Article pubs.acs.org/JPCC
Electronic States of Lead-Salt Nanosheets Jun Yang* and F. W. Wise School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, United States S Supporting Information *
ABSTRACT: The electronic states of lead-salt nanosheets are calculated with an effective mass model. In recognition of the anisotropic shape of the structure, the electron wave functions are separated into two parts. Motion along the narrow dimension is dominated by the quantum confinement, while the lateral motion is dominated by the Coulomb interaction. Strong contrast of the dielectric constant of the host environment further strengthens the Coulomb interaction. The calculated exciton binding energy for lead-salt nanosheets is smaller than that of nanowires of similar size but still larger than the thermal energy at room temperature, so it should be possible to observe excitons at room temperature. The calculation results are compared with existing data, and show good quantitative agreement.
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INTRODUCTION Anisotropically shaped semiconductor nanostructures have the potential to achieve very efficient charge transport while maintaining quantum confinement, due to the possibility of separating the dimensions for transport and confinement. Indeed, early measurements of single colloidally grown nanowires1,2 and nanosheets3 showed higher mobility than assemblies of the corresponding nanocrystals (NCs). In addition, the lowest exciton transitions in nanowires and nanosheets can have giant oscillator strengths connected with coherent center-of-mass motion.4,5 Combined with the high density of states near the energy gap and slow Auger recombination, nanosheets and nanowires have consistently exhibited improved stimulated emission6,7 and multiple exciton generation,8−10 thus showing promise for various optoelectronic device applications. Additional motivation to study nanosheets comes from the fact that they can be considered a limiting case of atomically coherent nanocrystal superlattices,11,12 which are sometimes referred to as having dimension slightly less than two.13 The rock salt crystal structure of the lead salts makes it difficult to colloidally synthesize an anisotropically shaped nanostructure. Schliehe et al. first synthesized thin lead sulfide nanosheets using two-dimensional oriented attachment.14 Later, Acharya et al. reported a similar method, via collective coalescence of nanowires.15 Very limited optical characterization of the structures was reported. Only recently, a few groups have reproduced and improved on Schliehe’s synthesis method,16,17 and reported absorption and fluorescence spectra of sheets with varying size.10,18−20 A detailed understanding of the electron states is a prerequisite to understanding the electron dynamics. The electron states of lead-salt NCs have been calculated previously by several methods, including envelope function theory,21 atomistic tight binding,22,23 atomistic pseudopotential,24 and ab initio density functional theory (DFT).25,26 Compared to the other methods, envelope function theory is the most intuitive, © 2015 American Chemical Society
and most efficient to implement. With proper envelope Hamiltonian and boundary conditions, envelope function theory produces quite accurate results in calculating the ground and excited states of NCs of different compositions,21,27,28 sizes, and shapes.5,29 It also provides the correct symmetry of the wave functions. Combined with the intuitive physical picture it provides, envelope function theory is frequently used to analyze more complex carrier dynamics, such as electron−phonon coupling,30 Auger recombination,31 multiple exciton generation,32 etc. Thanks to their highly anisotropic shape, the electronic structure of lead-salt nanosheets will be quite different from that of nanospheres or nanocubes. With strong quantum confinement in one direction (which we will refer to as the vertical or z direction) but no quantum confinement in the lateral directions, energy bands form. Allan and Delerue reported tight-binding calculations of the electronic structure of PbSe nanosheets.23 Their results show that the surfaces do not lead to states in the energy gap, and predict that the energy gap varies inversely with the sheet thickness. However, they only calculated the single electron states, and did not include the Coulomb interaction of the electron and hole. The Coulomb interaction between charges in thin nanosheets will be mediated by the host environment, which generally has a much smaller dielectric constant than the semiconductor and results in reduced screening of the Coulomb interaction compared to bulk materials. This strengthening of the Coulomb interaction should yield tightly bound excitons,29,33−35 and is an integral part of the electronic states of lead-salt nanosheets. Here we present an envelope-function calculation of the electron states in lead-salt nanosheets or nanoplatelets. To properly account for the strong Coulomb interaction, we Received: August 23, 2015 Revised: November 1, 2015 Published: November 6, 2015 26809
DOI: 10.1021/acs.jpcc.5b08207 J. Phys. Chem. C 2015, 119, 26809−26816
Article
The Journal of Physical Chemistry C separate the degrees of freedom into strongly confined motion in the vertical direction and Coulomb-coupled lateral motion. In the vertical direction, quantum confinement dominates, and we solve the Schrödinger equation without the Coulomb interaction, which is subsequently treated as a perturbation. The resulting energy bands are the dispersion relations for electrons in the lateral directions. The Coulomb interaction between electron and hole is calculated using results from the electrostatic potential of a charge in an infinite slab of dielectric. This interaction is averaged over the vertical wave functions to obtain the equivalent two-dimensional (2D) Coulomb interaction. Finally, the motion in 2D is solved numerically to obtain the exciton binding energy. Results of the calculations will be compared with existing optical data.
