Photoluminescence and the Franck-Condon Principle

1. What is photoluminescence

Photoluminescence is the phenomenon of a molecule emitting light. It occurs when an electron that has been excited relaxes back to its ground state, releasing its stored energy in the form of electromagnetic radiation.

Photoluminescence is divided in to two main subcategories: fluorescence and phosphorescence. Fluorescence involves a singlet-singlet relaxation and lasts nanoseconds, whereas phosphorescence involves a triplet-singlet relaxation and can last seconds to hours. Today we will be focusing on the photoluminescence of zinc sulfide (ZnS), which is a phosphorescence reaction.

2. The Franck-Condon Principle

The Franck-Condon Principle governs the wavelength of the absorbed and emitted light through photoluminescence. the absorbed light must have enough energy to excite an electron from the ground state, shifting its energy to another morse curve. After the electron relaxes to the lowest vibrational state in that morse potential curve, it drops back down to the ground potential, emitting a photon at a wavelength that is lower than the initial absorbed photon due to the loss in energy through the vibrational transition.

Morse Potential

Here, we explore some of the properties related to the Franck Condon principle. Although the presentation here uses the conventional example of a diatomic particle to illustrate the existence of discrete vibrational levels, the principle applies equally to the case of dopants in a crystal lattice. 

The Morse potential function is taken to be a function of the form:
V(r)=𝒟((((1-(e^(-β(r-(Subscript[r, e]))))))^2)-1)
which is minimum at the equilibrium distance (Subscript[r, e]) and a depth 𝒟 which represent the dissociation energy (conventionally expressed in wavenumbers ((cm^(-1))). The parameter β, together with the mass of the molecule and the dissociation energy detemine the fundamental vibrational frequency of the molecule.The potential function is normalized so that the potential goes to 0 as the distance r becomes large.

V[r_]:=𝒟 ((((1-Exp[-β (r-r0)]))^2)-1)

With[{morse=V[r]},
Manipulate[Plot[morse,{r,0.5r0,3r0},
PlotRange->{{0.5r0,3r0},{-3,1}},
Frame-> True, Axes->{True,False},
FrameLabel->{"Distance", "Potential Energy"},
BaseStyle->{FontSize-> 14,FontColor-> Black}
],
Grid[{{"𝒟", Control[{{𝒟,2,""},1,3,Appearance->"Labeled"}]},
{"β", Control[{{β,2,""},1,3,Appearance->"Labeled"}]},
{"(Subscript[r, e])", Control[{{r0,1.3,""},1,3,Appearance->"Labeled"}]}}]] ]

Vibrational States and Wavelength Shift

Unlike a harmonic oscillator, the Morse potential leads to a finite number of bound vibrational levels that result from solving Schrodinger’s equation. The solution was derived by Morse in closed form:
ψ(β,k,v,x)=(Sqrt[((k β)/(N(v)))])(((e^(((-k)/2)(e^(-βx))))(k (e^(-β x))))^((k-2v-1)/2))L(v,k-2v-1,k (e^(-β x)))
and vibrational energy levels
(Subscript[E, v])= (Subscript[ω, e])(v + (1/2)) - (((Subscript[ωχ, e])(v+(1/2)))^2), for v=0,1,2,..., (Subscript[v, max]) . (Subscript[ω, e]) is the harmonic constant, (Subscript[ωχ, e]) is the anharmonic constant, v is the vibrational quantum number and  x is the displacement from equilibrium r-r0. The parameter k is 2(Subscript[v, max]). When the energy equation is written in this way, it is based on wavenumbers ((cm^(-1))) rather than pure SI-unit energy.
(Subscript[ωχ, e]) can be written in terms of 𝒟 and β can be written in terms of (Subscript[ω, e]) and 
(Subscript[ωχ, e])=(((Subscript[ω, e])^2)/(4𝒟))
β=0.17ω(Sqrt[(( μ )/(2𝒟))])
(Subscript[v, max])=((Subscript[ω, e])/(2(Subscript[ωχ, e])))=2(𝒟/(Subscript[ω, e]))
μ=(Subscript[m, 1])(Subscript[m, 2])/((Subscript[m, 1])+ (Subscript[m, 2])) is the “reduced” mass of the diatomic molecule. (Subscript[N, v])is a normalization constant, N(v) = ((Γ(k-v))^2)(∑_(s=0)^v)((Γ(k-2v+s-1))/(Γ(s+1)))
and L(n,α,x) is the general Laguerre function.

e[𝒟_,ω_,v_]:= -𝒟+ω(v+(1/2))-((ω^2)/(4𝒟))(((v+(1/2)))^2)     (* Quantized energy levels *)

