Harnessing the Shape-Induced Optical Anisotropy of a Semiconductor

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Harnessing the Shape-Induced Optical Anisotropy of a Semiconductor Nanocrystal: A New Type of Intraband Absorption Spectroscopy Anvar S. Baimuratov, Ivan D. Rukhlenko, Vadim K. Turkov, Mikhail Yu. Leonov, Alexander V. Baranov, Yurii K. Gun'Ko, and Anatoly V. Fedorov J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 09 Jan 2014 Downloaded from http://pubs.acs.org on January 9, 2014

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The Journal of Physical Chemistry

Harnessing the Shape-Induced Optical Anisotropy of a Semiconductor Nanocrystal: A New Type of Intraband Absorption Spectroscopy Anvar S. Baimuratov,† Ivan D. Rukhlenko,∗,†,‡ Vadim K. Turkov,† Mikhail Yu. Leonov,† Alexander V. Baranov,† Yurii K. Gun’ko,†,¶ and Anatoly V. Fedorov† Saint Petersburg National Research University of Information Technologies, Mechanics and Optics, 197101 Saint Petersburg, Russia E-mail: [email protected]

KEYWORDS: Optical spectroscopy, quantum dots, linear optics, light–matter interactions.

∗

To whom correspondence should be addressed Saint Petersburg National Research University of Information Technologies, Mechanics and Optics, 197101 Saint Petersburg, Russia ‡ Advanced Computing and Simulation Laboratory (AχL), Department of Electrical and Computer Systems Engineering, Monash University, Clayton, Victoria 3800, Australia ¶ School of Chemistry and CRANN Institute, Trinity College, Dublin, Dublin 2, Ireland †

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Abstract In recent years there have been active developments of spectroscopic methods for the analysis of light absorption by individual nanocrystals. Here we provide a solid theoretical background for one of such methods via developing a uniform theory of anisotropic intraband absorption by a nonspherical semiconductor nanocrystal. The nanocrystal is assumed to be simultaneously excited by the linearly polarized pump and probe fields that are respectively resonant to the interband and intraband transitions of the nanocrystal’s electronic subsystem. Three relative arrangements of the excited electron– hole pair states are considered, covering all possible types of the transition schemes that can occur in experiment. The developed theory is then used to calculate the angular absorption spectra for the most common shapes of the nanocrystals, which essentially lays the foundation of stationary pump–probe polarization spectroscopy based on the shape-induced anisotropy of intraband absorption by a semiconductor nanocrystal. This spectroscopy may serve modern nanotechnology by facilitating the characterization of anisotropic nanostructures, as it allows one to reliably determine the shapes and orientations of individual nanocrystals, and find the symmetry of quantum states involved in the electronic transitions induced by the probe.

Introduction Experimental studies of individual semiconductor nanocrystals is of significance for both fundamental physics and the engineering of superior nanophotonics devices. Such nanocrystals— often referred to as quantum dots due to the size quantization of their elementary excitations energy spectra—already serve people in highly efficient single-photon sources, 1,2 single-electron transistors, 3 solar cells, 4,5 q-bits, 6 and lasers. 7 The study of the nanocrystals’ physical properties heavily rely on a variety of advanced stationary and transient spectroscopic methods. 8–13 The single-quantum-dot spectroscopy is one of such methods that enables avoiding the inhomogeneous broadening of optical spectra due to the size distribution of the


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nanocrystals. 14 The geometric characteristics (such as size, shape, and spatial orientation) and energy spectrum of an individual nanocrystal can be reliably determined using the polarization spectroscopy of intraband optical transitions. 15–17 This is possible owing to the pronounced anisotropy of the intraband transition rates and their strong dependence on the type of quantum states involved in the transitions. 18 The advantages of such spectroscopy can be fully utilized using the pump–probe experimental technique, where the information on the nanocrystal’s characteristics is contained in the angular absorption spectrum of a polarized probe interacting with a nanocrystal that was excited by an optical pump. A drawback of this technique is that it is quite a challenging experimental task to accurately measure the amount of light absorbed by a single nanocrystal. However, the recent advances in the experimental techniques for measuring absorption by individual molecules 19–21 and quantum dots 22–25 make this type of spectroscopy practicable. In this paper we present a uniform theory of intraband absorption spectroscopy of a semiconductor nanocrystal for three practical types of the steady-state regimes of excitation and detection, and provide simple guidelines of its using for experimentalists.