Table 1. Band Parameters Used in the Four-Band k·p Effective Mass Model for PbS and PbSe
FOUR-BAND EFFECTIVE MASS MODEL The most accurate effective mass model used to describe the band structure of the lead salts near the band extrema at the L points of the Brillouin zone is the k·p model developed by Mitchell and Wallis36 and Dimmock.37 This model includes the coupling between the highest valence band and the conduction bands as well as the coupling between the lowest conduction band and the valence bands in a second-order perturbation approximation. The spin−orbit interaction is also taken into account. The model has successfully explained the energy levels for lead-salt 0D NCs,21 1D quantum rods,29 and heterogeneous core−shell structures.28 In the four-band effective mass model, the wave functions are expanded as
Eg (T = 300 K) (eV) m−l /m0 m−t /m0 m+l /m0 m+t /m0 2Pl2/m0 (eV) 2Pt2/m0 (eV)
0.41 0.27 0.53 0.27 0.37 1.6 3.0
0.28 1.26 0.79 0.94 0.59 1.42 4.25
⎡⎛ E ⎤ 2 2 ⎞ ⎢ ⎜ g + pẑ + p⊥̂ ⎟I Pz p ̂ σ + P⊥ p̂ · σ ⎥ ⊥ z − − ⎜ ⎟ ⎢⎝ 2 ⎥ 2mz 2m⊥ ⎠ m0 z m0 ⊥ ⎢ ⎥ ⎢ 2 2 ⎞ ⎥ ⎛ Eg p̂ p̂ P⊥ ⎢ Pz ⎥ − ⎜⎜ + z + + ⊥ + ⎟⎟I ⎥ ⎢ m pẑ σz + m p⊥̂ · σ⊥ m m 2 2 2 ⊥⎠ ⎦ 0 z ⎝ ⎣ 0 (5)
The modified band parameters are29 P⊥ =
(1)
|L−6 ↑⟩
where and |L−6 ↓⟩ are the conduction band-edge Bloch functions with L−6 symmetry and up and down spin, while |L+6 ↑⟩ and |L+6 ↓⟩ are the valence band-edge Bloch functions with L+6 symmetry.38 The envelope functions -(r) = [Fi(r)] satisfy the Schrödinger equation (2)
where the Hamiltonian is
Pt P (1 + cos2 θ ) + l sin 2 θ 2 2
(6a)
Pz = Pt sin 2 θ + Pl cos2 θ
(6b)
1 1 1 (1 + cos2 θ ) + sin 2 θ = m⊥± 2mt± 2ml±
(6c)
1 1 1 = ± sin 2 θ + ± cos2 θ mz± mt ml
(6d)
where θ = 54.7° is the angle between the ⟨100⟩ and ⟨111⟩ directions. The ⟨111⟩, ⟨111̅ ⟩̅ , ⟨11̅ 1⟩̅ , and ⟨11̅ 1̅ ⟩ directions have the same angle relative to ⟨100⟩, and therefore, the four valleys have the same modified band parameters. Moreover, sin2 θ = 2/ 3, P⊥ = Pz ≜ P, and m±⊥ = m±z ≜ m±, so the Hamiltonian is completely isotropic. We define the vertical direction as the z direction. Given the translational symmetry in the x−y plane, we take as an ansatz
Ĥ 0(p̂ ) = ⎡⎛ E ⎤ 2 2 ⎞ ⎢ ⎜ g + pẑ + p⊥̂ ⎟I Pl p ̂ σ + Pt p̂ · σ ⎥ ⊥ z ⎢ ⎜⎝ 2 ⎥ m0 ⊥ 2ml− 2mt− ⎟⎠ m0 z ⎢ ⎥ ⎢ 2 2 ⎞ ⎥ ⎛ Eg p̂ p̂ Pt ⎢ Pl ⎥ − ⎜⎜ + z + + ⊥ + ⎟⎟I ⎥ ⎢ m pẑ σz + m p⊥̂ · σ⊥ m m 2 2 2 0 0 l t ⎝ ⎠ ⎣ ⎦
∞
(3)
Fj(r ) =
which is a 4 × 4 matrix. Here z denotes the ⟨111⟩ direction of the cubic lattice; Eg is the bulk band gap; Pl and Pt are the longitudinal and transverse Kane momentum-matrix elements between the extremal valence- and conduction-band states; m±l and m±t are the far-band contibutions to the longitudinal and transverse band-edge effective masses; m0 is the free electron mass; I is the 2 × 2 identity matrix; and σ is the Pauli matrix. Parameters used in the calculation are listed in Table 1. The boundary conditions for the nanosheet are assumed to be those of an infinite potential well -(z = 0, d) = 0
