Optical absorption at one-particle states of charge carriers in semiconductor quantum dots

Abstract

Abstract

It is shown that the oscillator strengths and dipole moments of the transitions for one-particle states of charge carriers that originated in semiconductor quantum dots take on large values, significantly exceeding the typical values of the corresponding values for semiconductor materials. In the framework of the dipole approach, it is shown that large values of the cross section of light in the studied quasi-zero-dimensional systems enable the use of such nanosystems as new strongly absorbing nanomaterials.

PACS

78. 67.Hc; 73. 30. Ly; 78.20. Dj

Correspondence/Findings

Introduction

At present, the optical[1–5] and electro-optical[6–8] properties of quasi-zero-dimensional structures are extensively studied. Such structures commonly consist of spherical semiconductor nanocrystals (the so-called quantum dots (QDs)) with a radius a ≈ 1 to 10² nm grown in semiconductor (or dielectric) matrices. The studies in this field are motivated by the fact that such heterophase systems represent as new promising materials for the development of new components of nonlinear optoelectronics to be used, specifically, for controlling optical signals in optical computers[9–12] or for manufacturing active layers of semiconductor injection lasers[5, 13–17].

The optical and electro-optical properties of such quasi-zero-dimensional structures depend on the energy spectrum of a spatially confined electron–hole pair (EHP), i.e., an exciton[1–8]. By the methods of optical spectroscopy, the effects of quantum confinement on the energy spectra of electrons[1, 2] and excitons[3, 4] were revealed in these heterophase structures.

Previously[18], the conditions for the localization of charge carriers near the spherical interface between two dielectric media were analyzed. In this case, the polarization interaction of a charge carrier with the surface charge induced at the spherical interface, U(r,a), depends on the relative permittivity ϵ = ϵ₁/ϵ₂. Here, r is the spacing between the charge carrier and the center of the dielectric particle, a is the radius of the particle, and ϵ₁ and ϵ₂ are the permittivities of the surrounding medium and of the dielectric particle embedded in the medium, respectively.

For the charge carriers in motion near the dielectric particle, there are two possibilities:

• 1. Due to the polarization interaction U(r,a), the carriers can be attracted to the particle surface (to the outer or the inner surface at ϵ < 1 or ϵ > 1, respectively), with the formation of outer [19, 20] or inner [21] surface states.

• 2. Due to the polarization interaction U(r,a), the carriers can be, at ϵ < 1, repelled from the inner surface of the dielectric particle, with the formation of bulk local states inside the particle bulk [22–24]; in this case, the spectrum of the low-energy bulk states is of an oscillatory shape.

It has been shown[18–24] that the formation of the above-mentioned local states is of a threshold-type nature and is possible if the radius of the dielectric particle a is larger than a certain critical radius α_c:

$$a\ge a_{\mathrm{c}}=6{\left|\beta \right|}^{-1}{a}_{\mathrm{B}i},$$

(1)

where β = (ϵ₁ − ϵ₂)/(ϵ₁ + ϵ₂) and α_Bi is the Bohr radius of a charge carrier in a medium with the permittivity ϵ₁ (i = 1, 2).

In[14–17], the optical properties of an array of InAs and InSb QDs in GaAs and GaSb matrices and the corresponding operational characteristics of injection lasers, with the active region on the basis of this array, were studied experimentally. In these studies, a large short-wavelength shift of the laser emission line was observed for the array of QDs. In such an array, the energy spectrum of charge carriers is completely discrete[21–24] if the QDs are smaller than a ≈ 1 to 7 nm in size. In the first-order approximation, the spectrum of such quantum-confined states can be described as a spectrum of a charge carrier in a spherically symmetric well with infinitely high walls.

Until now, there have been no theoretical investigations on optical absorption and scattering at such discrete states in arrays of QDs. To close the gap in this area, here we develop a theory of the interaction of an electromagnetic with one-particle quantum-confined states of charge carriers that originate in the bulk of a semiconductor QD.

In conclusion, we briefly discuss possible physical situations in which the results obtained above can be used for interpreting the experimental data.

