the electric charge perform an orbital motion in the plane perpendicular to the magnetic field direction. This motion is quantized, and equally spaced levels (called the Landau levels) separated in energy are formed. The Hamiltonian of the quantum mechanical system thus gets decoupled into a magnetic and an electric component, and the energy spectrum consists of a series of Landau ladders for each subband. In the presence of a magnetic field, the Drude conductivity, normalized to , is given by [38–42]:
where is the electron density, is the relaxation time constant and is the velocity of light in free space. Using Maxwell’s theory, the transmittance, , can be written as:
where is the plasma frequency. From the quantum mechanical description of such a system, when electrons in 2DEG are subjected to a space‐charge potential, and a magnetic field B tilted at an angle of with respect to the horizontal direction, the total Hamiltonian of the system is given by [4, 43]:
where and . The first two terms in the above equation describe the magnetic field quantization into Landau levels (similar to the harmonic potential). The third and the fourth terms illustrate the quantization due to the space‐charge potential within the triangular well approximation. The term results in the positive diamagnetic shift due to the parallel magnetic field component, . The last term, proportional to the product , couples the Landau and subband quantization at all angles .
Using the perturbation theory, one can solve the above Hamiltonian treating as the perturbation parameter. In order to solve the above problem, the product of Airy wavefunctions and Hermite functions is taken as the basis set. From the first order perturbation theory, there is no correction to the zeroth‐order energies, expect for the degenerate situation . This is commonly addressed as the full‐field coupling regime. In the non‐resonant regime, second‐order effects are present, and hence using perturbation theory of the second order for non‐degenerate levels, the total energy eigenvalues are obtained as :
where are the matrix elements for the ISTs from
5. Tuning and probing of intersubband transitions electrically
Intersubband transitions are the most fundamental optical transitions that can be excited in low‐dimensional semiconductor nanostructures. The observation of ISRs is a result of the fact that the component of the incident infrared electric field perpendicular to the semiconductor layers or parallel to the growth direction selectively couples, thus exciting the electrons from the lower occupied subband to the higher empty subband. By applying a voltage across the structure, it is possible to deplete and selectively populate the subbands. Thus, a more direct scheme of transmission measurement is proposed to study the intersubband spacing in such semiconductor nanostructures (viz. HEMT) even in the absence of an external magnetic field.
The change in the transmission, , due to the 2D space‐charge layer for a normally incident light polarized in the perpendicular direction (say ) is given by :
where is the dynamic dielectric function of the substrate, is the conductivity of the gate and is the conductivity tensor element of the 2D layer. and are the absolute permittivity and permeability of the free space, respectively. The Drude model very well describes the dynamic conductivity response of the quasi‐free charge carriers in the 2D space‐charge layer [41, 42]:
where is the scattering time, is the carrier density and is the effective mass. However, for the ISTs, observed under normally incident light, with the sample tilted at an angle, the conductivity can be described by replacing by [41, 42]:
where is the matrix element for the IST from the
6. An access to the interior of HEMT via artificial structures
Artificial structures such as metamaterials are engineered in the sub‐wavelength sizes for certain desired properties. They are designed in assemblies of multiple individual elements called unit cells. These structures possess unique properties such as negative  or zero  refractive indices, magnetism at optical frequencies , etc. The special properties are not inherent to the materials but the design of the structures and the way electromagnetic field interacts with them. They can also be treated as planar cavities with certain resonance frequencies. When electromagnetic radiation with a certain polarization is incident on these structures, the electric or the magnetic field couples to the cavity and exhibits a resonance that is known as the cavity resonance or resonance frequency of the metamaterials. In transmission measurements, this appears as a dip at that particular resonance frequency. An array of interconnected double split‐ring resonators (see Figure 6(a–d)) is adopted for the metamaterial design, whose dimensions and the characteristic frequency response are first simulated by the standard finite difference time domain solver (like CST microwave studio). For simplicity and small computation time, only one unit cell, as shown in Figure 6(e), is used for the simulation with a periodic boundary condition in the planar directions. Moreover, these meta‐atoms are placed far apart from each other to avoid any influence of inter‐meta‐atom interactions. For the right coupling of the electromagnetic radiation, the electric fields are confined in the two narrow capacitor arms of the double split‐ring resonator (see Figure 6(e)). Moreover, the fringing field effect ensures that there is a strong electric field component along the growth direction that extends over a few 100 nm . This component of the electric field couples with the HEMT to excite the ISRs in accordance with the polarization selection rule as discussed before.