β=0.17ω(Sqrt[(μ/(2𝒟))]);
(* The end points *)
r1=r0-(1/β)Log[1+(Sqrt[(1/𝒟)(ω(v+(1/2))-((ω^2)/(4𝒟))(((v+(1/2)))^2))])];
r2=r0-(1/β)Log[1-(Sqrt[(1/𝒟)(ω(v+(1/2))-((ω^2)/(4𝒟))(((v+(1/2)))^2))])];

With[{morse=V[r],R1=r1, R2=r2},
Manipulate[

Plot[morse,{r,0.5r0,3 r0},
PlotRange-> {-1.1 𝒟, 𝒟/2},
Frame-> True, Axes->{True,False},
FrameLabel->{"Internuclear Distance (Å)", "Potential Energy ((cm^(-1)))"},
ImageSize->Large,
BaseStyle->{FontSize-> 14,FontColor-> Black},
Axes->None,PlotRange->Automatic,
PlotLabel->"Discrete Energy Levels", Epilog->{Green, Thin,Line[Table[{{R1,e[𝒟,ω,v]},{R2,e[𝒟,ω,v]}},{v,0,Floor[2(𝒟/ω)]}]]}],
Style["Parameters:", FontSize->16],
{{μ,30,"Reduced atom mass μ"},1,150,1,Appearance->"Labeled"},{{ω,2000,"Vibrational frequency ω"},500,5000,Appearance->"Labeled"},
{{𝒟,50000,"Dissociation energy 𝒟"},(10^4),(10^5),Appearance->"Labeled"},{{r0,1.128,"Equilibrium distance r0"},.5,3,Appearance->"Labeled"}]]

Wavefunctions and Probability

Clear[β];

The wave functions are given by:

	ψ(v,x)=(Subscript[N, v])(((e^(((-k)/2)(e^(-βx))))(k (e^(-β x))))^((k-2v-1)/2))L(v,k-2v-1,k (e^(-β x)))

β[𝒟_,ω_,μ_] := 0.17ω(Sqrt[(μ/(2𝒟))]);
ψ[𝒟_,ω_,v_,μ_,x_]:=Module[{k,β1,Nv},
k =Floor[4(𝒟/ω)];
β1=β[𝒟,ω,μ];
Nv =(Sqrt[β1((v! (k-1-2v))/(Gamma[k-v]))]);
Nv  Exp[((-k)/2)Exp[-β1 x]](((k Exp[-β1 x]))^((k-1-2v)/2))LaguerreL[v,k-1-2v,k Exp[-β1 x]]]

x1[𝒟_,ω_,v_,μ_]:=Module[{β1},
β1=β[𝒟,ω,μ];
-(1/(β1))Log[1+(Sqrt[(1/𝒟)(ω(v+(1/2))-((ω^2)/(4𝒟))(((v+(1/2)))^2))])]];
x2[𝒟_,ω_,v_,μ_]:=Module[{β1},
β1=β[𝒟,ω,μ];
-(1/(β1))Log[1-(Sqrt[(1/𝒟)(ω(v+(1/2))-((ω^2)/(4𝒟))(((v+(1/2)))^2))])]];

To verify that the wavefunction is correctly normalized:

NIntegrate[((ψ[30000,5000,3,1,x])^2),{x,-∞, ∞}]

Display the wavefunctions on top of the discrete levels for vizualisation purposes (arbitrarily scaled so that the wavefunctions do not overlap)

V0[x_,𝒟_,ω_, μ_] := 𝒟 ((((1-Exp[-β[𝒟,ω,μ]  x]))^2)-1);

Manipulate[

Show[Plot[V0[x,𝒟,ω, μ],{x,-1.1/β[𝒟,ω,μ],5/β[𝒟,ω,μ]},
PlotRange-> {-1.1 𝒟, 𝒟/2},
Frame-> True, Axes->{True,False},
FrameLabel->{"Internuclear Distance", "Potential Energy ((cm^(-1)))"},
ImageSize->Large,
BaseStyle->{FontSize-> 14,FontColor-> Black},
Axes->None,PlotRange->Automatic,
PlotLabel->"Wave Functions"(*, Epilog->{Thin,Line[Table[{{x1[𝒟,ω,v,μ],e[𝒟,ω,v]},{x2[𝒟,ω,v,μ],e[𝒟,ω,v]}},{v,0,Floor[2(𝒟/ω)]}]],}*)],
Table[Plot[e[𝒟,ω,v]+100ψ[𝒟,ω,v,μ,x],{x,1.5x1[𝒟,ω,v,μ],1.5x2[𝒟,ω,v,μ]}],{v,0,Floor[2(𝒟/ω)]}]],
Style["Parameters:", FontSize->16],
{{μ,30,"Reduced atom mass μ"},1,150,1,Appearance->"Labeled"},{{ω,5000,"Vibrational frequency ω"},1000,10000,Appearance->"Labeled"},
{{𝒟,20000,"Dissociation energy 𝒟"},(10^4),(10^5),Appearance->"Labeled"}]