Theory of intraband absorption The excitation of an electronic subsystem by the pump field and the detection of the subsystem’s quantum state via the absorption of the probe field are the two basic processes of the pump–probe scanning of an individual nanocrystal. The first process involves generation of an electron–hole pair by an optical pump whose frequency, ωpu , is close to the resonance with an interband transition of the nanocrystal. The generation is accompanied by energy relaxation of the electronic subsystem due to its interaction with other subsystems of the nanocrystal and the environment. 26–34 On the detection stage, the system is illuminated by a probe of frequency ωpr , which is resonant to one of the intraband transitions of electrons or holes. This frequency is typically in the infrared 35 or terahertz 36 range and depends on



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Figure 1: Schemes of (a) cross, (b) enclosed, (c) adjacent, and (d) cascade transitions between states of electron–hole pairs excited by pump field of frequency ωpu and probe field of frequency ωpr ; |0i, |ii, |ji, and |f i are the vacuum and excited states of electron–hole pairs; the excited states are assumed to be ki -, kj -, and kf -fold degenerate. Wavy arrows indicate intraband relaxation processes due to interaction with a bath. the material and size of the nanocrystal. The described pump–probe scanning technique can be realized using four different experimental schemes 37 presented in Fig. 1. In the cross-transition scheme shown in Fig. 1(a), the electron–hole-pair state excited by the pump lies between states |ii and |f i involved in the intraband transition induced by the probe. In the enclosed-transition scheme in Fig. 1(b), the energies of states |ii and |f i are both smaller than the energy of state |ji excited by the pump. In the adjacent- and cascade-transition schemes shown in Figs. 1(c) and 1(d), the pump-excited state coincides with either final or initial state of the intraband transition, respectively. It is of significance that state |ii in schemes (a)–(c) is populated via the relaxation from the higher-energy states, whereas in scheme (d) it is directly excited by the pump. The cascade-transition scheme is therefore not practical in the stationary excitation regime, as the single-photon absorption of the probe is masked by the two-photon absorption of the pump and probe fields. 38–40


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The excited states of the electron–hole pairs are generally degenerate in energy. Let ki states |in i (n = 1, 2, . . . , ki ) of energy h ¯ ωi be excited by a probe into kf states |fm i (m = 1, 2, . . . , kf ) of energy h ¯ ωf . Using the density matrix formalism, 41 it is easy to show that the rate of intraband absorption of this field by a single nanocrystal is given by the expression (α) Wf,i

2 (α) (α) ρ X 2γfm ,in Vf(pr) in ,in − ρfm ,fm m ,in = , 2 2 + γ2 h ¯ ∆ pr fm ,in m,n

(1) (pr)

in which α = {a, b, c} denotes one of schemes in Figs. 1(a)–1(c), ∆pr = ωf −ωi −ωpr , Vfm ,in = −eEpr hfm |r · epr |in i is the intraband matrix element describing (in the dipole approximation) the interaction of the electron–hole pairs with the probe, 18 −e is the charge of a free electron, Epr and epr are the amplitude and polarization vector of the probe’s electric field; γfm ,in = (γin ,in + γfm ,fm )/2 + γ bfm ,in and γ bfm ,in are, respectively, the full and pure dephasing rates of transition |in i → |fm i; and γin ,in and γfm ,fm are the inverse lifetimes of states |in i and |fm i, (α)

(α)

which are characterized by populations ρin ,in and ρfm ,fm . (α)

(α)

In order to find ρin ,in and ρfm ,fm entering Eq. (1), one should solve the systems of ki (α)

and kf stationary master equations with source term gpν (pν = in or fm ) allowing for the population buildup of state |pν i due to the direct (dir) generation of electron–hole pairs by the pump field and/or its indirect (ind) population owing to the decay of the higher-energy states (we assume the temperature of the system to be low enough to neglect spontaneous processes leading to the increase in the energy of electron–hole pairs). For the cross- and enclosed-transition schemes, the populations of states |in i and |fm i depend on the populations of kj degenerate states |jr i (r = 1, 2, . . . , kj ) of energy h ¯ ωj . The master equations for the steady-state populations of states |in i, |fm i, and |jr i are:

γpν ,pν ρ(α) pν ,pν

−

kp X

(α) ξpν ,pµ ρ(α) pµ ,pµ = gpν ,

ν = 1, 2, . . . , kp

(p = i, j, f ),

(2)

µ=1, µ6=ν

where ξpν ,pµ is the dephasing rate of transition |pµ i → |pν i between a pair of degenerate



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states. Electron–hole pairs are generated directly in states |jr i for the cross- and enclosedtransition schemes, and in states |fm i for the adjacent-transition scheme. The linear approximation in pump intensity in these three situations yields the source term of the form

gp(α) ν

= ap ν ≡

(pu) 2 2γpν ,0 Vpν ,0 h ¯ 2 (∆2pu + γp2ν ,0 )

.

