UNIVERSIT `A DEGLI STUDI DI PISA dipartimento di fisica
corso di laurea magistrale in fisica
Grazia Raciti
InAs-based shallow single Quantum Well for Near-Field Spectroscopy of Intersubband Transitions
Master thesis
Supervisors:
Prof. A. Tredicucci Prof. A. Toncelli
Contents
1 Introduction 5
2 General theory of intersubband transitions 9
2.1 Intersubband transitions in quantum wells . . . 10
2.1.1 Intersubband absorption coefficient . . . 13
2.1.2 Absorption linewidth . . . 16
2.2 ISB Transitions: spatial dependence . . . 17
3 Simulations 21 3.1 Lineshape of the intersubband transitions . . . 21
3.2 Numerical integration . . . 25
3.2.1 QW width dependence . . . 27
3.2.2 Tip parameters dependence . . . 30
4 Experimental methods 33 4.1 Description of the samples . . . 33
4.2 Fourier Transform Infrared Spectroscopy . . . 36
4.2.1 Sample preparation for FTIR measurements . . . 40
4.3 Scattering Scanning Near-Field Optical Microscopy . . . 41
4.3.1 Working principle of s-SNOM . . . 42
4.3.2 Charge density modulation . . . 45 3
Chapter 1
Introduction
The manipulation of electronic states in artificial crystal structures is of primary importance for the development of many photonic technologies, such as lasers [1], LEDs [2] and photodetectors [3], [4]. For the Mid-IR and THz spectral regions the transitions between the quantized levels of the conduction band confined in nano-scale quantum wells (QW) is of particular interest [5].
The QW structure consists of a heterogeneous semiconductor in which charge carriers behave like a 2D electron (or hole) gas with quan-tized states in the direction orthogonal to the QW plane [5]. Transitions between these confined states are called intersubband transitions (ISB) and can be tuned by changing the QW thickness. The advancement in fabrication techniques, such as molecular beam epitaxy (MBE) [6], permits the fabrication of structures with atomic precision, thus allow-ing the realization of QW structures with suitably tailored properties. Thanks to such precise control of the QW thickness, ISB transitions are now at the heart of mid-IR and THz technologies like quantum well infrared photodetectors (QWIPs) [7] and quantum cascade lasers (QCLs) [8], [9], [10].
Typically, ISB transitions are characterized with the so-called Fourier Transform Infrared (FTIR) Spectroscopy [11], which is a far-field
shaped sample. In addition, the spatial resolution of the FTIR method is limited by the large typical radius of hundreds of microns of the col-limated light beam [13]. This makes FTIR unsuitable for applications where the examination of nano-objects is required. As an example, the peak absorption of a QW measured with FTIR is usually broadened by the average fluctuations in the thickness of the well, which varies on a nanometric scale [14].
Recently, another measurement technique has been developed, ex-ploiting the properties of near-field interaction [15]: scattering-Scanning Near Field Optical Microscopy (s-SNOM). This technique consists of an illuminated AFM tip which provides nanoscale resolution, well below the diffraction limit. Thanks to the near field of the tip, which provides an out-of-plane component of the electric field independently of the di-rection of the impinging light, s-SNOM can excite ISB transitions and in effect it was recently employed to measure ISB transitions in a few layer exfoliated 2D material [16].
Due to the evanescent nature of the near field, the s-SNOM tech-nique can probe only the superficial region of the sample [17], down to few tens of nanometres in depth. In order to measure the quantum well ISB absorption with the s-SNOM, the quantum well must therefore be placed as close to the surface as possible.
7 Bloch’s theorem describes what happens in the bulk of semicon-ductor crystals, where the translational symmetry is exploited to sim-plify the calculation of energy levels. However, when dealing with real crystals, surface effects must be taken into account. Common III-V semiconductors like GaAs/AlGaAs show charge depletion at the sur-face: it is a region of few tens of nanometres where the charge carriers are pushed away from the surface. For this reason, with this kind of semiconductors it is impossible to build surface nanostructures like nanodots, nanowires or shallow quantum wells [18]. On the other hand, InAs shows opposite behaviour: charges accumulate at the surface, thus creating a thin layer of charges. The accumulation of charges can be helpful for the operation of structures like nanowires [18]. However, it is detrimental for the confinement of charges in shallow QW. Moreover, a high charge density at the surface might shield the near field of the s-SNOM tip, thus preventing its interaction with deeper objects like the QW 2DEG.
The goal of this thesis is to characterize ISB transitions in two dif-ferent shallow single QWs of InAs/AlSb, in which a cap of InAlAs is able to suppress surface effects. This is designed to have the bottom of the conduction band aligned with the surface levels, so that Fermi level pinning does not cause major band-bending. Firstly we perform FTIR measurements, thus observing the presence of ISB transitions in QWs at only 6 nm in depth from the surface. We then designed and mea-sured a particular geometry that allows for carrier density modulation inside the QW. The modulation of the carrier density will be employed for more complex s-SNOM measurements. Preliminary s-SNOM mea-surements are finally discussed.
This thesis is organized as follow: in the second chapter, we overview ISB transitions theory, highlighting the main differences between far-field and near-far-field interactions with the heterostructure. In the third
to isolate the ISB contribution to the absorption from different mech-anisms, such as free-carrier absorption or background absorption. We then show our experimental findings in the fifth chapter and the issues remaining to be addressed with the described sample designs. Finally, we summarize the results and discuss the future outlook of this work.
Chapter 2
General theory of
intersubband transitions
In a semiconductor, the conduction and the valence bands are sepa-rated by an energy gap [19]. Transitions between states of different bands driven by a photon can happen by absorbing or emitting energy. These transitions are called interband transitions (see Figure 2.1a). The invention of the quantum well (QW) by Esaki and Tsu [20], represented a technological breakthrough in the development of high-performance semiconductor devices. A QW consists of a thin layer (orders of a few nm) of a semiconductor with a narrower gap embedded by two layers with wider band-gap [5]. Thanks to this so-called band engineering, an effective square well potential for electrons is created in the out-of-plane conduction of the fabricated heterostructure. Energy levels of electrons are quantized in the out-of-plane direction due to the confinement in the quantum well, while their in-plane free motion is not affected. As a result, one obtains a quasi-two-dimensional electronic gas with allowed energy levels quantized along the growth direction. The transitions that occur between such levels belonging to the same band are called intersubband transitions (see Figure 2.1b).
between two subbands within the conduction band: the energy separa-tion of two subbands can be controlled by the quantum well thickness and depth. Pictures adapted from [8]
2.1
Intersubband transitions in quantum
wells
Now we want to review the main characteristics of intersubband tran-sitions theory, using a simple single-band approximation [5].
Crystal structures like III-V semiconductors show a translational sym-metry due to their periodicity. For calculating the energy levels of such periodic crystal structures, it is possible to exploit Bloch’s theorem to easily solve the Schroedinger equation for electrons in a periodic poten-tial.