PbSe29
Ĥ 0(p̂ ) =
|Ψ(r)⟩ = F1(r)|L6− ↑⟩ + F2(r)|L6− ↓⟩ + F3(r)|L6+ ↑⟩
Ĥ 0(p̂ )-(r) = E -(r)
PbS39
The Hamiltonian has cylindrical symmetry with respect to the ⟨111⟩ lattice direction. However, this does not coincide with the normal to the nanosheet surface, which is in the ⟨100⟩ direction.14 Here we extract the main isotropic part21,29 and treat the anisotropic contribution as a perturbation. The isotropic Hamiltonian is
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+ F4(r)|L6+ ↓⟩
parameters
∑ Aj ,αeik xeik y sin x
α=1
y
απz , d
j = 1, ..., 4 (7)
Substitution of eq 7 into the Schrödinger equation (eq 2) and use of the orthogonality relations π π sin αx sin βx dx = δαβ (8a) 2 0
∫
∫0
π
⎧0 if α + β is even ⎪ sin αx cos βx dx = ⎨ 2α otherwise ⎪ 2 2 ⎩α − β
(4)
(8b)
where d is the thickness of the nanosheet.
yields four groups of coupled linear equations 26810
DOI: 10.1021/acs.jpcc.5b08207 J. Phys. Chem. C 2015, 119, 26809−26816
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The Journal of Physical Chemistry C ⎞ ℏP α·α 4ℏP ⎛ (kx − ik y)A4, α0 i⎜⎜ −∑ ′ 2 0 2 A3, α ⎟⎟ + md ⎝ α α0 − α m ⎠ ⎛ Eg ⎛ α0π ⎞2 ⎞⎞ ℏ2 ⎛ 2 2 ⎜ ⎟ ⎟⎟ A ⎜ = ⎜⎜E − − + + k k x y ⎝ d ⎠ ⎠⎟⎠ 1, α0 2 2m− ⎝ ⎝
energies to a parabolic function En(kx , y) =
(9a)
EFFECTIVE 2D COULOMB INTERACTION The potential of a point charge in an infinite dielectric slab can be found by the method of images.40 For a slab with dielectric constant ϵ2 embedded in a medium with dielectric constant ϵ1, the electric potential within the slab at any point (z, ρ) in cylindrical coordinates due to a point charge q at the origin is U (z , ρ ; a , b) = f (z , ρ) + Uimage(z , ρ ; a , b)
(9c)
∞
= f (z , ρ) +
⎞ ℏP α·α 4ℏP ⎛ i⎜⎜∑ ′ 2 0 2 A 2, α ⎟⎟ + (kx + ik y)A1, α0 md ⎝ α α0 − α m ⎠ ⎛ Eg ⎛ α0π ⎞2 ⎞⎞ ℏ2 ⎛ 2 2 ⎜ ⎟ ⎟⎟ A ⎜ k k = ⎜⎜E + + + + x y ⎝ d ⎠ ⎠⎟⎠ 4, α0 2 2m+ ⎝ ⎝
, assuming that
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(9b)
⎞ ℏP α·α 4ℏP ⎛ (kx − ik y)A 2, α0 i⎜⎜ −∑ ′ 2 0 2 A1, α ⎟⎟ + md ⎝ α α0 − α m ⎠ ⎛ Eg ⎛ α0π ⎞2 ⎞⎞ ℏ2 ⎛ 2 2 ⎜ ⎟ ⎟⎟A ⎜ k k = ⎜⎜E + + + + x y ⎝ d ⎠ ⎠⎟⎠ 3, α0 2 2m+ ⎝ ⎝
2meff, n
the dispersion relation at small kx,y is described by a simple oneband effective mass model. The fits are very good for kx,y ≲ 0.4 nm−1. Above that, the energy deviates from the simple parabolic model, particularly for the e1 and h1 bands. Parts c and d of Figure 1 show the thickness dependence of the energies and effective masses for different bands. With increasing thickness, the energy gaps and effective masses approach their bulk values.
