• Journal of Semiconductors
  • Vol. 40, Issue 9, 091102 (2019)
Yong Zhang
Author Affiliations
  • Department of Electrical and Computer Engineering, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
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    DOI: 10.1088/1674-4926/40/9/091102 Cite this Article
    Yong Zhang. Applications of Huang–Rhys theory in semiconductor optical spectroscopy[J]. Journal of Semiconductors, 2019, 40(9): 091102 Copy Citation Text show less

    Abstract

    A brief review of Huang–Rhys theory and Albrechtos theory is provided, and their connection and applications are discussed. The former is a first order perturbative theory on optical transitions intended for applications such as absorption and emission involving localized defect or impurity centers, emphasizing lattice relaxation or mixing of vibrational states due to electron–phonon coupling. The coupling strength is described by the Huang–Rhys factor. The latter theory is a second order perturbative theory on optical transitions intended for Raman scattering, and can in-principle include electron–phonon coupling in both electronic states and vibrational states. These two theories can potentially be connected through the common effect of lattice relaxation – non-orthonormal vibrational states associated with different electronic states. Because of this perceived connection, the latter theory is often used to explain resonant Raman scattering of LO phonons in bulk semiconductors and further used to describe the size dependence of electron–phonon coupling or Huang–Rhys factor in semiconductor nanostructures. Specifically, the A term in Albrechtos theory is often invoked to describe the multi-LO-phonon resonant Raman peaks in both bulk and nanostructured semiconductors in the literature, due to the misconception that a free-exciton could have a strong lattice relaxation. Without lattice relaxation, the A term will give rise to Rayleigh or elastic scattering. Lattice relaxation is only significant for highly localized defect or impurity states, and should be practically zero for either single particle states or free exciton states in a bulk semiconductor or for confined states in a semiconductor nanostructure that is not extremely small.
    $ {\psi _{{\rm{in}}}}\left( {x,Q} \right) = {\phi _i}\left( {x,Q} \right){\chi _{{\rm{in}}}}\left( Q \right), $ (1)

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    $ ({H_{\rm{e}}} + {H_{{\rm{eL}}}}){\phi _i}\left( {x,Q} \right) = {W_i}\left( Q \right){\phi _i}\left( {x,Q} \right). $ (2)

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    $ [{H_{\rm{L}}} + {W_i}\left( Q \right)]{\chi _{{\rm{in}}}}\left( Q \right) = {E_{{\rm{in}}}}{\chi _{{\rm{in}}}}\left( Q \right), $ (3)

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    ${H_{\rm{L}}} = \mathop \sum \nolimits_{ q} \frac{1}{2}\left( { -\hbar {{}^2}\frac{{{\partial ^2}}}{{\partial Q_{ q}^2}} + \omega _{ q}^2Q_{ q}^2} \right),$ (4)

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    $ {H_{{\rm{eL}}}} = u(x)Q. $ (5)

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    $ {W_i}\left( Q \right) = W_i^0 + \omega _0^2{\varDelta _{{i}}}Q, $ (6)

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    $ {\varDelta _i} = 1/\omega _0^2\left\langle {\phi _i^0\left( x \right)\left| {u\left( x \right)} \right|\phi _i^0\left( x \right)} \right\rangle , $ (7)

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    $ {E_{{\rm{in}}}} = {W_i} + (n + \frac{1}{2})\hbar {\omega _0}, $ (8)

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    $ {W_i} = W_i^0 - \frac{1}{2}\omega _0^2\varDelta _i^2. $ (9)

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    ${E_{fi}} = {W_f}\left( {{Q_f}} \right) - {W_i}\left( {{Q_i}} \right) = W_f^0 - W_i^0 - \frac{1}{2}\omega_0^2 \left( {\varDelta _f^2 - \varDelta _i^2} \right).$ (10)

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    ${S_{fi}} = \frac{{1/2\omega _0^2\varDelta _{fi}^2}}{{\hbar {\omega _0}}} = \frac{{{\omega _0}\varDelta _{fi}^2}}{{2\hbar }},\tag{11a}$ (11)

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    ${S_{fi}} = \frac{1}{N}\mathop \sum \nolimits_q \frac{{1/2\omega _0^2\varDelta _{fiq}^2}}{{\hbar {\omega _0}}} = \frac{1}{N}\mathop \sum \nolimits_q \frac{{{\omega _0}}}{{2\hbar }}\varDelta _{fiq}^2.\tag{11b}$ (11)

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    $ \begin{split} {E_{fi}} & = {W_f}\left( {{Q_i}} \right) - {W_i}\left( {{Q_i}} \right) - \frac{1}{2}\omega_0^2 \varDelta _{fi}^2 \\ & = {W_f}\left( {{Q_i}} \right) - {W_i}\left( {{Q_i}} \right) - \mathop \sum \nolimits_q {S_{fiq}}\hbar {\omega _q}. \end{split}$ (12)

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    $T\left( E \right) = \frac{{2{\text{π}} }}{\hbar }\left| {{\psi _{fn'}}\left( {x,{\rm{}}Q} \right)} \right|{H_{\rm{eR}}}{\left| {{\psi _{in}}\left( {x,{\rm{}}Q} \right) > } \right|^2}\delta \left[ {E - \left( {{E_{fn'}} - {E_{in}}} \right)} \right].$ (13)

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    ${M_{{{fi}}}}(Q) = \left\langle {{\phi _{{f}}}(x,Q)|{H_{{\rm{eR}}}}|{\phi _{{i}}}(x,Q)} \right\rangle\approx M_{fi}^0,$ (14)