According to Bloch’s theorem [19], the function which describes the electronic motion in a bulk crystal is the product of a plane wave times a lattice-periodic function. When putting together two semiconductors with different crystal structures, the translational symmetry at the base of Bloch’s theorem is broken. Usually these heterostructures vary on a larger scale than the crystal structures (i.e. a single layer is a few atoms thick at least). Since the perturbation varies slowly with respect to the
2.1. INTERSUBBAND TRANSITIONS IN QUANTUM WELLS 11 lattice potential, it is possible to use the formalism of the envelope function, which is a generalization of Bloch’s theorem. The electronic wavefunction can thus be expressed as the product between a lattice-period Bloch function uν(r) and the envelope function fi(r)
Ψi(r) = fi(r)uν(r) (2.1.1)
where i denotes the quantum numbers of the system and ν labels the band. We note that in the case of bulk crystal the envelope function fi(r) reduces to a plane wave eik·r. We can assume that the Bloch
function is the same in the materials constituting the quantum well, because, usually, heterostructures are built with materials with a small relative lattice mismatch. Large lattice mismatch are avoided in fabri-cation because they are challenging to grow (for example with epitaxial technique like molecular beam epitaxy (MBE) [21]) and induce defect formation at the interfaces. We can then write the Schroedinger equa-tion as follows: [︃ − ℏ 2 2m∗∇ 2 + V (r) ]︃ fi(r) = Eifi(r) (2.1.2)
where m∗ is the effective mass in the well material, and V (r) is the engineered band profile which acts as an effective potential. In the case of a QW structure, the perturbation of the potential interests only the out-of-plane direction. Therefore, indicating this direction as z, the envelope function can be further factorized in the product of a plane wave along the QW plane and a function depending only on z:
fk∥,n(r) = 1 √ Se ik∥,n·rχ n(z) (2.1.3)
where k∥ is the wave vector in the QW plane with area S (from which
we have the normalization factor √1
S in the formula above), and χn(z) is
out-of-plane contributions and can thus be written:
Ek∥,n = En+
ℏ2k2∥
2m∗ (2.1.5)
The energy En depends on the shape of the potential V (z). In
sim-ple cases, such as the finite, symmetric QW, Equation 2.1.4 can be solved analytically; for more complicated structures the solution must be obtained numerically. Transitions between the resulting quantized energy levels are the so-called ISB transition. In order to compute the absorption we need to evaluate the Fermi energy of the QW.
Fermi energy calculation
To compute the Fermi energy level, it is useful to introduce the density of the states (DoS) in the QW [19], given by:
ρ2D(E) = m∗ πℏ2 ∑︂ n Θ(E − En) (2.1.6)
where Θ is the Heaviside step function, and m∗ the electron effective mass. The effective sheet density for a QW is:
n2D =
∫︂ ∞
0
2.1. INTERSUBBAND TRANSITIONS IN QUANTUM WELLS 13 in which f (E) is the Fermi-Dirac distribution. By substituting the DoS and Fermi distribution we obtain:
n2D = m∗ πℏ2 ∫︂ ∞ 0 ∑︂ n Θ(E − En) 1 1 + e(E−EF)/kBTdE (2.1.8)
In our case (n2D = 1011-1012 cm−2) we assume only a single
two-dimensional subband is occupied, thus the Fermi energy is written:
EF = E1+ kBT log (︃ e πℏ2n2D kB T m∗ − 1 )︃ . (2.1.9)
For instance, in the case of an InAs QW with m∗ = 0.023m0 and
n2D = 1.0 x 1012 cm−2, the Fermi energy is EF = 103 meV.
2.1.1
Intersubband absorption coefficient
To study the optical excitation of ISB transitions, we start from Fermi’s Golden Rule and calculate the transition rate from an initial state i to a final state f , considering zero population in the final subband:
Wif = 2π ℏ gs ∑︂ k f (k)| ⟨Ψf|Hint|Ψi⟩ |2δ(Ef(k) − Ei(k) − ℏω) (2.1.10)
where gs is the spin degeneracy, f (k) is the Fermi distribution of the
electrons in the first subband, the second term in the sum is the matrix element of the interaction Hamiltonian and the last term is the energy conservation law. The interaction Hamiltonian is given by:
Hint =
e
2m∗(A · pˆ + pˆ· A) (2.1.11)
where e is the electron charge, A is the vector potential corresponding to an electric field E = −∂A∂t , pˆ is the momentum operator and m∗ is
by putting eq·r = 1 in Equation 2.1.13. This approximation is valid
if the wavelength of the radiation (λ = 2πq ) is much larger than any characteristic length of the system, which are the QW width d and the lattice constant of the crystal a. This approximation is accurate, since the QW thickness is at most a few tens of nm and the impinging light has a wavelength of few microns. Then pˆ commutes with A, which leads to Hint= me∗A · pˆ. Substituting in Equation 2.1.10, the transition
rate is: Wif = 2π ℏ e2E2 0 4m∗2ω2gs ∑︂ k f (k)| ⟨Ψf|e · pˆ|Ψi⟩ |2δ(Ef(k) − Ei(k) − ℏω) (2.1.14) For the purpose of estimating the optical absorption caused by ISB transitions, it is important evaluate the matrix element in Equation 2.1.18. By using the envelope function approximation, obtained in Equation 2.1.1, the matrix element can be write as follows:
⟨Ψf|e · pˆ|Ψi⟩ = e·⟨uνf|pˆ|uνi⟩ ⟨fnf|fni⟩+e·⟨uνf|uνi⟩ ⟨fnf|pˆ|fni⟩ (2.1.15)
where the indices νi,f and ni,f represent the band and subband numbers
2.1. INTERSUBBAND TRANSITIONS IN QUANTUM WELLS 15 nearly orthogonal set of functions, interband transitions occur between the same subbands (ni = nf), while intersubband transitions can only
occur within the same band (νi = νf). Focusing our attention on
intersubband transitions, the second term of Equation 2.1.15 becomes:
⟨fnf| e·pˆ|fni⟩ = 1 S ∫︂ d3re−ik∥,nf·rχ∗ nf(z)[expx+eypy+ezpz]e ik∥,ni·r χni(z) (2.1.16) The terms proportional to exand ey are multiplied by ⟨χnf|χni⟩, so that
the result is zero for ni ̸= nf, since the envelope functions are
orthog-onal. So, the only non zero remaining matrix element is ⟨χnf|pz|χni⟩.
Thus, we obtain that the electric field of the radiation must have a com-ponent along the out-of-plane direction to excite intersubband transi-tions. This condition is known as the polarization selection rule for intersubband transitions.
This strict selection rule requires specific geometries to allow the coupling between radiation and ISB transitions. For light impinging perpendicular to the surface, the electric field has components only in the QW plane, hence the ISB transitions cannot be excited. The simplest way to overcome this problem is the realization of waveguide geometries which allow for an oblique angle of incidence [22].
We proceed to calculate the ISB transition rate which we now can write as: Wif = 2π ℏ e2E2 0 4m∗2ω2gse 2 z ∑︂ k f (k)| ⟨χnf|pz|χni⟩ | 2 δ(Ef− Ei− ℏω) (2.1.17)
In order to express the matrix element in real space notation, we ex-ploit the commutation relation [H0, z] = ℏimpz∗, with H0 the unperturbed
and area S, ℏωWif/S, and the intensity of the incident radiation, I =
1/2ε0cηE02 , where η is the refractive index of the absorbing material,
ε0 is the vacuum permittivity and c the speed of light in vacuum. Thus,
we obtain this expression for the absorption coefficient:
α2D = ℏω SI|Ez(ℏω)dz| 22π ℏ gs ∑︂ k f (k)δ(Ef − Ei− ℏω). (2.1.19)
2.1.2
Absorption linewidth
We focus here on the line shape of ISB transitions. For simplicity we take into account the transition between the first two subbands, so that the ISB transition rate in Equation 2.1.18 can be written:
W12= 2π ℏ gs|Ez(ℏω)dz| 2∑︂ k f (k)δ(E2− E1− ℏω) (2.1.20)
What is important to notice is the δ line shape of the ISB transition, which is a consequence of the same curvature of the initial and final subband. Contrary to the bulk semiconductors, where the absorption (for direct transitions) between valence and conduction band has a square root dependence on the energy [19], [23], the ISB absorption,
2.2. ISB TRANSITIONS: SPATIAL DEPENDENCE 17 in an ideal case, is allowed only for photons whose energy is exactly the energy difference between the two subbands. However, in the real case, the absorption line is broadened by dephasing processes, like acoustic and optical phonon scattering, electron scattering, electron-impurity scattering and interface roughness [24], [25]. These processes give rise to an homogeneous broadening. The simplest way to take into account their effects is to introduce a broadening parameter Γ . The δ function in Equation 2.1.20 is thus replaced by a Lorentzian function so that the transition rate becomes:
W12= 2π ℏ gs|Ez(ℏω)|2 ∑︂ k f (k)Γ π (︃ 1 (E2− E1− ℏω)2 + Γ2 )︃ (2.1.21)
Furthermore, other processes such as well-to-well thickness fluctuation in multi-QW or the subband non-parabolicity are fundamental inho-mogeneous broadening mechanisms [26].