Assuming that the electronic wave functions are independent of the vibrational states (Condon approximation), the Franck-Condon factors are given by the square of the overlap integral between initial and final wave functions in the transition
(Subscript[F, 1,2])= |(Subscript[f, 1,2])((|)^2), where 
		(Subscript[f, 1,2])=⟨(Subscript[ψ, j1,v1]^(β1))(x-R1)|(Subscript[ψ, j2,v2]^(β2))(x-R2)⟩
and R1 and R2 are the equilibrium distances of the nuclei in the inital and final electronic states. Relative transition intensities in the Condon approximation are simply relates to the Franck-Condon factors by 
		(Subscript[I, 1,2])∝(v^4)|(Subscript[f, 1,2])((|)^2)
The integrals can be computed numerically.

Calculating the transition intensities gives us a photoluminescence emission spectrum, where the intensity is plotted against the wavelength. This can be interpolated to give us the color of the emitted light.

3. Zinc Sulfide

Zinc Sulfide is one of the most commonly used photoluminescent materials. It is the main ingredient in glow-in-the-dark paint, among other applications. Its photoluminescence spectrum is determined by the composition and concentration of dopants in the crystal.

Structure

Zinc sulfide has two primary structural types, sphalerite and wurtzite. Sphalerite, the more stable form, is a cubic cyrstal while wurtzite is a hexagonal structure.

SetDirectory[NotebookDirectory[]];
url="http://pruffle.mit.edu/3.016/Data/"
wurtziteData=Import[url<>"wurtzite.xyz"];
zincblendeData=Import[url<>"zincblende.xyz"];

wurtziteData

Manipulate[

If[crystalType==0,
zincblendeData,wurtziteData
],
{{crystalType,0," Crystal:"},{0->"Sphalerite",1->"Wurtzite"},Setter},
Grid[{{"Bond Strength",Control[{{t,15,""},0,30}]},
{"Atom Size",Control[{{r,.25,""},0,.75}]}}],
Control[{{showAtoms,True,"Atoms"},{True,False}}],
Control[{{showBonds,True,"Bonds"},{True,False}}]

]

Dopants cansit at interstitial sites or lattice sites and largely determine the end wavelength of the emitted light.

Spectra

In this section, emission spectra of copper-doped zinc sulfide was taken from research papers and converted to a glow color.

SetDirectory[NotebookDirectory[]]

As an example, the data in the plot below was obtained from an image in the article Characterization of ZnS nanoparticles synthesized by co-precipitation method, and converted using the Web plot digitizer. The data was saved in CSV (comma separated variable) format.

data = Import[url<>"ZnS_Spectrum.csv"];

ListPlot[data]

Turn the data into a function called spectrum...

spectrum[λ_] := Interpolation[data,InterpolationOrder->2][λ]

Plot[spectrum[λ],{λ,350,590}]

The idea now is to map the spectral intensity to a color. There is actually a standard CIE that tries to do exactly that. It converts the continuous spectrum to a tristimulus value (X,Y,Z) that corresponds to a unique color. The tri-stimulus values are obtained by integrating the spectrum multiplied my reference x, y, z functions. Some of it is already built into Mathematica, in the ChromaticityPlot package.

ChromaticityPlot["sRGB"];

{x,y,z}=Interpolation[Thread[{Image`ColorOperationsDump`$wavelengths,#}]]&/@Transpose[Image`ColorOperationsDump`tris];

Plot[{x[λ],y[λ],z[λ]},{λ,385,745},PlotStyle->{Red,Green,Blue}]

The X, Y and Z tristimulus values are obtained by integrating the functions over the spectrum, so define the spectrum and the integration:

This is how the tristimulus values are calculated:

xyzcolor := NIntegrate[Through[{x,y,z}[λ]]spectrum[λ],{λ,350,590}]/NIntegrate[spectrum[λ],{λ,385,590}]

The resulting color is shown in the disk:

Graphics[{XYZColor[xyzcolor],Disk[]}]

Using the same method, spectra of zinc sulfide at different dopant concentrations were generated from the article Structural, optical and photoluminescence properties of ZnS: Cu nanoparticle thin films as a function of dopant concentration and quantum confinement effect.