(3)

(pu)

Here Vpν ,0 = −eEpu hpν |r · epu |0i is the interband matrix element describing interaction of the nanocrystal with the pump’s electric field of strength Epu and polarization epu , and ∆pu = ωp − ωpu is the frequency detuning of pump from state |pν i. The indirect generation of electron–hole pairs in state |pν i occurs due to the transitions from all higher-energy states |sµ i and may be characterized by a set of phenomenological relaxation constants ξpν ,sµ , i.e.

gp(α) = b(α) pν ≡ ν

ks X X

ξpν ,sµ ρ(α) sµ ,sµ ,

(4)

s, ωs >ωp µ=1

where ks is the degeneracy of states |sµ i (µ = 1, 2, . . . , ks ) of energy h ¯ ωs . The system of linear algebraic equations (2) is formally solved using the Cramer’s rule. 42 (α,dir)

The solution can be written in the form Rpν (α,ind)

the form Rpν

(α,ind)

= Gpν

(α)

/Gp

(α)

/Gp

for direct excitation and in (α)

for indirect excitation, where Gp

(α,dir)

system (2) and determinants Gpν

(α,dir)

= Gpν

(α,ind)

and Gpν

is the determinant of (α)

are obtained from Gp by replacing its νth

(α)

column with the vectors of elements apν and bpν , respectively. It is seen from Eqs. (2) and (3) that the populations of states |in i and |fm i intricately depend on each other and the populations of states |jr i due to the indirect-generation mechanism. These populations are readily calculated in the first order of perturbation theory with respect to the probe’s intensity. The result reads: (a)

(a,ind)

ρin ,in = Rin

(a)

and ρfm ,fm ≈ 0, 6

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(5)

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(a,ind)

where Gin

is defined using elements

(a) bi n

kj X

=

(a,dir)

ξin ,jr Rjr

(6)

r=1

(a,dir)

and Gjr

is defined using ajr ; (b)

(b,ind)

(b)

ρin ,in = Rin (b,ind)

where Gin

(b,ind)

and Gfm

(b,ind)

and ρfm ,fm = Rfm

,

(7)

(b,ind)

(8)

are defined using

(b) bi n

=

kj X

(b,dir) ξin ,jr Rjr

r=1

+

kf X

ξin ,fm Rfm

,

m=1 kj (b)

bfm =

X

(b,dir)

ξfm ,jr Rjr

;

(9)

r=1

and (c)

(c,ind)

ρin ,in = Rin with (c) b in

=

(c,dir)

(c)

and ρfm ,fm = Rfm

kf X

(c,dir)

ξin ,fm Rfm

.

(10)

(11)

m=1

Equations (1), (3), and (5)–(11) constitute one of the main results of our work. They describe intraband absorption by an individual nanocrystal for the three pump–probe spectroscopic schemes shown in Figs. 1(a)–1(c), and are applicable to nanocrystals of arbitrary shape and dimensions. Although only those electron–hole pair states that are resonantly coupled through the pump and probe fields are shown on these schemes, the many other states that may lie between them are implicitly taken into account by the relaxation constants in Eq. (4). As it might have been expected, the expressions obtained predict stimulated emission at frequency ωf − ωi ≈ ωpr for the enclosed- and adjacent-transition schemes in the event that 7 ACS Paragon Plus Environment


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the pump field induces population inversion for some of degenerate states |in i and |fm i, i.e. (α)

(α)

if ρfm ,fm > ρin ,in .

Absorption by nanocrystals of different shapes In order to be able to use the results of the previous section in the polarization analysis of intraband absorption by a semiconductor nanocrystal, one needs to know the interband and intraband transition frequencies in the nanocrystals of different shapes and dimensions, and the corresponding matrix elements of the electron–photon interaction. The calculation of the optical matrix elements requires knowing the band structure of the nanocrystals, which can be found using k·p perturbation theory, 43–45 tight-binding model, 46–48 or pseudopotential method. 49–51 In what follows we use the simplest, two-band (one conduction band, e, and one valence band, h) version 52 of k·p perturbation theory and neglect both the spin effects and Coulomb interaction between electrons and holes inside the nanocrystals (implying they are in the regime of strong spatial confinement 44 ). We also assume the nanocrystals to be embedded in a dielectric host and use the infinite-potential-well model for the confined carriers, which is justified for colloidal quantum dots in organic and aqueous solutions, or glass. These simplifications lead to simple analytical expressions for the energy spectra, envelope wave functions, and matrix elements in cuboid, cylindrical, and spherical semiconductor nanocrystals.

Cuboid nanocrystals There are many types of semiconductor nanocrystals—including cubic quantum dots, 53–55 rectangular-cross-section nanorods, 56–58 square nanoplatelets and nanosheets 59–61 —whose shape can be approximated by a rectangular parallelepiped. If the lengths of the adjacent perpendicular edges of such parallelepiped are Lx , Ly , and Lz , then the energies and


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envelope wave functions of electron (hole) states inside it are given by 52

h ¯ ωne(h) x ,ny ,nz

π2h ¯2 Eg ± =± 2 2me(h)

n2y n2x n2z + + L2x L2y L2z

(12a)

and Ψe(h) nx ,ny ,nz (r) = Ψnx (x)Ψny (y)Ψnz (z),

(12b)

where the plus (minus) sign applies to electrons (holes) of effective mass me(h) , Eg is the band gap of the bulk semiconductor, nx , ny , and nz are positive integers, and s Ψnβ (β) =

2 × Lβ

(

sin (πnβ β/Lβ ) for even nβ ,

(12c)

cos (πnβ β/Lβ ) for odd nβ .