2.2
ISB Transitions: spatial dependence
Intersubband transitions are usually observed with classical optics, like Fourier-Transform Infrared Spectroscopy [11] (that we will discuss in detail below), which involves the use of far-field radiation to excite the ISB transitions. Although this technique provides a complete charac-terization of the optical properties of the QW structure, the spacial resolution is limited by diffraction to about half of the illumination wavelength λ. The impossibility, for far-field optics, to confine the light in a region smaller than λ/2, makes it unsuitable for the investigation of structures on the nanoscale size. The diffraction limit originates from the concept of free propagating wave. If we consider the possibility to have an imaginary component of the k-vector, the condition |kx| < |k| is
which generates a near field. This is the radiation that interacts with the sample below. The dotted arrow is the scattered light that contains information about the sample-tip interaction and therefore about the intersubband absorption. Adapted from [27]. (b) This is the field profile that forms between the tip and the sample. The different decay length depends on the different permettivities between air and sample; the penetration length is usually of the order of few ten of nm. The maximum of the electric field is located near the surface of the sample.
λ/2. Waves characterized by a complex wave-vector are called evanes-cent waves [13]. The near field is constituted by evanesevanes-cent waves that have an imaginary k-vector, and then decrease exponentially with the distance. This implies also that their Fourier transform is not a delta function and that they are not subjected to polarization selection rules, as the far field is. These properties of the near-field are optimally ex-ploited by scanning near-field optical microscopes (s-SNOM)[13],[28], [17]. The working principle of the type of s-SNOM here considered is shown in Figure 2.2a: a metallized tip, illuminated by a focused laser beam, generates a near field in the out-of-plane direction; after the in-teraction with the sample below, the scattered light is collected by a detector.
ISB transitions in a QW have yet to be studied with high spatial resolution near field technique. For the purpose of a theoretical
de-2.2. ISB TRANSITIONS: SPATIAL DEPENDENCE 19 scription, we introduce a modification in eq.2.1.20 to take into account the effects which the near-field cause in the transition rate. The new transition rate is:
W12 = 2π ℏ gs|Ez(ℏω)dz|2 ∑︂ k,ξ ρtip(ξ)f (k)δ(E2(k + ξ) − E1(k) − ℏω) (2.2.1) where ρtip describes the tip distribution in momentum ξ which we will
discuss in the next chapter. For the computation of this formula see Appendix A. We observe that the tip contributes a momentum ξ, break-ing the momentum conservation law that becomes k1+ ξ = k2. Thus,
the tip apex does not only provide the necessary out-of-plane polariza-tion component to permit ISB transipolariza-tions, but also provides in-plane momentum ξ which allows non vertical optical transitions.
Chapter 3
Simulations
We introduce here a model we customized to allow numerical simulation of the tip-sample interaction. Simulations presented in this chapter have the scope of studying the effects of the near-field measurement on the ISB absorption of the QW.
3.1
Lineshape of the intersubband
transi-tions
To simulate the intersubband absorption of a III-V semiconductor QW we need to examine in more detail the following equation (details on the theoretical calculation of this formula are given in Appendix A):
W12 = 2π ℏ gs|Ez(ℏω)dz| 2∑︂ k,ξ ρtip(ξ)f (k)δ(E2(k + ξ) − E1(k) − ℏω) (3.1.1) First of all, we need to discuss the broadening mechanism originating from the s-SNOM tip, which provides enough momentum to allow non vertical transitions in k space. To take into account this effect, we have to model the momentum distribution around the s-SNOM tip apex. In our work, we consider the optical response of the s-SNOM tip as the response of a point dipole placed at the centre of the sphere that best
Figure 3.1: A schematic picture of the simulated s-SNOM system.
approximates the tip apex (see Figure 3.1). In previous works, more complex models were studied, for example considering antenna theory [29], but no significant difference was highlighted [30]. We will therefore consider the tip as a sphere of radius a, tapping on the sample with frequency Ω and amplitude ∆z. Considering such a model yields the following momentum distribution, reported in [31]:
ρtip(ξ) = ξ2e−2ξzd (3.1.2)
where ξ is the momentum transferred by the tip and zd(t) = b + ∆z(1 −
cos Ωt) is the time-dependent distance between the tip apex and the active region of the quantum well. The parameter b is the distance of the point dipole from the active region, when the tip is touching the sample (∆z = 0):
b = a + dbarrier+
dQW
2 (3.1.3)
where dbarrier and dQW are the thickness of the top barrier and the
quantum well, respectively. In our simulations, we average out the time-dependence, since the pixel acquisition time usually takes a time much longer than the period. In Figure 3.2 the time-averaged distribution of
3.1. LINESHAPE OF THE INTERSUBBAND TRANSITIONS 23
Figure 3.2: Momentum tip distribution as function of 5 tip radii. The distribution broadens when the tip radius decreases.
a [nm] 1/a [107m−1] ξ0 [107m−1] F W HM [107m−1] 10 10.0 3.5 7.1 20 5.0 2.3 4.7 30 3.3 1.7 3.5 40 2.5 1.4 2.9 50 2.0 1.1 2.3
Table 3.1: Extrapolated parameters from the tip distribution.
Equation 3.1.2 is plotted for different values of the tip radius a. We find that the distribution centre ξ0 shifts towards higher values of the
momentum and the distribution width increases as the radius of the decreases. In fact, the distribution is centred at a value ξ0 which is
related to 1/zd and approaches 1/a for large values of a [31].
Considering the transition between the first two subbands of the QW, f (k) in Equation 3.1.1 is the Fermi distribution that describe the population of first subband.
in-Figure 3.3: Calculated energy for the ISB transition between the first two subband as a function of QW width in two different intervals.
side the delta are estimated by Equation 2.1.5. In this latter equation the parabolic term describes the in plane dispersion while the discretiza-tion derives from the energy levels of the QW. As a first approximadiscretiza-tion the in-plane energy can be described by a parabola with curvature pro-portional to the inverse effective mass of the electrons in the crystal. This model is valid for low values of in-plane momentum k∥[8]. On the
contrary, with the tip presence, a new momentum ξ is introduced ex-tending outside the range where the approximation is valid. Therefore, a new multiband model is used to better describe this phenomenon. Such a model [8] leads to the following more accurate, non parabolic band dispersion: Ei(k∥) = Eg 2 {︃ − 1 + √︄ 1 + 4(︃ Ei Eg (︃ 1 + Ei Eg )︃ + ℏ 2k2 ∥ 2m∗(0)E g )︃}︃ (3.1.4) where Ei is the energy of the i-th subband at k∥ = 0, Eg is the energy
gap of the well material and m∗is the effective mass at zero energy. The energy Ei is obtained by using a Schroedinger-Poisson code available
3.2. NUMERICAL INTEGRATION 25 and Poisson’s equation are solved iteratively until a self consistent solu-tion is obtained. In Figure 3.3 we show the energy separasolu-tion between the first two subbands evaluated for different widths of the QW. As we discussed in section 2.1.2, the linewidth of intersubband transitions can be broadened both by phonon and disorder. For this reason, we substitute the delta function in Equation 3.1.1 with a Lorentzian. The only broadening mechanism that we consider is the phonon one because performing near field measurements the effect of disorder should be sup-pressed. Thus, considering this fact and taking data from the literature [25], we set the width of the Lorentzian function at the phononic limit at T=0K (1.2 meV).