Hamiltonian for an electron–hole pair in a quantum dot

We consider a simple model in which a quasi-zero-dimensional system is defined as a neutral spherical semiconductor QD of radius a and permittivity ϵ₂, embedded in a surrounding with permittivity ϵ₁. Let an electron (e) and hole (h), whose effective masses are, correspondingly, m_e and m_h, be the motion in this QD. Let the spacing between the electron or hole and the QD center be r_e or r_h, respectively. We assume that the bands for electrons and hole are parabolic. Along with the QD radius a, the characteristic lengths of the problem are a_e, a_h, and a_ex, where

$$a_{\mathrm{e}}={ϵ}_2{ħ}^2/m_{\mathrm{e}}{e}^2,a_{\mathrm{h}}={ϵ}_2{ħ}^2/m_{\mathrm{h}}{e}^2,a_{\mathrm{ex}}={ϵ}_2{ħ}^2/\mu e^2$$

(2)

are the Bohr radii of the electron, hole, and exciton in the semiconductor with the permittivity ϵ₂, respectively, and μ = m_em_h/(m_e + m_h) is the exciton effective mass. All of the characteristic lengths of the problem are much larger than the interatomic spacing α₀:

$$a,a_{\mathrm{e}},a_{\mathrm{h}},a_{\mathrm{ex}}\gg a_0.$$

(3)

This allows us to treat the motion of the electron and hole in the QD in the effective-mass approximation.

In the context of the above-described model and approximations for a quasi-zero-dimensional system, the Hamiltonian of the EHP is

$$\begin{array}{l}H=-\frac{{ħ}^2}{2m_{\mathrm{e}}}{\mathrm{\Delta }}_{\mathrm{e}}-\frac{{ħ}^2}{2m_{\mathrm{h}}}{\mathrm{\Delta }}_{\mathrm{h}}+V_{\mathrm{eh}}\left(r_{\mathbf{e}},r_{\mathbf{h}}\right)\\ +U\left(r_{\mathbf{e}},r_{\mathbf{h}},a\right)+E_{\mathrm{g}},\end{array}$$

(4)

where the first two terms in the sum define the kinetic energy of the electron and hole, E_g is the energy bandgap in the bulk (unbounded) semiconductor with the permittivity ϵ₂, V_eh(r _e ,r _h ) is the energy of the electron Coulomb interaction:

$$V_{\mathrm{eh}}\left(r_{\mathbf{e}},r_{\mathbf{h}}\right)=\frac{e^2}{2{ϵ}_{2}a}\frac{2a}{{\left(r_{\mathrm{e}}^2-2r_{\mathrm{e}},r_{\mathrm{h}}\cos \theta +r_{\mathrm{h}}^2\right)}^{1/2}}$$

(5)

with the angle θ between the vector r _e and r _h , and U(r _e ,r _h ,a) is the energy of the interaction of the electron and hole with the polarization field induced by the electron and hole at the spherical interface between the two media. For arbitrary values of ϵ₁ and ϵ₂, the interaction energy U(r _e ,r _h ,a) can be represented analytically as[18, 24]

$$\begin{array}{l}U\left(r_{\mathbf{e}},r_{\mathbf{h}},a\right)=-\frac{e^2\beta }{{ϵ}_{2}a}\frac{1}{{\left[{\left(r_{\mathbf{e}}{r}_{\mathbf{h}}/a^2\right)}^2-2\left(r_{\mathbf{e}}{r}_{\mathbf{h}}/a^2\right)\cos \theta +1\right]}^{1/2}}\\ -\frac{e^2\beta }{2\left({ϵ}_1+{ϵ}_2\right)a}\underset{0}{\overset{\infty }{{\displaystyle \int }}}\frac{\mathrm{d}y{\left(a^2/r_{\mathbf{h}}y\right)}^{\alpha }\mathrm{\Theta }\left(y-a^2/r_{\mathbf{h}}\right)}{\left|r_{\mathbf{e}}-y\left(r_{\mathbf{e}}/r_{\mathbf{h}}\right)\right|}\\ -\frac{e^2\beta }{2\left({ϵ}_1+{ϵ}_2\right)a}\underset{0}{\overset{\infty }{{\displaystyle \int }}}\frac{\mathrm{d}y{\left(a^2/r_{\mathbf{e}}y\right)}^{\alpha }\mathrm{\Theta }\left(y-a^2/r_{\mathbf{e}}\right)}{\left|r_{\mathbf{h}}-y\left(r_{\mathbf{h}}/r_{\mathbf{e}}\right)\right|},\end{array}$$