It is seen that the quantum states of a cuboid nanocrystal are nondegenerate if Lx 6= Ly 6= Lz and the length of the adjacent perpendicular edges are not integer multiples of each other. Depending on the ratios between the parallelepiped edges, Eq. (12) can describe the spectra and wave functions of quantum dots (all edges are of similar length, Lx ∼ Ly ∼ Lz ), nanorods (one edge is much longer than the other two, Lz Lx ∼ Ly ), or nanoplatelets (one edge is much shorter than the other two, Lz Lx ∼ Ly ). In the case of two parallelepiped edges being alike, Lx = Ly ≡ L 6= Lz , the electron (hole) states h ¯ ωne(h) x ,ny ,nz

π2h ¯2 Eg =± ± 2 2me(h)

n2x + n2y n2z + L2 L2z

(13)

are nondegenerate for nx = ny and two-fold degenerate for nx 6= ny . If all three edges are of the same length, Lx = Ly = Lz ≡ L, then the states h ¯ ωne(h) x ,ny ,nz

Eg π2h ¯ 2 n2x + n2y + n2z =± ± 2 2me(h) L2

(14)

are nondegenerate for nx = ny = nz , two-fold degenerate for nx 6= ny = nz , nx = ny 6= nz or



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nx = nz 6= ny , and six-fold degenerate for nx 6= ny 6= nz . Using Eq. (12) and assuming that the initial and final states of charge carriers belong to the same band, yields the following explicit form of the intraband matrix elements in Eq. (1): 18

(pr) (y) (x) ynyf ,nyi δnxf ,nxi δnzf ,nzi xnxf ,nxi δnyf ,nyi δnzf ,nzi + Epr Vf,i = ∓ e Epr (z) + Epr znzf ,nzi δnyf ,nyi δnxf ,nxi , (15a) (β)

where the minus (plus) sign corresponds to electrons (holes), Epr is the βth Cartesian component of the probe’s electric field, and

βnβf ,nβi =

8nβf nβi Lβ π2

n2βf

−

2 n2βi

inβf +nβi +1

(15b)

for odd nβf + nβi and zero otherwise. Hence, the intraband transitions are possible upon preservation of two out of three quantum numbers nx , ny and nz , and the change of one of them by an odd number. In contrast to the intraband transitions, whose matrix elements critically depend on the orientation of the probe’s electric field with respect to the nanocrystal’s edges, the efficiency of the interband transitions is determined by the orientation of the pump’s electric field with respect to the crystallographic axes of the semiconductor material constituting the nanocrystal. If the semiconductor has Td or Oh symmetry and is characterized by Kane’s parameter P , 62 then the interband matrix elements entering Eq. (3) are of the form (pu) eEpu P Vp,0 = δnxf ,nxi δnyf ,nyi δnzf ,nzi . Eg


(16)

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Cylindrical nanocrystals The energies and wave functions of carriers confined in a circular-cylinder nanocrystal of radius R, height H, and axis z are characterized by a set of quantum numbers {p, l, nz } (p, nz = 1, 2, 3, . . . and l = 0, ±1 ± 2, . . .) and are given by 52 e(h) h ¯ ωp,l,nz

h ¯2 Eg ± =± 2 2me(h)

2 ζp,l π 2 n2z + R2 H2

(17a)

and e(h)

Ψp,l,nz (r) = Ψp,l (ρ, ϕ)Ψnz (z),

(17b)

where ρ, ϕ, and z are the cylindrical coordinates, Ψnz (z) is given by Eq. (12c) with Lz = H, 1 Jl (ζp,l ρ/R) ilϕ Ψp,l (ρ, ϕ) = √ e πR2 Jl+1 (ζp,l )

(17c)

and ζp,l is the pth zero of Bessel function Jl (x). The axial symmetry of quantum states with l 6= 0 is seen to result in the double degeneracy of the corresponding energy levels. The intraband matrix elements in cylindrical nanocrystals can be written as 18 ! (pr)

Vf,i = ∓ e δnzf ,nzi

X

(−µ)

(0) (µ) znzf ,nzi δpf ,pi δlf ,li Epr xpf ,lf ;pi ,li + Epr

,

(18a)

µ=±1

where the minus (plus) sign applies to electrons (holes), (µ)

xpf ,lf ;pi ,li

√ 2 2 ζpf ,lf ζpi ,li R = 2 δlf ,li +µ , ζp2f ,lf − ζp2i ,li

(18b)

(µ) (z) (x) (y) √ and Epr = Epr δµ,0 − µ(Epr + iµEpr )/ 2. According to these expressions, for example,

transitions {p, l, nzi } → {p, l, nzf } are allowed only for nzi + nzf = 2k + 1 with k being an integer. If a cylindrical nanocrystal is made of semiconductor with either Td or Oh symmetry,



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then the matrix elements of the interband transitions inside it are given by Eq. (16) with Kronecker deltas responsible for the preservation of quantum numbers p, l and nz . Similar to a cuboid, the circular cylinder can be used to model quantum dots if R ∼ H, nanorods if R H, or nonodisks if R H.