3.2
Numerical integration
The simulations are performed with a script in Python. The sum in Equation 3.1.1 is replaced by an integral that is calculated by the trape-zoidal rule through numpy.trapz. The transition rate becomes:
W12= ∫︂ dφ ∫︂ dξρtip(ξ)ξ ∫︂ dkf (k) Γ/π (E2(k∥+ ξ) − E1(k∥) − ℏω)2+ Γ2 . (3.2.1) The integration is performed over k, ξ and φ in order. The free pa-rameter in the simulations are the quantum well thickness (dQW), the
tip radius (a), the tapping amplitude (∆z) and the energy of the sub-band (Ei). For each simulation we calculate the ISB transition rate at
T=0K and T=300K to evaluate the thermal broadening, neglecting the tip (setting ξ = 0) and the transition mediated by the near-field. Typ-ical simulations obtained by the numerTyp-ical integration are displayed in Figure 3.4 for two QW thicknesses: 10.1 nm (in Figure 3.4a) and 15 nm (in Figure 3.4b). The thickness of the barrier is fixed at dbarrier = 6 nm.
In the pictures, three different lines are shown: the first, narrower one is the line of the transition at 0K; the second line is the far-field
tran-Figure 3.4: Simulated absorption for two different QW thickness. The blue and orange lines represent the absorption at different temperature. The green lines also takes into account the tip coupling.
T = 0K T = 300K Tip (300K)
FWHM [meV] 2.3 24.8 23.8
EM AX [meV] 201.9 197.9 200.9
Table 3.2: Line broadening and maximum energy obtained from the simulations for a QW thick 10.1 nm.
T = 0K T = 300K Tip (300K)
FWHM [meV] 2.3 18.7 21.1
EM AX [meV] 117.9 113.9 117.5
Table 3.3: Line broadening and maximum energy obtained from the simulations for a QW thick 15 nm.
3.2. NUMERICAL INTEGRATION 27 sition at 300K where thermal broadening arising from non-parabolicity is taken into account; the third line shows the transition mediated by the near-field photons of the tip with a radius of 20 nm at room tem-perature.
The main effect of the tip is the broadening of the absorption linewidth as a result of the transferred finite tip momentum. As visible in Table 3.4, the tip has a broadening effect of magnitude comparable to the thermal one.
Employing the near-field, it is possible make a QW ”map” not affected by the disorder and have more direct access to the natural transition linewidth and electron distribution. We thus explore the effects of s-SNOM tip momentum distribution on the absorption of QW by the simulations, which are organised as follows:
• QW width dependence: we fixed the tip radius a, the tapping amplitude ∆z and the barrier width dbarrier in Equation 3.1.3,
while the QW width dQW varies;
• tip parameters dependence: we fix the sample geometry and study the linewidth dependence, first on tip radius a and then on the tapping amplitude of the tip.
3.2.1
QW width dependence
An important type of disorder present in semiconductor quantum wells is interface fluctuation [32]. These fluctuations are characterized by two parameters: the thickness fluctuation (in out-of-plane direction), ∆ and the lateral correlation length (in plane), Λ, as it is shown in Figure 3.5. We want to test if this model is able to appreciate small fluctuations of the QW thickness. In order to check this during the simulations we varied the thickness of the QW of few Angstrom, typically one monolayer [34]. This is the typical uncertainty on the QW thickness
From [33]
achieved by MBE fabrication [14] [35].
We simulated the absorption peaks for seven values of the QW thickness from 10 nm to 10.6 nm in steps of 0.1 nm (Figure 3.6 ). In all the simulations, we fixed the barrier thickness at 6 nm and the amplitude of the tapping frequency at 70 nm. We can observe how the centre of the peak moves with the variation of QW thickness. For example, a variation of 0.1 nm corresponds to a shift of the absorption peak of ∼ 3 meV. Therefore, if the experimental set-up can resolve this energy separation, in principle we are able to evaluate variations in QW thickness of the order of 0.1 nm.
By modifying the tip radius in the simulations, the peaks shift as well, while the linewidth changing is more evident. Absorption broadening is lower when the tip radius increases: for a QW 10 nm-thick, FWHM goes from 23.3 meV to 16.1 meV when the tip radius passes from 10 nm to 40 nm. As already mentioned above, this could be explained with the transferred tip momentum. Referring to Figure 3.2, we can see that at lower tip radius corresponds a flatter momentum distribution, allowing non-vertical transition over a larger momentum range. On the other hand, tips with bigger radius show a narrower momentum distribution,
3.2. NUMERICAL INTEGRATION 29
Figure 3.6: Quantum well thickness dependence. We show the result of simulations for four values of tip-radius. In each graph we represent the absorption peak of seven different values of quantum well thickness.
dQW [nm] 10.0 10.1 10.2 10.3 10.4 10.5 10.6
E12 [meV] 206 202 199 196 194 192 190
a=10 [nm] Emax[meV] 206.1 201.9 198.9 195.9 193.9 191.9 189.9
FWHM[meV] 23.3 28.4 28.8 28.4 28.4 28.4 28.8 a=20 [nm] Emax[meV] 204.9 200.9 197.9 194.9 192.9 190.9 188.9
FWHM[meV] 23.8 23.8 23.8 23.8 23.8 23.4 23.4 a=30 [nm] Emax[meV] 204.2 200.2 197.2 194.2 192.2 190.2 188.5
FWHM[meV] 22.4 22.4 22.1 22.4 22.1 22.1 22.1 a=40 [nm] Emax[meV] 203.9 199.9 196.9 193.9 191.9 189.9 187.9
FWHM[meV] 22.1 21.7 21.8 21.4 21.4 21.1 21.4 Table 3.4: Parameters obtained from simulations. We report the energy
max Emax of the absorption peak and its FWHM. The QW thickness
Focusing on the tip parameters, decreasing the tip radius, a blueshift of the ISB absorption is revealed (see Figure 3.7b). This can be also explained by the broadener momentum distribution introduced by the tip. On the contrary, the tapping amplitude seems not to influence the absorption and all peaks appear superimposed (see Figure 3.7a).
Summarizing, even though the presence of the tip broadens the linewidth, it seems that it is possible to distinguish clearly monolayer fluctuations in the QW thickness.
3.2. NUMERICAL INTEGRATION 31
Figure 3.7: On the right the absorption peaks of a QW wide 10.1 nm as a function of the tip radius: the peak is blueshifted and broadens when the tip radius decrease. On the left absorption peak dependence on the tapping amplitude: this parameter does not influence the absorption and all peak are superimposed.
dQW=10.1 nm, E12=202 nm
a[nm] Emaxdist[meV] FWHM[meV]
10 201.9 28.4
12 201.6 27.1
20 200.9 23.7
30 200.2 22.4
40 199.9 21.7
Chapter 4
Experimental methods
In this chapter we give a detailed description of the samples and the experimental techniques used in this Master thesis project. Firstly we address the design of the QW structure that we want to study, point-ing out the issues derivpoint-ing by the use of a shallow spoint-ingle QW and our workarounds. Then, we introduce the Fourier Transform Infrared Spec-troscopy (FTIR) method and the preparation of the samples needed to perform the measurement. In the last part of the chapter, we explain the working principle of a scattering- Scanning Near-Field Optical Mi-croscopy (s-SNOM) and describe some practical aspects on how the sample was prepared for near field measurements. The samples have been fabricated at the University of Montepellier, in collaboration with the group of A. Baranov.
4.1
Description of the samples
We will work with two different devices sketched in Figure 4.1: one with a 10.1 nm InAs quantum well layer (Figure 4.1 a) and the other with a 15.0 nm InAs quantum well layer (Figure 4.1 b). The sample with a narrower quantum well consists of, in order of growth:
• Substrate of nominally undoped InAs; 33
The wider well sample consists of the same layers with a 15.0 nm InAs (Si-doped n = 6.6x1017 cm−3) quantum well. The doping of the
active layer is chosen to have an effective sheet density of n2D = ndQW ≃
1012 cm−2 for both the designs. Among all III-V semiconductors, we
selected InAs/AlSb because of their particular properties at the surface of the heterostructure [38] [39], [40]. In contrast with semiconductors like GaAs and AlGaAs which show charge depletion at the surface, InAs shows an opposite behaviour: a sheet of charges is accumulated on the surface. This characteristic makes InAs and AlSb excellent semi-conductors for the realization of a shallow quantum well [39], [40]. On the other side, the presence of surface charges could be detrimental to perform optical measurements, because this sheet of charges might shield the near field of the s-SNOM tip, preventing its interaction with deeper objects like the QW 2DEG. However, this surface effect can be avoided by depositing a top layer of In0.75Al0.25As, designed to have the
bottom of the conduction band aligned with the surface levels so that Fermi level pinning does not cause major band bending (see Figure 4.2). Summing up, to be measurable with a near field technique, our QW should have two major properties: it should be as close as possible to the surface, because of the small penetration length of the evanes-cent field of the s-SNOM and, at the same time, it should not have an
4.1. DESCRIPTION OF THE SAMPLES 35
Figure 4.1: Layer structures for 10.1 nm (a) and 15 nm (b) quantum well heterostructures, indicating the compositions and dimensions in the growth direction. The active layer of the QW is highlighted in red.