(6)

where Θ(x) is the unit step function and

$$\alpha =\frac{{ϵ}_1}{{ϵ}_1+{ϵ}_2}.$$

(7)

The spectrum of charge carriers in a quantum dot

In the bulk of a QD, the electron (hole) energy levels can originate. Their energies are defined as[25]

$$E_{n,l}\left(a\right)=\frac{{ħ}^2}{2m_{\mathrm{e}\left(\mathrm{h}\right)}{a}^2}{\phi }_{n,l}^2,$$

(8)

where the subscripts (n,l) refer to the corresponding quantum size-confined states. Here, n and l are the principal and azimuthal quantum numbers for the electron (hole), respectively, and φ_n,l are the roots of the Bessel function, i.e.,$J_{1+\frac{1}{2}}\left({\phi }_{n,l}\right)=0$. For the quantum-confined levels to originate, it is necessary that in the Hamiltonian (4), the electron (hole) energy E_n,l(a) (8) be much larger than the energy of the interaction of the electron (hole) with the polarization field U(a) (6) generated at the spherical QD-dielectric (semiconductor) matrix interface:

$$E_{n,l}\left(a\right)=\frac{{ħ}^2}{2m_{\mathrm{e}\left(\mathrm{h}\right)}{a}^2}{\phi }_{n,l}^2\gg U\left(a\right)\approx \frac{e^2\beta }{2{ϵ}_{2}a}.$$

(9)

Condition (9) is satisfied for QDs of radii

$$a\ll a_{\mathrm{s}}^{\mathrm{e}\left(\mathrm{h}\right)}=\frac{{\phi }_{n,l}^2}{\beta }{a}_{\mathrm{e}\left(\mathrm{h}\right)}.$$

(10)

At room temperature T₀, the discrete levels of the electron (hole) E_n,l (8) in the QD are slightly broadened if the energy separation between the levels is[20–22]

$$\mathrm{\Delta }{E}_{n,l}\left(a\right)=E_{n^{`},l}-E_{n,l}\left(a\right)\ll k{T}_0.$$

(11)

Taking into account (8), we can rewrite inequality (11) as

$$\frac{{ħ}^2}{2m_{\mathrm{e}\left(\mathrm{h}\right)}{a}^2}\frac{{\phi }_{n^{`},l}^2-{\phi }_{n,l}^2}{k{T}_0}=\eta \left(a\right)\ll 1.$$

(12)

Formula (8), describing the spectrum of charge carriers in a QD, is applicable to the lowest states (n,l) that satisfy the inequality

$$\mathrm{\Delta }{E}_{n,l}\left(a\right)\ll \mathrm{\Delta }{V}_0\left(a\right),$$

(13)

where ΔV₀(a) is the depth of the potential well for electrons in the QD. For example, for the CdS QDs whose sizes satisfy inequality (10), the value ΔV₀ is 2.3 to 2.5 eV[26].

If condition (10) is satisfied, we can use, for the electron (hole) wave function in a QD, the wave function of an electron (hole) in a spherical quantum well with infinitely high walls[27]:

$${\mathrm{\Psi }}_{n,l,m}\left(r,\theta ,\phi \right)=Y_{l,m}\left(\theta ,\phi \right)\frac{J_{1+1/2}\left({\phi }_{n,l}\right)\sqrt{2}}{J_{1+3/2}\left({\phi }_{n,l}\right)a\sqrt{r}},$$

(14)

where r = r_e or r = r_h is the distance of the electron or hole from the QD center, respectively, θ and φ are the azimuthal and polar angles that define the orientation of the radius vector of the electron (hole), respectively, Y_l,m are the normalized spherical functions (m is the magnetic quantum number of the electron or hole), and J _ν (x) are the Bessel functions that can be expressed in terms of the spherical Bessel functions j _ν (x)[27] as

$$J_{1+1/2}\left({\phi }_{n,l}\right)=\sqrt{2/\pi }{\phi }_{l,n}{j}_l\left({\phi }_{l,n}\right)$$

$$J_{1+3/2}\left({\phi }_{n,l}\right)=\sqrt{2/\pi }{\phi }_{l,n}{j}_{l+1}\left({\phi }_{l,n}\right).$$