⎞ ℏP α·α 4ℏP ⎛ (kx + ik y)A3, α0 i⎜⎜∑ ′ 2 0 2 A4, α ⎟⎟ + md ⎝ α α0 − α m ⎠ ⎛ Eg ⎛ α0π ⎞2 ⎞⎞ ℏ2 ⎛ 2 2 ⎜ ⎟ ⎟⎟ A ⎜ k k = ⎜⎜E − − + + x y ⎝ d ⎠ ⎠⎟⎠ 2, α0 2 2m− ⎝ ⎝
ℏ2kx , y 2
∑ β 2n+ 1f (z − 2an − 2b(n + 1)) n=0
∞
+
∑ β 2n+ 2f (z − 2(a + b)(n + 1)) n=0 ∞
+
(9d)
∑ β 2n+ 1f (z + 2bn + 2a(n + 1)) n=0 ∞
The sums run over all terms (indexed by α) that have parity opposite that of α0. The electron states are calculated with a finite number of terms, α ≤ N = 16. The resulting fractional error in the energy gap is estimated as 0. Equation 10 can be rewritten 1
2
in a coordinate system with the origin shifted to one dielectric interface and in terms of the sheet thickness d V (z1 , z 2 , ρ) = U (z = z 2 − z1 , ρ ; a = z1 , b = d − z1) (11)
First, we consider the self-interaction of a point charge with its image charges at the interfaces. For a charge q embedded in the medium at position (z0, 0), the electric potential at the same point is V(z0, z0, 0). The energy of the electric field stored 1 in the medium is 2 qV (z 0 , z 0 , 0). Using eq 10 and eq 11, the 1
contribution from the first term in eq 10 is equal to 2 qf (0, 0), which is the energy of a bare charge embedded in an infinite medium. This energy is already accounted for in the work function of the semiconductor. The rest of the terms 1 qV (z , z 0 , 0) account for the energy of the interaction 2 image 0 between the charge and its polarized charges at the dielectric interface, or the interface polarization energy. Figure 2a shows the shape of this potential energy for an electron (or hole) placed at position z. Due to the smaller dielectric constant in the host medium, this potential pushes the electron away from the interface, so it is called the dielectric confinement energy. The first-order correction to the energy of an electron due to the interface polarization energy is
Figure 1. (a) Energies for 12 states close to the energy gap for a 3 nm PbS nanosheet. (b) Energy of electron with different lateral momentum kx,y. The band structure in the x,y plane is isotropic. Lines are parabolic fits, with inferred effective masses indicated. (c) Energy levels and (d) 2D effective masses of PbS nanosheets of different thicknesses. 26811
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The Journal of Physical Chemistry C lim V2D,eff (ρ) = −
ρ→0
1 e 2 −(1 − α) ρ 4πϵ1 ρ0α
with ρ0 being the transition point from one limit to the other. For a fixed medium dielectric constant of 2.0, the best-fit value for α is consistently 0.80, while ρ0 is proportional to the sheet thickness; for fixed thickness at 3 nm, with increasing medium dielectric constant, the interface dielectric effect gets weaker, and α increases slightly while ρ0 decreases (Figure 2c,d).