Figure 4.2: Energy band diagram of a shallow InAs/AlSb quantum well. The presence of the In0.75Al0.25As cap is needed to avoid any
charge accumulation of the surface of the sample due to the Fermi pinning. The orange dashed lines indicates the effects of the doping (n2D ≃ 1012 cm−2 for both the designs) on the band structure of our
Figure 4.3: A FTIR spectrometer schematic. It is composed by a broad-band source, an interferometer, a sample chamber, a detector and a computer. Figure adapted from [41].
accumulation of charges at the surface, in order not to shield any near field interaction. Thanks to the described design, our samples match both these features, thus being suitable for the optical characterization of a single QW placed at only 6 nm in depth from the surface.
4.2
Fourier Transform Infrared Spectroscopy
As we said in the introduction of this chapter, FTIR spectroscopy is a measurement technique that allows to record infrared spectra [42],[11]. The schematic of a FTIR is shown in Figure 4.3. The working process can be summarized as follow:
• Source: FTIR spectrometers use an infrared broadband source. This light is collimated and injected in the interferometer. • Interferometer: The interferometer used in FTIR spectrometry
4.2. FOURIER TRANSFORM INFRARED SPECTROSCOPY 37 of which oscillates, and a beam splitter as shown in Figure 4.3. Collimated light from the light source is split into two parts by the beam splitter. One beam is reflected by the fixed mirror, and the other is reflected by the moving mirror and both return to the beam splitter, which recombines them in a single beam. The oscillating mirror periodically changes one beam optical path length, resulting in an oscillating relative phase shift between the two recombined beams. The result is an interference pattern. • Sample: After passing through the interferometer, the light is
focused on the sample placed in the sample chamber and then directed onto the detector.
• Detector: The detector collects the resulting signal, called in-terferogram, which represents light output as a function of mirror position.
• Computer: The digital signal is sent to the computer where the Fourier transform is computed.
We use two FTIR spectrometers in our measurements: a thermo NICOLET MAGNA-IR-860 and a JASCO FT/IR-6800, both operating in transmission mode.
The NICOLET FTIR is equipped with an IR light source, a series of beam splitters and detectors. We use a Mercury Cadmium Telluride (MCT) detector which operates at nitrogen temperature (∼ 77K) and a Kr beam splitter. The Omnic spectra software allows us to specify the moving mirror velocity and the spectral range we want to explore. We can also set the resolution in cm−1 and the number of scan repetitions. In this spectrometer it is possible to install a cryostat to perform mea-surements at low temperatures. The cryostat is put inside the sample chamber between the light source and the detector.
by air absorption peaks as can be seen by comparing the noisy regions from ∼ 1300 cm−1 to 200cm−1 in Figure 4.4. These spectra are mea-sured in two different polarization by using a linear polarizer which is employed to measure the ISB absorption as we will explain below. Both interferometers employ a HeNe reference laser which also allows to control the alignment of the sample in the system.
As we have shown in Section 2.1, the electric field of the radiation must have a component along the out-of-plane direction to excite the ISB transitions. Light impinging perpendicular to the surface has compo-nent only in the QW plane, thus we need to employ a specific geometry to drive light with a non zero angle of incidence. To allow the oblique incidence, the sample is lapped at 45◦ on both sides so that the light impinges perpendicular to the bevelled facet. If the electric field lays in the plane of incidence, the mode is called Transverse Magnetic (TM, with reference at Figure 4.5). If the electric field is perpendicular to the plane of incidence, the mode is called Transverse Electric (TE, with reference at Figure 4.5). As visible from the picture, the TM mode has a polarization component along the z axis. In order to maximize the ISB absorption, we exploit a linear polarizer that is placed in front of the sample in the sample chamber.
4.2. FOURIER TRANSFORM INFRARED SPECTROSCOPY 39
Figure 4.4: Background spectra of the the light source in two different polarizations for the JASCO (a) and NICOLET (b) spectrometers. The main difference between the JASCO and NICOLET lamp is that the first is taken in vacuum and the second in air: the presence of air yields the increased evidence of absorption lines (b). The difference between TE and TM polarization spectra in (b) can be explained by a non perfectly unpolarized light emission from the source or by a slight polarisation dependence of the internal FTIR optics.
two separate spectroscopic measurements: one for TM polarization and another for TE polarization. In order to minimize the effects caused by air absorption or by other spurious absorptions, we also recorded the background spectra of the lamps in two different polarizations (as shown in Figure 4.4). On top of that, we normalize the transmission spectra acquired with TM polarization with the spectra acquired with TE polarization, thus obtaining a relative absorption. As a result, the final signal is the ratio of spectra for the two polarizations, each normalized with the corresponding background:
T⊥ T∥ = sample⊥ lamp⊥ sample∥ lamp∥ (4.2.1)
Figure 4.5: Schematic representation of the lapped sample inside FTIR sample chamber. Light impinging at 90◦ on the lapped facet undergoes multiple reflections inside the sample. The yellow layer represents the metallized coating while the green layer is the QW active region. Figure adapted from [43].
sample while lamp is the spectrum of the light source, ⊥ is TM polar-ization and ∥ is TE polarpolar-ization.
4.2.1
Sample preparation for FTIR measurements
Before the optical characterization, the sample surface is coated with 150 nm thick layer of gold. This metallic layer on the semiconductor leads to a maximum of the TM electric field in correspondence of the surface, allowing a better coupling between light and the active region of the QW. This is a crucial aspect for a single QW as shallow as in our samples with respect to the radiation wavelength. Referring to Figure 4.5, the lateral facets of the sample are lapped to obtain an angle of 45◦. For that, the sample is glued to a proper metallic support and lapped using several sandpaper sheets with different grains. The coarse-grained sheet allows to shape the facet, while the finer ones smoothen and polish
4.3. SCATTERING SCANNING NEAR-FIELD OPTICAL MICROSCOPY41 the surface in order to limit the diffusion of the incident radiation.
The result of a lapping process suitable for optical characterization measurements is sketched in Figure 4.5. Each sample is attached on a copper bar through two different approaches: silver paste and kapton tape. The bar is then mounted on a sample holder which is oriented at 45◦ with respect to the incident beam and put into the sample chamber. In addition, the sample is surrounded by an aluminium foil to make sure that the detected light is only the one transmitted through the sample. The sample holder is also equipped with one micrometre screw that can change the position of the sample in the vertical direction. Before the measurement, the micrometre screw is moved in order to maximize the output signal on the detector. For low temperature measurements, we designed a sample holder which is mounted inside the cryostat. This is realized to again allow the beam to impinge at 90◦ with respect to the lapped facet.
4.3
Scattering Scanning Near-Field
Opti-cal Microscopy
Scattering-Scanning Near Field Optical Microscopy (s-SNOM) is a pow-erful microscopic technique with a nanometer spacial resolution, well below the diffraction limit (Keilmann and Hillenbrand [17]),[30]. It consists of a metallic tip of an Atomic Force Microscope (AFM) illu-minated by a focused laser beam. The near field generated at the tip interacts with the sample below. The backscattered light, that car-ries local optical information about the sample, is collected and sent a detector, as shown in Figure 4.6a.
is collected and detected. Adapted by [27].(b) Model of the near-field interaction close to the s-SNOM. Adapted by [44].