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    $ T(E = {E_{fi}} - p{{{\hbar }}}{\omega _0}) = \frac{{2{\text{π}} }}{\hbar }{\left| {M_{fi}^0} \right|^2}\frac{{{S^p}}}{{p!}}{e^{ - s}}. $ (15)

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    ${\varPsi _{{\rm{ex}}}} = \mathop \sum \nolimits_{{k}} {A_k}{\psi _{vk,ck}}.$ (16)

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    ${\varDelta _{{\rm{ex}}}}\left( {{q}} \right) = \frac{1}{{\omega _0^2}}\mathop \sum \nolimits_k {\left| {{A_k}} \right|^2}\left(\left\langle {ck\left| {{u_q}} \right|ck} \right\rangle - \left\langle {vk\left| {{u_q}} \right|vk} \right\rangle\right).$ (17)

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    ${\varDelta _{{\rm{ex}}}} = \frac{1}{{\omega _0^2}}\left( {\left\langle {{\phi _i}\left| {{u_q}} \right|{\phi _i}} \right\rangle - \mathop \sum \limits_{{k}} {{\left| {{A_k}} \right|}^2}\left\langle {vk\left| {{u_q}} \right|vk} \right\rangle } \right),$ (18)

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    ${\varDelta ^2} = \frac{{{e^2}}}{a}{\left( {\frac{{24}}{{\text{π}} }} \right)^{1/3}}\frac{{\left( {\varepsilon _\infty ^{ - 1} - \varepsilon _0^{ - 1}} \right)}}{{{\omega _{\rm LO}}}}\frac{1}{w}\mathop \int \nolimits_0^w \frac{{{x^4}{{\left( {2 + {x^2}} \right)}^2}}}{{{{\left( {1 + {x^2}} \right)}^4}}}{\rm{d}}x,$ (19)

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    ${\alpha _{nm}} = \frac{1}{\hbar }\mathop \sum \nolimits_r \left(\frac{{{M_{nr}}{M_{rm}}}}{{{\omega _{rm}} - {\omega _0}}} + \frac{{{M_{rm}}{M_{nr}}}}{{{\omega _{rn}} + {\omega _0}}}\right),$ (20)

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    ${\alpha _{gj,gi}} = \frac{1}{\hbar }\mathop \sum \nolimits_{ev} \frac{{{M_{gj,ev}}{M_{ev,gi}}}}{{{\omega _{ev,gi}} - {\omega _0}}}.$ (21)

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    ${M_{rm}} = {M_{ev,gi}} = \left\langle {{\chi _{ev}}\left| {{M_{e,g}}\left( Q \right)} \right|{\chi _{gi}}} \right\rangle ,$ (22)

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    ${M_{e,g}}\left( Q \right) = \left\langle {{\phi _e}\left| {{H_{\rm eR}}} \right|{\phi _g}} \right\rangle .$ (23)

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    ${M_{e,g}}\left( Q \right) = M_{e,g}^0 + \mathop \sum \nolimits_s {\lambda _{e,s}}\left( Q \right)M_{s,g}^0,$ (24)

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    ${M_{ev,gi}} = M_{e,g}^0\left\langle {{\chi _{ev}}|{\chi _{gi}}} \right\rangle + \sum\nolimits_s {M_{s,g}^0} \langle {\chi _{ev}}|{\lambda _{e,s}}\left( Q \right)\left| {{\chi _{gi}}} \right\rangle .$ (25)

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    $ {\alpha _{gi,gj}} = A + B, $ (25)

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    $A = \frac{1}{\hbar }\sum\nolimits_{ev} {\frac{{|M_{e,g}^0{|^2}\left\langle {{\chi _{gj}}|{\chi _{ev}}} \right\rangle \left\langle {{\chi _{ev}}|{\chi _{gi}}} \right\rangle }}{{{\omega _{ev,gi}} - {\omega _0}}}} ,$ (26)

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    $B = \frac{1}{\hbar }\sum\nolimits_{ev} {\sum\nolimits_s {\frac{{M_{g,e}^0M_{s,g}^0\left\langle {{\chi _{gj}}} \right.\left| {\left. {{\chi _{ev}}} \right\rangle \left\langle {{\chi _{ev}}} \right.} \right|{\lambda _{e,s}}|\left. {{\chi _{gi}}} \right\rangle + M_{e,g}^0M_{s,e}^0\left\langle {{\chi _{ev}}} \right.\left| {\left. {{\chi _{gi}}} \right\rangle \left\langle {{\chi _{gj}}} \right.} \right|{\lambda _{s,g}}|\left. {{\chi _{ev}}} \right\rangle }}{{{\omega _{ev,gi}} - {\omega _0}}}} } .$ (27)

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    $A''' = \frac{1}{\hbar }\mathop \sum \nolimits_v \frac{{|M_{e,g}^0{|^2}\left\langle {{\chi _{gj}}} \right|\left. {{\chi _{ev}}} \right\rangle \left\langle {{\chi _{ev}}} \right|\left. {{\chi _{gi}}} \right\rangle }}{{{\omega _{ev,gi}} - {\omega _0} + i{\gamma _e}}},$ (28)

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    $A = |M_{e,g}^0{|^2}\mathop \sum \nolimits_m \frac{{\left\langle {n\left| m \right\rangle \left\langle m \right|0} \right\rangle }}{{{E_{e,i}} + m\hbar {\omega _{\rm LO}} - {\omega _0} + i{\gamma _{\rm e}}}}.$ (39)

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    Yong Zhang. Applications of Huang–Rhys theory in semiconductor optical spectroscopy[J]. Journal of Semiconductors, 2019, 40(9): 091102
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