(15)

Dipole moments of charge carrier transitions in a quantum dot

In the frequency region corresponding to the above-considered states of charge carriers in the QD bulk, (n,l), the wavelength of light is much larger than the dimensions of these states (~a_e, a_h), that is the behavior of approximation. In this case, the operator of the dipole moment of the electron (hole) located in the QD bulk is expressed as[28]

$$D\left(r\right)=\frac{3{ϵ}_1}{2{ϵ}_1+{ϵ}_2}{D}^0\left(r\right),D^0\left(r\right)=\mathit{er}.$$

(16)

To estimate the value of the dipole moment, it is sufficient to consider the transition between the lowest discrete states (8), e.g., between the ground states |1s = (n = 1, l = 0, m = 0) and |1p = (n = 1, l = 1, m = 0). To calculate the matrix element of the dipole moment of the charge carrier transition from the 1s state to the 1p state, D_1,0(a), we assume that the uniform field of the light wave E(ω,t) is directed only along the axis Z (ω is the wave frequency). In this case, we take the dipole moment D(r) (16) induced by the field E(ω,t) as the perturbation responsible for such dipole transition. The expression for the dipole moment of the transition D_1,0(a) follows from formulas (16) and the expression for the dipole moment of the transition in free space:

$$D_{1,0}^0\left(a\right)=1s\left|D^0\left(r\right)\right|1p=e1s\left|r\right|1p.$$

(17)

Taking into account (14) and (15), we can write the wave functions of the |1s and |1p states as

$$\left|1s\right.={\mathrm{\Psi }}_{1,0,0}\left(r,\theta \right)=Y_{0,0}\left(\theta \right)\frac{2}{a^{3/2}}\frac{j_0\left({\phi }_{0,1}\left(r/a\right)\right)}{j_1\left({\phi }_{0,1}\right)},$$

(18)

$$\left|1p={\psi }_{1,0,0}\left(r,\theta \right)=Y_{1,0}\left(\theta \right)\right.\frac{2}{{\alpha }^{3/2}}\frac{j_1\left({\phi }_{1,1}\left(r/\alpha \right)\right)}{j_2\left({\phi }_{1,1}\right)}.$$

(19)

On substitution of (18) and (19) into formula (17) and integration, we obtain the expression for the dipole moment of the transition in free space as follows:

$$\begin{array}{l}{D}_{1,0}^0\left(a\right)=\frac{2\pi \sqrt{6}}{3{\phi }_{1,1}{j}_2\left({\phi }_{1,1}\right)\left({\phi }_{1,1}^2-{\pi }^2\right)}\\ \times \left[\cos {\phi }_{1,1}-\frac{\left(3{\phi }_{1,1}^2-{\pi }^2\right)}{{\phi }_{1,1}\left({\phi }_{1,1}^2-{\pi }^2\right)}\sin {\phi }_{1,1}\right]\mathit{ea}\\ =0.433\mathit{ea}.\end{array}$$

(20)

Then, according to (20) and (16), the dipole moment of the transition in the QD with the permittivity ϵ₂ in the surrounding matrix with the permittivity ϵ₁ is

$$D_{1,0}\left(a\right)=\mathrm{\Lambda }{D}_{1,0}^0\left(a\right)=\mathrm{\Lambda }0.433\mathit{ea},$$

(21)

where

$$\mathrm{\Lambda }=\frac{3{ϵ}_1}{2{ϵ}_1+{ϵ}_2}.$$

(22)

Optical absorption on light at one-particle states of charge carriers in quantum dots

Using the above results for the matrix element of the dipole moment of the transition D_1,0 (formulas (21) and (22)), we can elucidate the behavior of the semiconductor quasi-zero-dimensional systems on absorbing the energy of the electromagnetic field in the frequency region corresponding to the energies of the quantum-confined states in the QD E_n,l (8). The absorption cross section of a spherical QD of radius a can be expressed in terms of the polarizability of QD,$A^{``}\left(\omega ,a\right)$A, as[28]

$${\sigma }_{\mathrm{abs}}\left(\omega ,a\right)=4\pi \left(\omega /c\right)A^{``}\left(\omega ,a\right),$$

(23)

where ω is the frequency of the external electromagnetic field.