4.3.1
Working principle of s-SNOM
In order to describe the tip-sample interaction we need to model the s-SNOM tip. We assume the S-SNOM tip is approximated by a point dipole at the center of a polarizable sphere [17],[45],[46] with radius a, polarizability α and complex dielectric function εt. It has been shown
that refining the model, for example by considering a finite-dimension dipole or antenna theory, does not lead to qualitative changes in the results [47],[48]. The polarizability of the sphere can be written as follows:
α = 4πa3εt− 1 εt+ 2
. (4.3.1)
Referring to Figure 4.6b, the sphere is at distance z = r − a above the sample which is placed in the half space z < 0. The sample has complex dielectric function εs. We further assume that the sample
can only be polarized by the sphere’s dipolar field, and not directly by the incident field. When applying an electric field Ei, the sphere
4.3. SCATTERING SCANNING NEAR-FIELD OPTICAL MICROSCOPY43 the probe dipole placed at the center of the sphere induces a surface
charge in the sample. The field generated by this surface charge can be described by an image dipole with dipole moment p′ = βp, with β = (εs− 1)/(εs + 1), located in the sample at distance 2r from the
probe tip dipole (see Figure 4.6b). This image field adds up to the incident field and the effective polarizability of the tip-sample system is given by: p = α (︃ E + p ′ 16πr3 )︃ = α 1 − αβ 16πr3 E. (4.3.2)
The scattered field is a superposition of both probe and image dipole fields and we can therefore use a complex effective polarizability αef f,
which takes into account the coupling between the two dipoles. In the end, the scattered field can be written:
Esca ∝ αef fEi (4.3.3)
with αef f being the effective polarizability:
αef f =
α(1 + β)
1 − αβ
16π(z + a)3
. (4.3.4)
Equation 4.3.4 is a central result since it contains the main character-istic of the s-SNOM technique. First, we note that the scattered field depends on the tip parameters a and εt, on the tip-sample distance z
and on the dielectric function of the sample εs. Therefore, the s-SNOM
is suitable to measure the local dielectric function of the sample, since all the other parameters can be known in advance. Second, we note that αef f is a complex value αef f = seiφ, characterized by a relative
amplitude s and a phase shift φ between the incident and scattered light. As it is shown in Figure 4.7, a Michelson interferometer is used to detect both amplitude and phase. The backscattered light from the
Figure 4.7: A sketch of the s-SNOM experimental setup. We can em-ploy one of the three laser lines. The beam is injected into an interfer-ometer where the beam is split into two lines. One of the two beams is focused on the tip and interacts with the sample, the other is reflected by a mirror oscillating at fosc ∼ 300 Hz. The interference of the two
beams is detected and processed by a lock-in amplifier. This system eliminates the background and isolates only the signal referring to the tip-sample coupling. Picture is taken from [49].
4.3. SCATTERING SCANNING NEAR-FIELD OPTICAL MICROSCOPY45 AFM tip, placed at the end of one arm, interferes with a reference beam
modulated by an oscillating mirror. After recombining, the two beams are detected by a nitrogen-cooled MCT detector.
In order to suppress the background signal, which is the light directly reflected by the tip shaft or the sample, we exploit the non linear de-pendence of αef f on z. Due to this non linearity, the tip oscillation at
frequency Ω yields higher harmonics in the backscattered signal. While the near-field interaction depends strongly on z, the background signal contains a constant contribution and at most a component modulated directly at Ω. Thus, a lock-in amplifier at higher harmonics can be used to suppress the background signal [50], [51]. A map of amplitude and phase of the complex signal is shown on the PC while the tip scans the surface of the sample. In experiments, the phase of the detected signal is typically related to the imaginary part of the εs, while the
amplitude is related to the real part of the εs [52]. For our work we
will employ a s-SNOM from Neaspec, present in the labs of the group of Frank Koppens, at ICFO, Barcellona (Spain). A schematic of the setup is shown in Figure 4.7.
4.3.2
Charge density modulation
An additional modulation technique allows to modulate the doping in the sample [16]. It consists of a modulation of the back-gate voltage VBG that, exploiting a FET geometry of the QW sample, results in a
modulation of the charge inside the well (see Figure 4.8). By using this technique, the signal/noise increases thanks to the reduction of the noise which we can be attributed to drifts of the interferometer arm or within the laser cavity. Furthermore only the signal originating from the ISB transition is retained, eliminating other possible contributions. In our case, applying VBG modulates the doping of the QW between
than the pixel time of the s-SNOM (the acquisition time for pixel), we alternatively acquire some consecutive pixels with no charge in the QW followed by some consecutive pixels with charge inside the QW. This way, when the QW is empty of electrons, it does not exhibit ISB absorption and it can be used as a reference signal. By considering the difference of the average of each bunch of pixels, we can isolate the ISB absorption from the laser drift effects. The phase signal of the s-SNOM depends strongly on the initial phase of the laser which changes during the experiment. Therefore, it is of primary importance to have a reference that can be used to suppress effects such as laser phase fluctuations.
Chapter 5
Experimental results
In this chapter we present the experimental measurements of the in-tersubband transitions in two different InAs/AlSb QWs. Firstly, we observe the optical absorption of these devices at room temperature with FTIR measurements, presenting all experimental issues resulting from dealing with a single shallow QW. We also study the temperature dependence of the QW absorption. Then, we report the results of the measurements with the s-SNOM, with particular attention to the issues related to the charge modulation technique.
5.1
FTIR: Transmission spectra of ISB
tran-sitions
We discuss the transmission spectra of samples with geometry shown in Figure 4.5 measured in the FTIR setup. All the spectra are measured at room temperature by using the JASCO FTIR spectrometer. As we said in Section 4.2, in our measurements we normalized the transmis-sion through the sample with the lamp spectrum taken with the same polarization. We analysed two different structures: Sample A, with a QW thickness of 10 nm and Sample B, with a QW thickness of 15 nm. We coated with a 100 nm-thick gold layer the top surface of a
Figure 5.1: Transmission spectra of the two designs Sample A and B bare (left column) and coated with gold (right column). All the samples are contacted with silver paste to the copper plate.
sample from each structure and we measured them in comparison with uncoated ones. The measured spectra are shown in Figure 5.1. With regards to Sample A, we can notice that the spectrum of the uncoated sample (Figure 5.1a) does not show any absorption matching the ISB transition at energy E12= 202 meV. On the other hand, the spectrum
for the same Sample A coated with gold (Figure 5.1b) shows a weak absorption peak emerging at the resonant energy. However, since the relative absorption is so small, the peak strength is within the signal to noise ratio and it is therefore not distinguishable from spurious ab-sorption.
We found a similar behaviour also for Sample B, where we observe an even worse signal to noise ratio (Figure 5.1c,d).
5.1. FTIR: TRANSMISSION SPECTRA OF ISB TRANSITIONS 49
Figure 5.2: Transmission spectra for Sample A. The contact with the copper bar is made with kapton. a) Uncoated sample. No distinguish-able peak is visible at the expected resonant energy region, highlighted in yellow. b) Coated sample. A strong peak is visible in the resonant energy region, highlighted in yellow. The expected energy resonance E12= 202 is calculated with Schroedinger-Poisson method and is
high-lighted in the picture with an orange solid line. The centre of the peak visible in b) is slightly red shifted from the computed resonance energy.
The possible cause of this low contrast can be attributed to the use of silver paste in order to mount the samples on the copper holders. Due to its conductive nature and dishomogeneity, we cannot exclude an important dissipative effect of the silver paste itself or diffusion of light at the interface.
To overcome this problem, we attached the samples on the copper bar with kapton tape instead of silver paste. Spectra taken through these samples are visible in Figure 5.2. They clearly show a much higher signal to noise ratio, thus confirming the detrimental effect of the silver paste on the absorption.