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2D LATERAL MOTION AND EXCITON FORMATION Last, we solve for the lateral motion of the electron and hole in the effective 2D Coulomb potential. The Schrödinger equation is ⎡ ℏ2 ∂ 2 ⎤ ℏ2 ∂ 2 ⎢− − + V2D,eff (rh − re)⎥Ψ(re, rh) 2 2 ⎢⎣ 2me1 ∂re 2mh1 ∂rh ⎥⎦
Figure 2. (a) Interface polarization energy potential of a 3 nm PbS nanosheet. Parameters: ϵ1 = ϵenv = 2.0, ϵ2 = ϵPbS = 17.0. (b) Electron− hole Coulomb energy, averaged over the vertical wave functions of the lowest energy electron and hole bands for PbS nanosheets of indicated thickness. The solid curves are fits to the model potential 2
1 e 1 α. 1 ρ 1 + (ρ0 / ρ)
V2D,eff (ρ) = − 4πϵ
= E Ψ(re, rh)
The motion can be separated into the center of mass motion and the relative motion Ψ(r, R) = Ψcm(R)ψrel(r), where the center of mass wave function Ψcm(R) is a plane wave. Since the interaction is also independent of the relative direction, the relative motion can be further separated as ψrel(ρ, ϕ) = 1 R(ρ)Φ(ϕ), where Φ(ϕ) = 2π eimϕ , with orbital angular momentum perpendicular to the sheet mℏ. The equation for the radial wave function R(ρ) becomes
The dashed curve corresponds to the
Coulomb interaction with ϵ = ϵenv, without the interfaces. (c) Fitting parameters α (blue) and ρ0 (red) for the effective 2D potential for different thicknesses. α is consistently around 0.8, and ρ0 is proportional to the thickness. (d) Dependence of fitting parameters α and ρ0 for the effective 2D potential on the dielectric constant of the environment.
Vpol =
∫ dz |-e (z)|2 12 qVimage(z , z , 0) 1
(15)
d2 1 d R (ρ ) + R (ρ ) 2 ρ dρ dρ ⎡ ⎛ ⎞ m2 ⎤ 2μ 1 e2 1 ⎟⎟ − 2 ⎥R(ρ) = 0 + ⎢ 2 ⎜⎜Erel + ⎢⎣ ℏ ⎝ 4πϵ1 ρ 1 + (ρ0 /ρ)α ⎠ ρ ⎥⎦
(12)
The polarization energy of the electron and hole are both 51 meV, for a total of 102 meV, in 3 nm PbS nanosheets. The 2D effective Coulomb potential between an electron and a hole in the lowest energy bands is
(16)
V2D,eff (ρ) =
∬
dze |-e1(ze)|2 dzh |-h1(zh)|2 ( −V (zh , ze , ρ))
where μ =
(13)
1 e2 , 1 ρ
ρ →∞
1 e2 1 4πϵ1 ρ 1 + (ρ0 /ρ)α
Figure 3. (a) First few exciton energies for m = 0, 1, 2. (b) Radial wave functions for the four lowest energy states. Material: PbS, 3 nm.
shows the energy levels for the first few bound states. The binding energy for the lowest energy exciton in a 3 nm PbS nanosheet is 83 meV. The radial wave functions for the four lowest energy states are plotted in Figure 3b. Figure 4a summarizes the self-energies (calculated with eq 12), the exciton binding energy, and the total energy with varying sheet thicknesses. Figure 4b summarizes the same quantities but for varying dielectric constant of the environment. The exciton binding energy partially cancels out the selfpolarization energy, and as a result, the total energy shift due to Coulomb interaction is much smaller than either one. Parts c
(14)
which goes to the correct limiting forms lim V2D,eff (ρ) = −
ρ→∞
is the reduced mass, and the analytic form
so it approaches the direct Coulomb
interaction mediated by the dielectric constant of the environment, ϵ1 (dashed curve in Figure 2b). For small ρ, the 2D Coulomb potential approaches another power law with an exponent smaller than 1. We fit the effective 2D Coulomb potential to an analytic form (solid curves in Figure 2b) V2D,eff (ρ) = −
me1 + mh1
of the 2D Coulomb interaction (eq 14) is included. Equation 16 is solved numerically (Supporting Information). Figure 3a
Here we neglect the exchange interaction, which is much smaller than the direct Coulomb interaction.41,42 The negative sign is due to the opposite charges. The 2D Coulomb potentials for 3, 5, and 8 nm PbS nanosheets are plotted in Figure 2b as the dots. Thinner nanosheets produce stronger Coulomb interactions. The 2D effective Coulomb potential has the limit lim V2D,eff (ρ) = − 4πϵ
me1mh1
1 e2 4πϵ1 ρ
and 26812
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Figure 5 shows the transition dipoles calculated for 3 nm PbS nanosheets. The transition band that starts near 0.8 eV is the h1
Figure 5. Interband optical transition dipoles of 3 nm PbS nanosheets. The solid line is the absorption spectrum calculated assuming broadening δ = 25 meV; the dashed line corresponds to δ = 100 meV. Figure 4. (a, b) Interface polarization energy (electron plus hole), exciton binding energy, and total energy and (c, d) average electron and hole distance ⟨ρ⟩ for the lowest exciton state (m = 0, n = 1) for (a, c) different thickness, assuming ϵenv = 2.0 and (b, d) different dielectric constant for the medium, assuming thickness d = 3.0 nm.