Two transmission minima (or dips) are observed—one at 1.2 THz and the other at 2.4 THz (see Figure 6(f)). The experimental characterization of the metamaterial array is performed by a standard THz time‐domain spectroscopy at room temperature, where the thermal energy, (= 25 meV), is higher than the subband spacings. This thermal occupation of higher subbands consequently prevents us from observing the ISR (ground‐to‐first excited state) in the 2DEG layer. Hence, the response from the sample is purely due to metamaterials. A Ti:Sa laser with an 80 fs pulse duration (a centre wavelength of 800 nm) and a repetition rate of 80 MHz is used to generate the THz radiation by exciting an inter‐digitated photoconductive antenna  processed on a GaAs substrate. A fixed DC bias is applied on the antenna. The THz generation is obtained under the transmission geometry of the antenna. Four 90° off‐axis parabolic mirrors are used for the collection and collimation of the THz beam. The detection is based on free space electro‐optic sampling [8, 29] of the THz electric field by using a birefringent, 2‐mm‐thick ZnTe crystal. As compared to the simulation, the transmission is normalized with respect to the orientation of the metamaterial. The electric field component of the THz source is in the plane parallel to the optical table. Hence, when the metamaterials are oriented at 0° (solid-black arrow in Figure 6(f)), the incident infrared radiation couples into the structures. When the metamaterials are oriented at 90° (black-dashed arrow in Figure 6(f)), the field does not couple. By normalizing the transmitted spectrum of the metamaterial at 0° with respect to the one at 90°, two transmission dips are obtained—one at 1.2 THz and the other at 2.4 THz as shown in Figure 6(f).
Once the sample is cooled down to liquid helium temperatures, at first, the characterization of the voltage range is performed over which the density‐chopping measurements are to be taken. The change in capacitance with the gate voltage is measured by capacitance‐voltage spectroscopy, mentioned before, as shown in Figure 7(a). A typical charging spectrum of 2DEG has a capacitance close to zero in the beginning and then as the gate voltage is increased, the conduction band is pulled below the quasi‐Fermi level and subsequently the 2DEG subbands are filled with electrons. The filling is observed as a steep increase in the capacitance. The region of interest is the steep slope, where increasing the gate voltage increases the 2DEG ISRs. This is due to the fact that with more positive gate voltage, the slope of the triangular potential confinement steepens and hence increases the subband spacings, thus shifting the ISRs to higher energies. This phenomenon is well known as the They are isolated donor atoms, which can be occupied by the electrons in connection with a large lattice relaxation also known as a deep-donor complex (DX center) in Al
They are isolated donor atoms, which can be occupied by the electrons in connection with a large lattice relaxation also known as a deep-donor complex (DX center) in Al
where is the oscillator strength of the ground‐to‐first excited state ISR with an energy of . is the mode volume of the microcavity, given by (to a very good approximation) :
where is the electric field. The coupling strength depends on three important parameters. First, it is proportional to , implying that the cavity mode volume should be small for higher coupling strength. Second, the higher the transition energy, the smaller the coupling strength (). And finally, the coupling strength scales as , which is a characteristic feature of the fermionic systems. The higher the carrier density, the greater is the coupling. The voltage tuning of our device is based on the quantum‐confined Stark effect. The dependence of the coupling strength, , on the number of quantum wells as shown Gabbay et al.  can be written as:
where is the average absorption coefficient, is a constant, which is proportional to the average light‐matter interaction and is the distance between the QWs. For a single QW, as in the present investigation, the coupling strength is proportional to the value of . In the theoretical studies, Gabbay et al. found the value of to be 1, which implies that for a single QW, the coupling strength is 0.5. The splitting in our experiments is found to be 0.47 THz, which agrees well with the theoretical value. It is well known that if the splitting is significantly above the sum of the full width at half maximum of both the ISR and the metamaterial resonance, then the coupling can be assigned to be in the strong coupling regime. Thus, an ultra‐strong light‐matter interaction regime is achieved by employing a single triangular quantum well in a HEMT heterostructure with a normalized coupling ratio of 0.19.