Spherical nanocrystals In a spherical nanocrystal of radius R, the energy states and wave functions of electrons and holes are: 52 e(h) h ¯ ωn,l,m

2 h ¯ 2 ςn,l Eg =± ± 2 2me(h) R2

(19a)

and e(h) Ψn,l,m (r)

r =

2 jl (ςn,l r/R) Ylm (ϑ, ϕ), R3 jl+1 (ςn,l )

(19b)

where n = 1, 2, 3, . . ., l = 0, 1, 2, . . ., and m = −l, −l+1, . . . , l−1, l are the principle quantum number, angular momentum, and its projection, respectively; r, ϑ and ϕ are the spherical coordinates; jl (u) and Ylm (ϑ, ϕ) are the spherical Bessel function and spherical harmonics; and ςn,l is the nth zero of function jl (u). It is seen that the states with nonzero angular momentum are (2l + 1)-fold degenerate due to the spherical symmetry of the nanocrystal. In this case, the intraband matrix elements can be written as 18 (pr)

Vf,i = ∓ e

X

(−µ)

(µ) (−1)µ Epr xnf ,lf ,mf ;ni ,li ,mi ,

(20a)

µ=±1,0

where 4ςnf ,lf ςni ,li R 2 χµ δmf ,mi +µ , ςn2f ,lf − ςn2i ,li s s (li ± mi + 1)(li ± mi + 2) (li ∓ mi − 1)(li ∓ mi ) δlf ,li +1 − δlf ,li −1 , = 2(2li + 1)(2li + 3) 2(2li − 1)(2li + 1) (µ)

xnf ,lf ,mf ;ni ,li ,mi =

χ±1


(20b)

(20c)

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and

s χ0 =

(li + 1)2 − m2i δl ,l +1 + (2li + 1)(2li + 3) f i

s

li2 − m2i δl ,l −1 . (2li − 1)(2li + 1) f i

(20d)

As before, the dipole approximation allows generation of electron–hole pairs only if the states of electrons and holes are characterized by the same sets of quantum numbers n, l and m. The interband matrix elements of the dipole-allowed transitions are all alike and equal eEpu P/Eg in semiconductors of Td or Oh symmetry.

Intraband absorption spectroscopy The concept Unlike the interband matrix elements [in Eqs. (3) and (16)], which depend on the orientation of the pump’s polarization with respect to the crystallographic axes of the nanocrystal, the intraband matrix elements entering Eq. (1) are the functions of the probe’s polarization solely due to the shape-induced anisotropy of the nanocrystal [see Eqs. (15), (18), and (20)]. 18 The resulting anisotropy of the intraband absorption enables one to determine the geometric parameters and spatial orientation of the nanocrystal from its three-dimensional angular absorption spectrum, which can be measured by varying the polarization of the probe. Since the angular absorption spectrum also drastically depends on the quantum numbers of the electronic states involved in the transitions, this type of intraband absorption spectroscopy is capable of providing valuable information on the electronic subsystem of the nanocrystal. Equation (1) describes absorption by a nanocrystal of any shape and for the most general case of intraband transitions between electron–hole pair states of arbitrary degeneracies. It takes a fairly simple form for the typical situation where the initial and final states involved (α)

(α)

(α)

(α)

in the transition are populated with equal probabilities (i.e. ρfm ,fm ≡ ρf,f and ρin ,in ≡ ρi,i ) and the total dephasing rates of all transitions are alike. In this instance, the absorption rate of the probe field (polarized in the direction set by angles ϑ and ϕ) is proportional to



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the sum of squared matrix elements of the intraband transitions between different pairs of degenerate states, (α)

Wf,i (ϑ, ϕ) ∝

X (pr) 2 V fm ,in (ϑ, ϕ) .

(21)

m,n

Note that the angular spectrum obtained does not depend on the type of transition scheme that is realized in the experiment. The analysis of the experimental absorption spectra of real nanocrystals is possible if (α)

one knows the angular absorption patterns, Wf,i (ϑ, ϕ), for the most common shapes of the nanocrystals. We calculate such patterns for the nanocrystals described in the previous section, while assuming that the intraband transitions occur in their conduction band (the same patterns are obtained if the intraband transitions occur in the nanocrystal’s simple valence band).