→ e1 transition. There are a few discrete excitonic states, but they quickly blend into a continuum. The seemingly discrete lines are due to the finite overall size (200 nm in this case) in the numerical calculation (see the Supporting Information for more details). The second band that starts near 1.75 eV is the h2 → e2 transition. The lowest-energy exciton carries the largest dipole moment. We can calculate the linear optical susceptibility
and d of Figure 4 show the average distance between the electron and hole ⟨ρ⟩ for the lowest exciton state. The exciton size is on the order of a few nm and increases roughly linearly with the sheet thickness and quadratically with the medium dielectric constant.
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χ (ω) ∝
INTERBAND OPTICAL TRANSITION DIPOLES For comparison with the absorption spectrum, we calculate the oscillator strength of the interband optical transition dipoles for lead-salt nanosheets. The oscillator strength is proportional to the square of the dipole operator, which can be written as21
∫ dr- †e (r)(σx ⊗ σz)-h (r)
= (efield ̂ ·e valley ̂ )2 Pl 2
n1
∑ n
∫ dρFe† (ρ)Fh (ρ) n1
n1
2
2
n2
2
(17)
n2
n2
|R n(n1, n2)(r = 0)|2 En(n1, n2) − ω − iδ
(18)
and the absorption coefficient α(ω) = 4πω/(cn(ω)) Im[χ(ω)]. The absorption spectra of 3 nm PbS nanosheets with δ = 25 meV and δ = 100 meV are plotted in Figure 5. When the broadening (which includes homogeneous and inhomogeneous broadening) is smaller than the exciton binding energy, the first exciton peak is visible (in the case of δ = 25 meV), similar to the case of high quality cadmium-salt nanoplatelets.5 If the sample inhomogeneity is large, then only an indistinct step is observed (δ = 100 meV), which is close to what is currently attainable with lead-salt nanosheets.10,14,18,19
n2
∫ dz - †e (z)(σx ⊗ σz)-h (z)
n1
2
n1, n2
Mc , v = ⟨Ψc(r) e·p|Ψv(r)⟩|2 = (efield ̂ ·e valley ̂ )Pl
∫ dz - †e (z)(σx ⊗ σz)-h (z)
∑
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where we only keep the largest dipole moment contributed by the Bloch functions, and we also separate the vertical and lateral envelope wave functions. Pl is the longitudinal Kane momentum-matrix element between the conduction and valence band-edge Bloch functions.
LEAD SELENIDE NANOSHEETS The electronic states of lead selenide nanosheets are calculated using exactly the same method. The results are summarized in Figure 6 for reference. The band parameters used are from the literature,29 and are listed in Table 1.
2
The term ∫ dz - †en1(z)(σx ⊗ σz)-hn2(z) determines the selection rules of optical transitions between different subbands. As a consequence of inversion symmetry, only transitions with n1 and n2 of the same parity are allowed, and the strongest transitions have n1 = n2.
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COMPARISON WITH EXPERIMENTS Experimental data on the size-dependent optical properties of lead-salt nanosheets is currently quite limited, due to challenges in the synthesis of high-quality samples and reliable measurements of the thickness. Ideally, the thickness can be measured directly with an atomic-force microscope.10,18 However, the organic ligands on the surfaces will obscure the measurement, and special surface treatments to strip off these ligands are needed for reliable measurement. The thickness can also be estimated by applying the Scherrer analysis to the line widths in the X-ray diffraction pattern.10,14 However, nanosheet thicknesses are generally not uniform, and this inhomogeneous broadening naturally complicates the analysis.