In conclusion, we have reviewed the quantum mechanical phenomenon that governs various electrical and optical properties in the low‐dimensional semiconductor nanostructures such as a HEMT. We have demonstrated how one could electrically, or in combination with magnetic fields, probe and tune the intersubband transitions in the heterojunction of a HEMT structure. Such structures primarily have a triangular confinement potential. In the presence of a magnetic field, each subband is further split into a series of Landau levels or cyclotron orbits. Upon optical excitation with an infrared source, the intersubband resonances couple to the cyclotron resonance under tilted magnetic fields. This leads to the appearance of satellite peaks at the anti‐crossing point. From the values of splitting at the anti‐crossing points, the spacing between the corresponding subbands can be evaluated. Experiments performed in the absence of magnetic fields demonstrated that it is also possible to directly measure and tune these spacings via density‐chopped infrared transmission spectroscopy. The subband spacings are measured directly and found to be in the far‐infrared region (wide electrically tunable from 6 to 12 meV) of the electromagnetic spectrum. New epitaxial, complementary‐doped, semi‐transparent electrostatic gates that have better optical transmission are introduced . The integrated device with a 2DEG in a high electron mobility transistor structure and artificial metamaterials forms a strongly coupled system that can be electrically driven from an uncoupled to a coupled and again back to the uncoupled regime. In the strongly coupled regime, a periodic exchange of energy between the two systems is observed as a splitting of 0.47 THz at the point of avoided crossing. This is a very high‐energy separation, considering the fact that only one quantum well is employed and thus the achievement of a strong coupling regime can be safely claimed. The tuning mechanism is attributed to the quantum‐confined Stark effect. This device architecture is particularly interesting in designing devices like modulators and detectors specifically in the THz regime. The integrated device has the high‐speed dynamic characteristics of the HEMT design and the appropriate frequency‐controlling ability of the metamaterials. From the design perspective of the metamaterials, they can be made particularly for the THz regime with appropriate dimensions (like the one used in this chapter). Upon excitation with a broadband source, this layer selects the desired frequency for which it is designed, and under the application of an external electrical field across the structure, the transmission of this frequency can be controlled and also modulated. This control dynamics can be very fast, simply owing to the fast dynamics of the HEMT design [52, 53]. Furthermore, this design can also be used to detect THz frequencies. Various other 2D materials (like graphene [54–56] or black phosphorous [57, 58]) are also used these days in the transistor configuration for developing THz detectors, simply utilizing the fast dynamics of the transistor design. These novel devices have thus helped to reduce the long‐debated THz gap in the electromagnetic spectrum, where there is a severe lack of fast electronic devices.
- In optical experiments with wavelengths λ in the order of micrometers, the width of the quantum well, which practically sets the scale of the electronic wavefunctions, is much shorter than λ. To a good approximation, the momentum of the photon can be neglected and the electric field can be assumed as constant across the electronic states (with or at k→=0). This approximation is also known as the electric-dipole approximation.
- They are isolated donor atoms, which can be occupied by the electrons in connection with a large lattice relaxation also known as a deep-donor complex (DX center) in AlxGa1−xAs.