Even with the use of kapton tape to fix the sample to the holder, we do not observe any ISB absorption on uncoated Sample A, as shown in Figure 5.2a. On the other hand, if the sample is gold coated, a strong peak is visible (Figure 5.2b). We compared the spectrum with the simulation scheme described in Chapter 3, as shown in Figure 5.3a.
between the 1→2 subbands. The red curve is the simulation for the absorption of a 10.1 nm thick QW in the same experimental condi-tions. The simulated peak is centred at Esim
12 = 197.9 meV with a
FWHMsim = 24.8 meV. b) Lorentzian fit of the right part of the data
in order to estimate the peak central energy E1 = 195.5 ± 0.1 meV. The
FWHM extracted by the experimental data is FWHM1 = 28.8 ± 0.7
meV. The Q factor associated to this transition is Q = 6.8 ± 0.2.
The peak obtained with the simulation is centred at Esim
12 = 197.9 meV
with a FWHMsim = 24.8 meV. This is in good agreement with the
mea-sured peak that is centred at E1 = 195.5 ± 0.1 meV with a FWHM1 of
28.8 ± 0.7 meV. The central position of the peak is estimated by fitting with a Lorentzian, as shown in Figure 5.3b:
L(x) = 1 − A γ
(x − x0)2 + (γ2)2
(5.1.1) where x0 is the energy position of centre of the peak, γ is a FWHM
and A is the multiplication factor which guarantees the normalization of the Lorentzian. The fit is performed considering the right part of the curve only, since the left part is affected by the from inhomogeneous thermal broadening. For this reason, the FWHM is extracted by the experimental data and not by the Lorentzian fit.
5.1. FTIR: TRANSMISSION SPECTRA OF ISB TRANSITIONS 51
Figure 5.4: Simulated total reflectivity of the heterostructure as a func-tion of the depth of the QW. The gold metallizafunc-tion forces the out-of-plane component of the field to be maximum at the semiconductor-metal interface. This results in a maximum of the absorption for a QW right below the surface, as visible in the colormap. Due to cavity effects, inside the heterostructure a standing wave establishes. This explains the periodicity visible in the picture. No dissipation effect is taken into account in this simulation.
and the experimental FWHM is probably due to the fact that in the simulations we considered only the thermal broadening, while disorder in the QW is another important broadening mechanism. In fact, con-sidering fluctuations of the order of 0.3 nm (more or less the thickness of one monolayer), compared to the dimension of the QW (10.1 nm), the resulting relative error is of the order of 3%. The Q factor associ-ated to the transition is Q = Epeak
F W HM = 6.8 ± 0.2.
The difference in absorption spectra between uncoated (Figure 5.2a) and coated (Figure 5.2b) samples, can be understood considering the electric field boundary conditions imposed by the gold on the surface. In fact, when a metallic layer is deposited on the sample surface, the
Figure 5.5: Transmission spectrum of Sample B at T=300K. The spec-trum shows two distinct peaks matching the ISB transitions between the 1→2 subbands and 2→3 subbands. The expected energy resonances E12= 118 meV and E23 = 153 meV are calculated with
Schroedinger-Poisson method and are highlighted in the picture with an orange and a green solid line, respectively.
boundary conditions lead to a maximum of the electric field at the semiconductor-metal interface [5]. The simulation in Figure 5.4 shows the total reflection of the heterostructure as a function of the QW depth in the structure. We note that the out-of-plane component of the field inside the cavity has a maximum at the metal-semiconductor interface. Since the QW is only 6 nm from the metallic layer, the gold deposition becomes crucial to the light-ISB transition coupling.
Measurements analogue to the previous ones were performed on Sample B. Figure 5.5 shows two distinct absorption peaks. Comparing these measurements with the Schroedinger-Poisson calculation, it turns out that the first peak corresponds to the expected energy E12 = 118
5.1. FTIR: TRANSMISSION SPECTRA OF ISB TRANSITIONS 53
Figure 5.6: a) Zoom on the transition 1→2 for the Sample B. The red curve is the simulation for the absorption for a 15 nm thick QW in the same experimental conditions. The simulated peak is centred at Esim
12 = 117.9 meV with a FWHMsim = 18.7 meV. b) Lorentzian fit of
the right part of the data in order to estimate the peak central energy E1 = 118.1 ± 0.7 meV. The FWHM extracted by the experimental data
is FWHM1 = 20.4 ± 0.8 meV. The Q factor is Q1 = 5.8 ± 0.3.
Figure 5.7: a)Zoom on the transition 2→3 for the Sample B. The red curve is the simulation for the absorption for a 15 nm thick QW in the same experimental conditions. The simulated peak is centred at Esim
23 = 147.5 meV with a FWHMsim = 11.7 meV. b) Lorentzian fit of
the right part of the data in order to estimate the peak central energy E2 = 146.56 ± 0.03 meV. The FWHM extracted by the experimental
one.
The absorption peak correspond to the 2→3 transition is shown in Fig-ure 5.7. The measFig-ured peak is centred at E2 = 146.56 ± 0.03 meV
with a FWHM2 = 11.6 ± 0.2 meV while the simulated one is centred
at Esim
23 = 147.5 meV with a FWHMsim2 = 11.7 meV. The second peak
appears less affected by the thermal broadening effects. This could be due to the lower overall population in the second subband, which yields a reduced effect of the non-parabolicity of the bands.
The Q factors are Q1 = 5.8 ± 0.3 for the transition 1→2 and Q2 =
12.6 ± 0.2 for the transition 2→3.
To understand the observation of the transition 2→3, we must consider the population inside the QW. We computed Fermi energy EF = 103
meV above the first subband energy for both structures (see Chapter 2). While for Sample A EF lays ∼100 meV below the second subband, for
sample B the second subband is only ∼ 15 meV above the Fermi energy. We can therefore conclude that at room temperature also the second subband is populated due to thermal fluctuations, which at T = 300 K are of the order of kBT ≈ 25 meV.
This population is also at the base of the intensity difference between the two peaks, visible in Figure 5.5. The transition between the first and the second subband is partially inhibited by electrons already present
5.1. FTIR: TRANSMISSION SPECTRA OF ISB TRANSITIONS 55
Figure 5.8: Schematic of subband dispersion of samples A (a) and B (b). In both cases the Fermi energy is 103 meV (dashed line). The yellow shaded area in (b) shows the electron distribution in the subbands, highlighting that there is a non trascurable thermal population in the second subband for Sample B.
in the final one. Furthermore, the transition dipole matrix element is a factor of two larger for the 2→3 transition with respect to the 1→2 transition [5]. Figure 5.8 shows schemes of the energy subbands of sam-ple A and B with the alignment of EF. The red shifts are maybe due to
band filling (as shown in Figure 5.8) and to the strong non-parabolicity of the QW subbands inside InAs [53].
We have to remark here, however that a simple single-particle picture of the ISB excitation might not be accurate enough to explain the detailed features of the absorption spectra. Many-body effects, like the depolar-isation shift (usually producing a red shift of the excitation), have to be taken into account to accurately predict the transition energy. These aspect go beyond the scope of the present thesis, also considering the experimental uncertainties of the structural parameters.
We also note that for highly-doped wide QWs with multiple subband occupation, the character of the transition shifts more and more away from a combination of single particle transitions towards an actual
sin-formed in this setup at two different temperatures (90K and 300K) for both Samples A and B. The measured spectra are shown in Figure 5.9 and 5.11, respectively.
Sample A (see Figure 5.10) actually shows a shoulder close to the main ISB peak for both temperatures. The origin of this shoulder is presently not understood. To estimate the peak central energy, we fit the right part of the curve with a Lorentzian function. We found E1,300K = 196.7 ± 0.1 meV at T = 300 K and E1,90K = 197.1 ± 0.1
meV at T = 90 K, as shown in Figure 5.10a and Figure 5.10b. The FWHMs are evaluated from the experimental data and not from the fit, in order to consider also the inhomogeneous broadening. As expected, the width of the peaks decreases with temperature [53]. We found FWHM300K = 20 ± 2 meV at T = 300K and FWHM90K = 19 ± 2 meV
at T = 90K. The magnitude of the narrowing effect depends on the fact that the thermal phonons decreases with the temperature, while the disorder remains constant. It is therefore possible that the broaden-ing induced by disorder overcomes the thermal broadenbroaden-ing, thus result-ing in a reduced dependence of the linewidth on temperature. Figure 5.11 shows the dependence on the temperature of the two peaks of Sample B. With the same fit method used for Sample A, we found E1,300K = 121.3 ± 0.1 meV and E2,300K = 143.2 ± 0.1 meV at T = 300
5.1. FTIR: TRANSMISSION SPECTRA OF ISB TRANSITIONS 57
Figure 5.9: Transmission spectra of Sample A at two different temper-atures. The expected peak is at energy E12= 202 meV, computed with
the Schroedinger-Poisson solver.