Basic angular absorption patterns Using in Eq. (21) the analytical expressions for the matrix elements given in Eqs. (15) and (20), it may be readily shown that the intraband absorption by a spherical or cubic nanocrystal is independent of the probe’s polarization regardless of the quantum numbers characterizing the states involved in the intraband transition. The angular absorption pattern for both nanocrystal shapes is a sphere, as is schematically shown in Fig. 2(a). The energy-level diagrams in Fig. 2(b) show four possible types of the dipole-allowed intraband transitions between electronic states of different degeneracies (solid and dashed energy levels correspond to the nondegenerate and degenerate states, respectively). Using the selection rules for the intraband transitions in a circular-cylinder nanocrystal following (α)

from Eq. (18), we find that Wf,i ∝ cos2 ϑ for transitions between the nondegenerate states with quantum numbers li = lf = 0, pi = pf , and nzi 6= nzf . Hence, in this case, the absorption pattern is a symmetric dumbbell along the z axis. Analogously, for the transitions between the nondegenerate and degenerate states with


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(a)

Circular cylinder

(b)

Sphere

L

L L

Cube

(d)

(c)

L

Ly

z

L Lz

Lx Lz y

x

Square parallelepiped

Rectangular parallelepiped

Figure 2: Energy-level diagrams and angular absorption patterns for dipole-allowed intraband transitions between electronic states of semiconductor nanocrystals in the forms of (a) sphere or cube, (b) circular cylinder, (c) square parallelepiped, and (d) rectangular parallelepiped. Solid and dashed energy levels represent nondegenerate and degenerate states, respectively. In all cases, the selection rules for the transition between states with nβi 6= nβf (β = x, y, z) require nβi + nβf to be an odd number. See text for details and notations. (α)

nzi = nzf and either li = 0 and lf = ±1 or li = ±1 and lf = 0, we obtain Wf,i ∝ sin2 ϑ. If the initial state of the electron is degenerate, then pi and pf must satisfy the inequality ζpf ,1 > ζpi ,0 . Otherwise, ζpf ,0 > ζpi ,1 and pf > pi . The corresponding absorption patterns are horn toruses with the z axis. 15 ACS Paragon Plus Environment


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The lowest diagram in Fig. 2(b) corresponds to the transitions between the doubly degenerate states of the nanocrystal. These transitions occur between electronic states with either different or equal quantum numbers nzi and nzf . In the first case, li = lf , pi = pf , and the dumbbell-shape absorption pattern is observed; in the second case, lf = li ± 1, ζpf ,li ±1 > ζpi ,li , and the absorption pattern is the horn torus. These two angular patterns allow one to distinguish between different types of electronic states involved in the intraband optical transitions, and make it possible to find the spatial orientation of the nanocrystal, which is determined by the rotational symmetry axis of the absorption pattern. The analysis of the electronic states based on the absorption spectroscopy data is facilitated by taking into account the energy-level structure of the nanocrystal given by Eq. (17a). In the case of a square parallelepiped L × L × Lz , the energy states with quantum numbers nx 6= ny are doubly degenerate, whereas the other states are nondegenerate [see Eq. (13)]. Owing to the fourth-order axial symmetry of this parallelepiped, its absorp(α)

tion patterns are similar to those of a circular cylinder. Indeed, Wf,i ∝ cos2 ϑ for transitions between either a pair of nondegenerate states with nxi = nxf = nyi = nyf and nzi 6= nzf or a pair of degenerate states with nxi = nxf 6= nyi = nyf and nzi 6= nzf . On the other hand, the horn torus absorption pattern occurs for transitions with nzi = nzf in the following three cases: (i) nxi = nyi and either nxi 6= nxf or nyi 6= nyf ; (ii) nxf = nyf and either nxi 6= nxf or nyi 6= nyf ; and (iii) nxi 6= nyi , nxf 6= nyf , and either nxi 6= nxf or nyi 6= nyf . These results are summarized in Fig. 2(c). In the general case of Lx 6= Ly 6= Lz , all electronic states of the nanocrystal are generally nondegenerate [see Eq. (12a)] and the transitions between them lead to the dumbbell-shape absorption patterns. The orientation of the dumbbell corresponding to a particular transition coincides with the axis (β) of the nonequal quantum numbers (nβf and nβi ), as shown in Fig. 2(d).


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Figure 3: (a) Cylindrical nanocrystal probed by a linearly polarized beam propagating in the Z direction and (b) energy-level diagram showing intraband transitions {1, 0, 1} → {1, 0, 4} and {1, 0, 1} → {2, ±1, 1} between electronic states of the nanocrystal. Angular absorption spectra are obtained by rotating beam’s polarization epr in the XY plane.