undoped = Import[url<>"ZnS_Undoped.csv"];
cu001 = Import[url<>"ZnS_Cu001.csv"];
cu007 = Import[url<>"ZnS_Cu007.csv"];
cu01 =  Import[url<>"ZnS_Cu01.csv"];
cu1 = Import[url<>"ZnS_Cu1.csv"];

These files contain data for ZnS doped with 0, 0.001, 0.007, 0.01, and 0.1% Copper.

cuSpectrum[λ_,data_] := Interpolation[data,InterpolationOrder->2][λ]

Show[Plot[cuSpectrum[λ,undoped],{λ,360,650}, PlotStyle -> Red], Plot[cuSpectrum[λ,cu001],{λ,360,650}, PlotStyle -> Orange],Plot[cuSpectrum[λ,cu007],{λ,360,650}, PlotStyle -> Yellow],Plot[cuSpectrum[λ,cu01],{λ,360,650}, PlotStyle -> Green],Plot[cuSpectrum[λ,cu1],{λ,360,650}, PlotStyle -> Blue], PlotRange->Automatic]

Intensity of the emission spectrum peaks at 0.01 molar percent copper, though 0.007, 0.01, and 0.1% copper ZnS all have similar characteristic peaks.

cuColor[data_] := NIntegrate[Through[{x,y,z}[λ]]cuSpectrum[λ,data],{λ,360,650}]/NIntegrate[cuSpectrum[λ,data],{λ,360,650}]

color0 = cuColor[undoped];
color001 = cuColor[cu001];
color007 = cuColor[cu007];
color01 = cuColor[cu01];
color1 = cuColor[cu1];

colorDisk[color_] := Graphics[{XYZColor[color],Disk[]}]

Manipulate[colorDisk[molarPercent],{molarPercent,{color0-> "0%",color001->"0.001%",color007->"0.007%",color01->"0.01%",color1->"0.1%"}},
 ControlType->RadioButton]

The color emitted by the Zinc Sulfide crystal changes as it becomes increasingly copper-doped. Different dopants can produce different colors, including emission spectra in the infrared range. This is due to the changing Franck-Condon factor of the crystal with increasing co

4. Conclusion

The color of emitted light in a photoluminescent material is determined by its photoluminescence spectrum, the intensity of light at each wavelength that it emits when its electrons return to ground state. This intensity is governed by the Franck-Condon factor of the crystal, which depends on the probablity that the electron will be at a given vibrational state, as calculated by the waveform squared. Because an atom is most likely to be present at the lowest vibrational state, the wavelength of the photons emitted by a material through photoluminescence is often larger than the photons that charged them. This can be justified by the spontaneous relaxation of an electron to the lowest vibrational state after being charged. 

One of the most commonly used photoluminescent materials in industry is Zinc Sulfide, and its most common activator is copper. This is the primary ingredient in glow-in-the-dark paint. The emitted photoluminescence spectrum of ZnS:Cu changes with dopant concentration, with 0.01% copper being the brightest emission. ZnS can be activated by other dopants, but many emit in the infrared or were studied in the 1950s and have less accurate graphs.

Citations

Anderson, W. W., S. Razi, and D. J. Walsh. “Luminescence of Rare-Earth-Activated Zinc Sulfide.” The Journal 	of Chemical Physics J. Chem. Phys. 43.4 (1965): 1153. Web.

Bube, Richard H. “Luminescence and Trapping in Zinc Sulfide Phosphors with and without Copper Activator.” 	Phys. Rev. Physical Review 80.4 (1950): 655-66. Web.

“Crystallography Open Database ” http://www.crystallography.net/ 

Fox, Mark. Optical Properties of Solids. Oxford: Oxford UP, 2001. Print.

Iranmanesh, Parvaneh, Samira Saeednia, and Mohsen Nourzpoor. “Characterization of ZnS Nanoparticles 	Synthesized by Co-precipitation Method.” Chinese Physics B Chinese Phys. B 24.4 (2015): 046104. 	Web.

Jayanthi, K., S. Chawla, H. Chander, and D. Haranath. “Structural, Optical and Photoluminescence Properties 	of ZnS: Cu Nanoparticle Thin Films as a Function of Dopant Concentration and Quantum Confinement 	Effect.” Crystal Research and Technology Cryst. Res. Technol. 42.10 (2007): 976-82. Web.

V., J. C. Lopez, A. L. Rivera, Yu. F. Smirnov, and A. Frank. “Simple Evaluation of Franck-Condon Factors and 	Non-Condon Effects in the Morse Potential.” Int. J. Quantum Chem. International Journal of Quantum 	Chemistry 88.2 (2002): 280-95. Web.