Characterization of lead bromide perovskite colloidal nanocrystals : stability and optical properties

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12

• 𝑡 = (𝑟𝐴+ 𝑟𝑋) √2⁄ (𝑟𝑃𝑏+ 𝑟𝑋)

𝑟_𝐴, 𝑟_𝑃𝑏, 𝑟_𝑋

≤ ≤

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14 𝛹(𝑟⃗) = 𝑒𝑥𝑝(𝑖 ∙ 𝑘⃗⃗ ∙ 𝑟⃗) 𝜓(𝑟⃗) 𝑓(𝐸) = 1 1 + 𝑒𝑥𝑝 (𝐸 − 𝐸_𝐾 𝑓 𝐵𝑇 ) 𝐸_𝑓 𝐾_𝐵

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15 𝐸_𝑐(𝑘⃗⃗) = 𝜀_𝑔+ ℏ 2_𝑘2 2𝑚_𝑐 𝐸_𝑣(𝑘⃗⃗) = − ℏ 2_𝑘2 2𝑚_𝑣 𝑘⃗⃗ 𝑚_𝑐,𝑣 𝜀_𝑔 𝜌(𝜀) 𝜀 ÷ 𝜀 + 𝑑𝜀 𝜌(𝑘) (2𝜋)3_⁄_𝑉 𝜌(𝑘)𝑑𝑘 𝑘 ÷ 𝑘 + 𝑑𝑘 𝑘 𝑘 + 𝑑𝑘

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16 𝑘 𝑘 + 𝑑𝑘 4𝜋𝑘2_𝑑𝑘 8𝜋3 𝑉 = 𝑘2 2𝜋2𝑉𝑑𝑘 𝜌(𝑘) = 𝑘 2 𝜋2 𝜌(𝜀)𝑑𝜀 = 𝜌(𝑘)𝑑𝑘 𝑑𝜀 =ℏ 2_𝑘 𝑚∗ 𝑑𝑘 𝜌(𝜀) = 1 2𝜋2( 2𝑚∗ ℏ2 ) 3 2 √𝜀 − 𝜀𝑔 𝑓(𝜀) ̅̅̅̅̅̅ = 1 − 𝑓(𝜀)

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17 𝑓(𝜀) 𝑛 = ∫ 𝜌_𝑐(𝜀)𝑓(𝜀)𝑑𝜀 ∞ 𝐸𝑐 𝑝 = ∫ 𝜌_𝑣(𝜀)𝑓(𝜀)̅̅̅̅̅̅𝑑𝜀 𝐸𝑣 −∞ 𝜌(𝜀) 𝑓(𝜀): 𝑛 = 1 2𝜋2( 2𝑚_𝑐 ℏ2 ) 3/2 ∫ √𝜀 − 𝐸𝑐 1 + 𝑒𝑥𝑝 (𝜀 − 𝜀𝐹 𝑘𝐵𝑇 ) 𝑑𝜀 ∞ 𝐸𝑐 𝑥 = (𝜀 − 𝜀𝑐) 𝑘⁄ 𝐵𝑇 𝑦 = 𝜀_𝐹− 𝜀_𝑐⁄𝑘_𝐵𝑇 𝑛 ∝ ∫ √𝑥 1 + exp(𝑥 − 𝑦)𝑑𝑥 ∞ 0 3𝑘_𝐵𝑇 𝑓_𝑐(𝜀) ≈ 𝑒𝑥𝑝 (−𝜀 − 𝜀𝐹 𝑘_𝐵𝑇 ) 𝑛 = 𝑁_𝑐𝑒𝑥𝑝 (−𝜀𝑐 − 𝜀𝐹 𝑘_𝐵𝑇 ) 𝑝 = 𝑁_𝑣𝑒𝑥𝑝 (−𝜀𝐹− 𝜀𝑣 𝑘_𝐵𝑇 ) 𝑁_𝑐 =1 4( 2𝑚𝑐∗𝑘𝐵𝑇 𝜋ℏ2 ) 3/2 ; 𝑁_𝑣 = 1 4( 2𝑚𝑣∗𝑘𝐵𝑇 𝜋ℏ2 ) 3/2 𝑁_𝑐 𝑁_𝑣

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18 𝑛 = 𝑝. 𝑛 = 𝑝 ≡ 𝑛_𝑖 = √𝑁𝑐𝑁𝑣𝑒𝑥𝑝 (− 𝜀_𝑔 2𝑘𝐵𝑇 ) 𝑛𝑖 𝑁𝐶 𝑁𝑉 𝑛_𝑖 = 𝑁_𝑐𝑒𝑥𝑝 (−𝜀𝑐 − 𝜀𝑖 𝑘_𝐵𝑇 ) = 𝑁𝑣𝑒𝑥𝑝 (− 𝜀_𝑖 − 𝜀_𝑣 𝑘_𝐵𝑇 ) 𝑁_𝑐 ≈ 𝑁_𝑣 𝜀_𝑐 − 𝜀_𝑖 ≈ 𝜀_𝑖− 𝜀_𝑣 𝑛 = 𝑛_𝑖𝑒𝑥𝑝 (𝜀𝐹− 𝜀𝑖 𝑘_𝐵𝑇 ) ; 𝑝 = 𝑛𝑖𝑒𝑥𝑝 ( 𝜀𝑖 − 𝜀𝐹 𝑘_𝐵𝑇 ) 𝐺 = 𝑓₁(𝑇) 𝑓₁