Figure 5.10: a) Lorentzian fit of the peak at T = 300K: the estimated peak central energy is E1,300K = 196.7 ± 0.1 meV. The FWHM300K =
20 ± 2 meV and the Q factor is Q = 9.8 ± 0.9. b) Lorentzian fit of the peak at T = 90K: the estimated peak central energy is E1,90K =
197.1 ± 0.1 meV. The FWHM90K = 19.5 ± 0.5 meV and the Q factor
Figure 5.11: Transmission spectra of Sample B at two different tem-peratures. The expected peaks are at energies E12 = 118 meV and
E23= 153 meV. We observe that the second peak almost disappears at
low temperature.
Figure 5.12: a) Lorentzian fit of the peaks at T = 300K: the estimated peak central energies are E1,300K = 121.3 ± 0.1 meV and E2,300K =
143.2 ± 0.1 meV. The FWHM is FWHM1,300K = 14.4 ± 0.4 meV and
the Q factor is Q1 = 8.4 ± 0.2 for the peak 1→2; while FWHM2,300K =
7.6 ± 0.3 meV and Q2 = 18.9 ± 0.9 are evaluated for the peak 2→3
b) Lorentzian fit of the peak at T = 90K: the estimated peak central energies are E1,90K = 123.0 ± 0.1 meV and E2,90K = 146.2 ± 0.3 meV.
The FWHM is FWHM1,90K = 10.2 ± 0.2 meV and the Q factor is
5.1. FTIR: TRANSMISSION SPECTRA OF ISB TRANSITIONS 59
Figure 5.13: On the right of this picture we illustrate the subband levels of the QW with thickness 15 nm and the centred at the same Fermi en-ergy, value of the Fermi energy in our experimental conditions (dashed line). On the left we plot the Fermi distribution for two different tem-peratures. At low temperature the population in the second subband decreases because of the Fermi distribution being sharper than at room temperature.
K and at E1,90K = 123.0 ± 0.1 meV and E2,90K = 146.2 ± 0.3 meV at
T = 90 K.
The peak corresponding to the E23 ISB transition reduces at low
tem-perature. As illustrated in Figure 5.13, by lowering the temperature, the second subband starts to depopulate because the energy associated to the thermal fluctuations becomes smaller. The thermal fluctuation in fact are of the order of kBT = 8 meV at T = 90 K and the second
subband is much less populated. Again, the peak matching the 1→2 transition becomes narrower with the temperature decrease. In fact the FWHMs estimated by the Lorentzian fit are: FWHM1,300K = 14.4 ± 0.4
meV for at room temperature and FWHM1,90K = 10.2 ± 0.2 meV at
low temperature. The Q factor associated to the transition 1→2 is Q1 = 8.4 ± 0.2 at T = 300K while it is Q2 = 18.9 ± 0.9 for the transition
5.2. S-SNOM MEASUREMENTS 61
Figure 5.14: Electrical contacts: VBG is the applied voltage between
top and the back of the structure; VSD is the voltage between source
and drain, employed to measure the in plane conduction.
5.2
s-SNOM measurements
We will now discuss the results obtained by s-SNOM measurements. As already explained in Section 4.3.2, we want to use the charge mod-ulation to isolate the ISB absorption in the signal measured with the s-SNOM tip. To verify if we are able to modify the carrier density inside the QW, we performed preliminary transport measurements.
5.2.1
Transport measurements
Three contacts are fabricated on the InAlAs surface as shown in Figure 5.14 in order to apply a voltage VSD between two generic points on the
surface, that serve as source and drain. We want to check how ISD,
at VSD fixed, depends on the back-gate voltage (VBG) applied between
the top and the bottom layers of the heterostructure. By apply-ing VBG, we expect to measure a variation of ISD as a consequence of
ge-5.16. (b) Variation of ISD as function of VBG.
ometry. The characterization of the samples’ electrical properties are performed in a probe station with three needles as probes. The needles are mounted on micro-manipulators which allow a precise positioning of the probes on the sample. First, we studied the electrical conduction of the sample shown in Figure 5.16. The gold patch are made with the optical lithography technique. We measured the current flowing in the plane of the QW, by applying a VSD from 0 mV to 1 mV (see Figure
5.15a). By performing a linear fit, we obtained an electrical resistance of 584 ± 1 Ω for the in-plane conduction. Then, we measured ISD
while varying VBG from 0 mV to 50 mV. As it is shown in Figure 5.15
(b), the current immediately reached the instrument compliance. So it is likely that this current is due to out-of-plane conduction. In fact, additional transport measurements between top and back-gate showed electrical resistances of the same order of the in plane resistance (∼ 100 Ω). Thus, we can conclude that this heterostructure is conductive even in the growth direction, contrary to what we expected. This undesired out-of-plane conduction can be attributed to many possible causes. The first one could be the presence of charges in the superlattice that can contribute to the transport. To verify the presence of charge, we extend
5.2. S-SNOM MEASUREMENTS 63
Figure 5.16: Four gold patches realized with optical lithography on the sample A.
the Schroedinger-Poisson calculation, taking into account the presence of the superlattice. Figure 5.17 shows that some minibands form at energies Em1 = 0.6 eV and Em2 = 1.61 eV, but these energies are well
above the Fermi energy EF = 103 meV. We can therefore conclude that
no thermal excitation can populate these minibands and the bending of the band caused by the small bias applied through the sample is not enough to allow conduction through the superlattice.
Another problem may be due to the transport of thermally generated holes because there are no barriers for the holes in the superlattice. However, the bipolar conduction showed very symmetric characteris-tics. Additional measurements at low temperature confirmed that the conduction cannot be due to holes population.
Another possibility is lateral conduction along the heterostructure cleaved edge. As we said in Section 4.1 Fermi pinning causes a charge accumu-lation at the surface of InAs. This surface charge can yield a conduction in the out-of-plane direction, with a resistance Rz inversely proportional
Figure 5.17: Calculation of the QW heterostructure energy subbands taking into account the presence of the superlattice for Sample B. Some minibands form within the superlattice but their energies Em1 = 0.6
eV and Em2 = 1.61 eV are very larger than the Fermi energy EF = 103
meV.
to the lateral surface S of the sample. To overcome this problem, a new design was realized. In order to increase the resistance Rz in the
out-of-plane direction we aimed at reduce the lateral surface of the sample. We etched the sample with a solution of H3P O4 : H2O2 : H2O (1:1:15)
down into the substrate, leaving a small square of 100 µm x 100 µm untouched, as shown in Figure 5.18. We deposited small gold patches on the pillars so created. We measured the conduction between the gold patch and the back of the sample. The best out-of-plane resis-tance obtained for this sample is Rz = 16.3 ± 0.1 KΩ. The data are
shown in Figure 5.19. We can thus conclude that all samples mea-sured until now show conduction in the direction perpendicular to the QW plane. Therefore, at the moment it is not to possible to use the modulation technique to control the charge of the QW in the studied heterostructures.
5.2. S-SNOM MEASUREMENTS 65
Figure 5.18: a) The four big squares represent the etched part of the sample, while the small squares are untouched. The gold patches are deposited on the resulting pillars to create electrical contacts. b) SEM image of the heterostructure: The big grey square is the etched zone while the smaller one is the untouched area.
Figure 5.19: I-V curve for the out-of-plane conduction of the sample in Figure 5.18.