Spectroscopy in use To illustrate how the proposed new type of optical spectroscopy can be used to study the properties of isolated nanocrystals, we consider the following thought experiment. Let a semiconductor nanocrystal with a symmetry axis set by a unit vector n be simultaneously excited by a pump beam of fixed linear polarization and a probe beam, whose linear polarization vector epr makes an angle η with the Y axis. For simplicity, we first assume that n lies in the Y Z plane and makes an angle κ with the Z axis, as shown in Fig. 3(a). By choosing the frequency of the probe beam close to one or, sequentially, several of the nanocrystal’s intraband transitions, one can measure the corresponding angular absorption spectra via rotating the probe’s polarization in the XY plane. As we have seen earlier, the intraband absorption by a randomly oriented cubic nanocrystal is independent of η, which means that its angular absorption spectra cannot be used to determine the spatial orientation of the nanocrystal or study its electronic subsystem without allowing for the complex band structure of the nanocrystals’ material. The situation is different for a nanocrystal in the form of a circular cylinder, whose absorption patterns are symmetric dumbbells or horn toruses depending on the quantum states involved in the transitions. By way of example we consider a cylindrical nanocrystal with R = 5 nm and H = 10 nm, and assume it to be made of InAs characterized by Eg = 354 meV and me = 0.023 m0 , with m0 being the mass of free electron. 63 We focus on the intraband



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transitions between ground state |ii = {1, 0, 1} and excited states |f1 i = {1, 0, 4} and |f2(3) i = {2, ±1, 1} of electrons [see Eq. (17a) and Fig. 3(b)]. By assuming the full dephasing rates to be the same for all transitions, γf1(2,3) ,i ≡ γ, and using Eq. (1), we can write the following proportionality for the intraband-absorption cross section of the nanocrystal: (pr) 2 (pr) 2 2ωpr Vf2 ,i [ϑ(κ, η), ϕ(κ, η)] ωpr Vf1 ,i [ϑ(κ, η), ϕ(κ, η)] + , D(ωpr , κ, η) ∝ 2 2 ωf1 − ωi − ωpr + γ 2 ωf2 − ωi − ωpr + γ 2

(22)

where ωf1 = 3.42 eV, ωf2 = 3.85 eV, ωi = 0.97 eV, and factor 2 in the second term is due to the degeneracy of states {2, ±1, 1}. The intraband matrix elements entering this expression are given in the reference frame of the nanocrystal by Eq. (18), and are essentially the functions of angles κ and η. In Fig. 4 we plot spectra D(ωpr , κ, η) for γ = 100 meV and four different orientations of the cylindrical nanocrystal. To the right of the spectra, we show cross sections of the dumbbell and horn torus absorption patterns corresponding to transitions {1, 0, 1} → {1, 0, 4} and {1, 0, 1} → {2, ±1, 1}. These cross sections represent the angular absorption spectra that are measured in real experiment. Only the transitions between the states with quantum numbers nzf = nzi = 1 are seen to be manifested in the absorption spectrum for κ = 0 in Fig. 4(a). Since the probe beam propagates in this case along the nanocrystal’s axis, its absorption is independent of polarization. As κ increases [see Figs. 4(b) and 4(c)], the absorption peak due to transition |ii → |f1 i starts to appear in the nanocrystal’s spectrum; the corresponding angular absorption spectrum is the figure of eight elongated along the X axis. At the same time, the angular spectrum for the high-energy transition gets compressed along directions η = ±π/2 and starts to resemble a figure-of-eight without self-crossing. Finally, if the axis of the cylinder is along the Y direction (κ = π/2), then the low- and highenergy transitions are characterized by angular dependencies sin2 η and cos2 η, respectively [see Fig. 4(d)]. By measuring the angular absorption spectrum for different intraband transitions of


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8

6

2

4

0

η

2π

(a)

π

η

0 2π

(b)

π

η

0 2π

(c)

π

0 2π

η

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(d)

π

0

2.2 2.4 2.6 2.8 ωpr, eV

3

3.2

Figure 4: Intraband absorption spectrum in the (ωpr , η) domain for four orientations of cylindrical nanocrystal with respect to the propagation direction of probe: (a) κ = 0, (b) κ = π/4, (c) κ = π/3, and (d) κ = π/2 (see Fig. 3). Cross sections of three-dimensional absorption patterns on the right show angular absorption spectra for intraband transitions |ii → |f1 i (green pattern) and |ii → |f2 i (grey pattern). See text for details.