2
The term ∫ dρFc†(ρ)Fυ(ρ) is the overlap integral of the electron and hole lateral wave functions. Due to the negligible photon momentum compared with electron momentum, optical transitions from the ground state will be allowed only if the center of mass momentum of the exciton is zero. Thus, in the center of mass coordinate, the overlap integral is reduced to |Rm,n(r = 0)|2.43 This has the consequence that direct transitions only create exciton states with m = 0. All other transitions are forbidden. 26813
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Figure 7. (a) Optical bandgap for PbS nanosheets. The solid line is the four-band model result, corrected by Coulomb interaction. Data points are collected from the literature.10,18,19 (b) Optical bandgap for PbSe nanosheets.
Figure 6. Summary of results for lead selenide nanosheets. (a) 2D band energies without Coulomb interaction. (b) 2D effective mass. (c, e) Self-polarization energy and exciton binding energy, and exciton size as a function of thickness. (d, f) Same quantities as a function of medium dielectric constant.
energy at room temperature. This indicates that, at room temperature, the electron and hole in the nanosheets are tightly bound, and move as a single entity within the lateral dimension. If the inhomogeneous broadening is small enough (such as measurement on a single sheet of uniform thickness), one should observe the exciton peak above the flat plateau in the absorption spectrum. The absence of a sharp exciton peak in past experiments is due to large inhomogeneity of the sample. At the same time, the formation of an exciton is expected to affect charge dynamics, such as multiple exciton recombination,44,45 and carrier multiplication.10 On the other hand, compared with 1D lead-salt nanowires with the same diameter, the exciton binding energy is a lot smaller. Three major effects are responsible. First, in general, lower-dimensional excitons have larger binding energies than those in higher dimensions. Second, since the electron and hole are only confined in one direction, they are relatively more spread out than the electron and hole in a nanowire. This reduces the effective Coulomb interaction, especially at short distances. Third, the dielectric screening is stronger in 2D, and this also decreases the Coulomb interaction, especially at long distances. To summarize, we have presented effective-mass calculations of the electronic states of lead-salt nanosheets. The exciton binding energies are generally larger than the thermal energy at room temperature, and the electron and hole form tightly bound excitons and move together in the lateral dimensions. With monodisperse samples, one should observe sharp optical absorption peaks from the 2D excitons. The calculations agree with the limited experimental data that are available, and we expect that the electronic states determined here will be valuable to many aspects of future nanosheet research.
On the other hand, owing to the 2D electronic structure, the optical absorption spectra of nanosheets will have much less prominent absorption peaks than 1D or 0D nanocystals, which makes it harder to determine the exact position of the optical bandgap. In ref 10, the optical bandgap energy is assumed to be the peak position of the bleach signal in the transient absorption spectrum; in ref 18, the optical bandgap energy is assumed to be the absorption edge of the CW absorption spectrum. The scattering background from nanosheets is large owing to their micron-scale lateral dimensions. We plot all the available data in the literature on sizedependent optical bandgap in Figure 7a. Our model (solid line) agrees reasonably well with available data, but better control of the synthesis and more systematic study of the size-dependent optical properties are sorely needed. The calculated optical bandgap of PbSe nanosheets is shown in Figure 7b for reference. To our knowledge, there is no report of successful synthesis of strongly confined PbSe nanosheets yet.
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DISCUSSION AND CONCLUSION The assumed separation of the electron motion into vertical and lateral motions requires that the energy difference between subbands be much larger than the binding energy of the exciton. The subband separation energy ranges from 400 to 100 meV for 3−8 nm PbS or PbSe nanosheets. The binding energy ranges from 80 to 30 meV over the same range of sizes. Therefore, the approximation is reasonable for this size range. For much thicker nanosheets, coupling between subbands due to the Coulomb interaction will be too large to be treated as a perturbation. For colloidal PbS or PbSe nanosheets in common organic solvents, the exciton binding energy is between 30 and 80 meV, depending on the size. This energy is larger than the thermal
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b08207. 26814
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Numerical calculation of the 2D exciton wavefunction (PDF)
AUTHOR INFORMATION
Corresponding Author
*Phone: +1 (607) 227-2781. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank Liangfeng Sun for valuable discussions. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Division of Materials Sciences and Engineering, under Award DE-SC0006647.
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DOI: 10.1021/acs.jpcc.5b08207 J. Phys. Chem. C 2015, 119, 26809−26816