The polarizability A^{``</sup> can be easily determined if the QD is considered as a single giant ion. Let the radius of QD be a (10). In such QD, the quantum-confined states of charge carriers (n,l) (8) are formed. At room temperature, these states are slightly broadened, satisfying inequality (12). In this case, the polarizability of the charged QD, A<sup>``}, can be expressed in terms of the matrix element of the dipole moment of the transition D_1,0(a) (21) between the lowest 1s and 1p states[29, 30]:

$$A^{``}\left(\omega ,a\right)=\frac{e^2}{m_{\mathrm{e}\left(\mathrm{h}\right)}}\frac{f_{0,1}}{{\omega }_1^2\left(a\right)-{\omega }^2-\mathit{i\omega }{\mathrm{\Gamma }}_1\left(a\right)},$$

(24)

where

$$f_{0,1}=\frac{2m_{\mathrm{e}\left(\mathrm{h}\right)}}{ħe^2}\left[{\omega }_1\left(a\right)-{\omega }_0\left(a\right)\right]{\left|D_{1,0}\left(a\right)\right|}^2$$

(25)

is the oscillator strength of the transition of a charge carrier with the effective mass m_e (or m_h) from the ground 1s state to the 1p state,${\omega }_1=E_{1.1}$ and${\omega }_0=E_{1.0}$ are, correspondingly, the energies of the discrete 1s and 1p levels by formula (8), and Γ₁(a) is the width of the 1p level[20, 22]. Taking into account formulas (8) and (21), we can express the oscillator strength (25) of the transition as

$$f_{0,1}=\left({\phi }_{1,1}^2-{\pi }^2\right)\frac{{\mathrm{\Lambda }}^2{\left(D_{1,0}^0\right(a\left)\right)}^2}{e^2{a}^2}=0.967\times {10}^{-1}{\mathrm{\Lambda }}^2$$

(26)

We assume that the frequency ω of the wave of light is far from the resonance frequency ω₁ of the discrete 1p state and, in addition, that the broadening Γ₁ of the 1p level is small, i.e., Γ₁/ω₁ ≪ 1[20–22]. Then, for the qualitative estimate of the QD polarizability A^``(ω, a) (24), we obtain, with regard to (8), the following expression:

$$A^{``}\left(a\right)=\frac{4f_{0,1}}{{\phi }_{1,1}^4}\frac{m_{\mathrm{e}\left(\mathrm{h}\right)}}{m_0}{\left(\frac{a}{a_{\mathrm{B}}}\right)}^4{a}_{\mathrm{B}}^3,$$

(27)

where$a_{\mathrm{B}}={ħ}^2/m_0{e}^2$ is the Bohr radius of an electron in free space. Now we write the expression for the cross section of elastic scattering of the electromagnetic wave with frequency ω by the QD of radius a[28] as

$${\sigma }_{\mathrm{sc}}\left(\omega ,a\right)=2^7{3}^{-3}{\pi }^3{\left(\omega /c\right)}^4{\left|A^{``}\left(a\right)\right|}^2.$$

(28)

The theory developed here for QDs is applicable only to the intraband transitions of electrons (holes) whose spectrum$E_{\mathit{nl}}^{e\left(h\right)}\left(a\right)$ is defined by formula (8). The optical attenuation coefficient, which involves both the absorption and scattering of light by one-particle quantum-confined states of charge carriers in the QD bulk of radius a, can be written, for the concentration of QDs N, as[29, 30]

$$\gamma \left(\omega ,a\right)=N\left[{\sigma }_{\mathrm{abs}}\left(\omega ,a\right)+{\sigma }_{\mathrm{sc}}\left(\omega ,a\right)\right]$$

(29)

Formula (29) is applicable to an array of QDs that do not interact with each other. The condition, such that the QDs of radius a and concentration N do not interact with each other, is reduced to the requirement that the spacing between the QDs (~N^−1/3) is much larger than the dimensions of the above-considered one-particle states (~a_e(h)), i.e.,

$$a_{\mathrm{e}\left(\mathrm{h}\right)}{N}^{-1/3}\ll 1.$$

(30)

With a_e(h) ≈ 5.0 nm criterion (30) is satisfied for the concentrations of QDs N ≤ 10¹⁵ cm⁻³ achievable under the experimental conditions of[13–17] for a number of III-V semiconductors.