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a cylindrical nanocrystal using three propagation directions of the probe beam, one can fully restore the dumbbell and horn torus absorption patterns, determine the nanocrystal’s orientation, and receive evidence that its shape is a circular cylinder. The latter requires detailed examination of the nanocrystal’s energy spectra using Eqs. (13) and (17a), since the circular cylinder has absorption patterns identical to those of a square parallelepiped [see Fig. 2(b) and 2(c)]. It is even easier to experimentally distinguish between a cylindrical nanocrystal and the nanocrystal in the form of a rectangular parallelepiped. To demonstrate this, we consider intraband transitions {1, 1, 1} → {1, 4, 1}, {1, 1, 1} → {2, 1, 1}, and {1, 1, 1} → {1, 1, 2} between the nondegenerate electronic states of a 5 × 15 × 4-nm3 cuboid nanocrystal made of InAs. The intraband-absorption cross section of this nanocrystal is described by the proportionality (pr) 2 (pr) 2 ωpr Vf1 ,i (n, η) ωpr Vf2 ,i (n, η) + D(ωpr , n, η) ∝ 2 2 ωf1 − ωi − ωpr + γ 2 ωf2 − ωi − ωpr + γ 2 (pr) 2 ωpr Vf3 ,i (n, η) + , (23) 2 ωf3 − ωi − ωpr + γ 2 where ωf1 = 3.31 eV, ωf2 = 4.18 eV, ωf3 = 5.28 eV, ωi = 2.21 eV, and the matrix elements are given by Eq. (15). Figure 5(a) illustrates how the cross section given in Eq. (23) depends on the probe’s frequency and polarization. It was assumed that γ = 100 meV and the nanocrystal’s ori√ entation was set by vector n = ( 3/4, 1/2, 3/4), which is passing through the center of the parallelepiped and aligned with its edges Lz ; bringing n in coincidence with the Z axis makes edges Lx , Ly , and Lz parallel to the axes X, Y , and Z, respectively [see Fig. 5(b)]. According to Fig. 2(d), the absorption patterns of a rectangular parallelepiped are symmetric dumbbells for all intraband transitions. The cross sections of the dumbbells corresponding to the transitions shown by red arrows in Fig. 5(c) are plotted in Fig. 5(d). Their figure-of-eight shapes are indicative of the nanocrystal being a cuboid. 20 ACS Paragon Plus Environment

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30

25

20

15

10

5

0

LxZ

2π

Lz

{1,1,2}

Ly

{2,1,1} {1,4,1}

η

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Y

π

X

(a) 0

1

ωpr

1.5

2 2.5 ωpr, eV

3

{1,2,1} {1,1,1}

ωpu

(b)

(c)

(d)

Figure 5: (a) Intraband absorption spectrum in the (ωpr , η) domain for cuboid nanocrystal oriented as shown in (b); (c) and (d) show three probe-induced intraband transitions (red arrows) between electronic states of the nanocrystal and their angular absorption patterns. Left, middle, and right patterns correspond to final states {1, 4, 1}, {2, 1, 1}, and {1, 1, 2}, respectively. See text for details. It should be noted that the intraband absorption spectrum of a QD is a mixture of the absorption peaks of electrons and holes. These peaks strongly overlap at room temperature for QDs made of lead-chalcogenide-type semiconductors (such as PbS and PbSe), with close values of the effective masses of electrons and holes. 64 The overlapping of the absorption peaks can lead to the intermixture of the angular absorption patterns due to the optical transitions in the conduction and valence bands, which could significantly hamper the use of the proposed spectroscopic method. To prevent the smearing out of the absorption patterns, the experiments with QDs have to be conducted at sufficiently low (liquid nitrogen or liquid helium) temperatures—such that the homogeneous widths of the absorption peaks are in a few millielectronvolt range and the peaks are well resolved in frequency. 21 ACS Paragon Plus Environment


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As a concluding remark, we would like to note that the advancement of the proposed polarization spectroscopy requires considering complex models of the nanocrystal’s band structure and confinement potential, as well as abandoning the assumption of equal populations of the degenerate electronic states. Furthermore, to make possible characterization of the nanocrystals whose size is exceeding the exciton Bohr radius, one needs to treat the intraband absorption using the approximation of weak confinement. It is of significance that these changes can be done without modifying the developed theory of intraband absorption based on the density matrix formalism. The theoretical treatment of the intraband absorption spectroscopy of a semiconductor nanocrystal will be advanced in our further publications.

Conclusions We have developed a uniform theoretical basis for the stationary intraband absorption spectroscopy of a semiconductor nanocrystal, which enables a reliable determination of the nanocrystal’s shape and spatial orientation and provides significant information on the types of quantum states involved in the absorption process. In particular, using the density matrix formalism we calculated absorption by the nanocrystal that is exposed to the linearly polarized pump and probe fields. In order to cover all possible types of transitions that can be induced by the pump and probe fields, we considered three relative arrangements of the excited electron–hole pair states and calculated the angular absorption patterns for five common shapes of the nanocrystals. The significance of our work is in using a somewhat involved mathematics of the intraband absorption to derive simple guidelines for analysis of experimental data.

Acknowledgments The authors gratefully acknowledge the financial support from the Ministry of Education and Science of the Russian Federation through its Grant No. 14.B25.31.0002. The Ministry of 22 ACS Paragon Plus Environment

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Education and Science of the Russian Federation also supports A.S.B. and M.Y.L., through its scholarship of the President of the Russian Federation for young scientists and graduate students (2013–2015).

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82x44mm (300 x 300 DPI)

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Harnessing the Shape-Induced Optical Anisotropy of a Semiconductor

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