Comparison of the theory with experiments

Similar to[26], we can assume that under the experimental conditions of[13–17], the annealing of the arrays of InAs and InSb QDs in the GaAs and GaSb matrices at the temperature T = 273 K induces the thermal emission of a light electron, so that a hole alone remains in the QD bulk. In this case, the electron may be localized at a deep trap in the matrix. If the distance d, from this trap to the QD center, is large compared to the QD radius a (d ≫ a), the Coulomb electron–hole interaction V_eh(r_e,r_h) (5) in the Hamiltonian (4) can be disregarded. As a result, the one-particle quantum-confined hole states (n,l), with the energy spectrum E_n,l(a) described by formula (8), appear in the QD bulk. Now we roughly estimate the cross sections of optical absorption σ_abs ((25) and (29)) and σ_sc (28) at the quantum-confined hole state in the QDs for the selected 1s → 1p transition under the experimental conditions of[13, 17]. For the rough estimation of the cross sections of optical absorption and scattering, we use expressions (23), (27), and (28) on the assumptions that the frequency of the light wave ω is far from the resonance frequency ω₁ of the discrete hole state in the QD and that the broadening Γ₁ of the energy level$E_{1.1}={\omega }_1$ (8) is small[20–22] (Γ₁/ω₁ ≪ 1). In this case, the absorption cross section σ_abs and the scattering cross section σ_sc take the following forms:

$${\sigma }_{\mathit{abs}}\left(\omega ,\alpha \right)=\frac{16\pi f_{0,1}}{{\phi }_{1,1}^4}\frac{\omega }{c}\frac{m_h}{m_0}{\left(\frac{\alpha }{{\alpha }_B}\right)}^4{\alpha }_B^3,$$

(31)

$${\sigma }_{\mathrm{sc}}\left(\omega ,a\right)=\frac{1^{11}{\pi }^3{f}_{0,1}{2}^{}}{3^3{\phi }_{1,1}^8}{\left(\frac{\omega }{c}\right)}^4{\left(\frac{m_{\mathrm{h}}}{m_0}\right)}^2{\left(\frac{a}{a_{\mathrm{B}}}\right)}^8{a}_{\mathrm{B}}^6.$$

(32)

The estimated parameters of the hole states in QDs dispersed in the III-V semiconductors are listed in Table 1. It is worth noting that at room temperature, the quantum-confined hole states are slightly broadened; the parameter η(a) (14) does not exceed 35%.

Table 1 Parameters of quantum-confined states of hole in motion in quantum dots

The quasi-zero-dimensional systems considered here are essential nonlinear media with respect to infrared radiation. In fact, the dipole moments of the transitions in QDs of radii a ≈ 20 to 40 Å are rather large in magnitude: D_1,0 ≈ 10 D (see table), being many times larger than the values D ≈ 0.1 D typical for bulk III-V semiconductors[31, 32]. In addition, according to the selection rules for QDs in the electromagnetic field, the dipole transitions between the nearest levels E_n,l (8) are allowed[27]. Under these transitions, the azimuthal quantum number changes by unity.

From the estimates presented in the table, it follows that for QDs of radii a ≈ 2.0 to 4.0 nm, the absorption cross section can be as large as σ_abs ≈ 10¹⁶ cm². This value is 8 orders of magnitude larger than the typical absorption cross sections for atoms[33]. Since the scattering cross sections σ_sc (32) under the experimental conditions of[13–17] are negligible compared to the corresponding absorption cross sections σ_abs (31) (σ_sc/σ_abs ≈ 10⁻¹²), the estimates for σ_sc are not included in the table.

Thus, the optical attenuation coefficient γ (31) for QDs is controlled mainly by the absorption process at the quantum-confined states of charge carriers (n,l) (8). If the concentration of QDs of radii a satisfying condition (38) is N ≈ 10¹⁵ cm⁻³, the optical attenuation coefficient, on absorption at the QDs with the parameters listed in the table, takes the value γ ≈ 0.1 cm⁻¹.

The large optical absorption cross section and attenuation coefficients in the quasi-zero-dimensional systems treated above allow the use of such heterophase structures as new efficient absorbers of electromagnetic waves in a wide wavelength range variable over wide limits in accordance with the nature of materials in contact[34].