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19 𝑅 = 𝑛𝑝𝑓₂(𝑇) 𝐺 = 𝑅 𝑛𝑝 = 𝑛_𝑖2 = 𝑓₁(𝑇) 𝑓⁄ ₂(𝑇) = 𝑓₃(𝑇). 𝑛_𝑖2 = 𝑁_𝑐𝑁_𝑣𝑒𝑥𝑝 (− 𝜀𝐺 𝑘_𝐵𝑇) 𝑛𝑝 = 𝑛_𝑖2 𝑛 = 𝑛₀+ Δ𝑛 𝑝 = 𝑝₀+ Δ𝑝 𝐸 = ℎ𝜈 − 𝜀_𝐺 = ℎ 2_𝑘2 2𝑚𝑐,𝑣∗ 𝑚𝑐,𝑣∗ 𝐸𝐹𝑐 𝐸𝐹𝑣 𝑓_𝑐(𝐸_𝑐) 𝑓_𝑣(𝐸_𝑣)

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20 𝐸_𝐾_𝑋_,𝑛= 𝜀_𝐺− 𝑅 𝑛2 + ℎ2_𝐾 𝑋2 2(𝑚_𝑒∗ _{+ 𝑚} ℎ∗)

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21 𝜀_𝐺 _{− 𝑅 𝑛}⁄ ₂ 𝐻(𝑟⃗) = 1 4𝜋𝜀₀𝜀_𝑟 𝑒2 𝑟 − ℏ2 2𝜇∇ 2 𝜇 𝜇 = ( 1 𝑚_𝑒+ 1 𝑚_ℎ) −1 ℏ2_𝐾 𝑋2⁄2(𝑚𝑒∗+ 𝑚ℎ∗) 𝑅 = 𝑅0 𝜇 𝑚0 1 𝜀𝑟2 𝑅₀ 𝑅₀ = 13.6 𝑒𝑉

𝜀

_𝑟 𝐸𝐵 = 𝑅 − 𝐸𝐺 𝑎_𝐵= 𝑎_𝐵0𝜀 (𝑚0 𝜇 ) 𝑛 2 𝑚₀ 𝑎_𝐵0 = 0.05𝑛𝑚 𝑎_𝐵 𝑥2 (1 − 𝑥) = 1 𝑛( 2𝜋𝜇𝑘_𝐵𝑇 ℎ2 ) 3 2 𝑒 𝐸𝐵 𝐾𝐵𝑇

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22 𝑥 = 𝑛_𝑒ℎ/𝑛

𝐸₂ 𝐸₁

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23 𝜀₁ 𝜀₂ 𝜀₂− 𝜀₁ = 𝜀₀ = ℏ𝜔₀ 𝜌_𝑗(𝜀₀) 𝜌_𝑗(𝜀₀)𝑑𝜀₀ 𝜀0÷ 𝜀0 + 𝑑𝜀0. 𝑘_𝑐 = 𝑘_𝑝ℎ+ 𝑘_ℎ; Δ𝑆 = 0; 𝑘𝑒 = 𝑘ℎ 𝜀₀ = 𝜀₂− 𝜀₁= 𝜀_𝐺+ℏ 2_𝑘2 2 ( 1 𝑚_𝑐∗+ 1 𝑚_𝑣∗) = 𝜀𝐺 + ℏ2_𝑘2 2𝑚_𝑟∗ 𝑚𝑟∗ 𝜌_𝑗(𝜀0)𝑑𝜀0 = 𝜌(𝑘)𝑑𝑘 = 𝑘2 𝜋2𝑑𝑘 𝑑𝜀₀ = ℏ2𝑘 _𝑚 𝑟 ∗ ⁄ 𝑑𝑘 𝜌_𝑗(𝜀0) = 1 2𝜋2( 2𝑚𝑟∗ ℏ2 ) 3 2 (𝜀0− 𝜀𝐺) 1 2 𝜀₂ 𝜀₁ 𝑟_𝑠𝑝(𝜖) = 𝐴𝑐𝑣𝑓𝑐(𝜀2)[1 − 𝑓𝑣(𝜀1)] 𝜀₂ 𝜀1 𝐴_𝑐𝑣_{≡ 1 𝜏}⁄ _𝑟𝑎𝑑 𝜏_𝑟𝑎𝑑

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24 𝑅_𝑠𝑝(ℏ𝜔) = ∫ 𝑟𝑠𝑝(𝜀)𝜌𝑗(𝜀)𝑔(𝜀 − ℏ𝜔)𝑑𝜀 ∞ 𝜀𝑔 𝑅_𝑠𝑝(ℏ𝜔) = 𝑟𝑠𝑝(ℏ𝜔)𝜌𝑗(ℏ𝜔) 𝑅_𝑠𝑝(ℏ𝜔) = 1 𝜏_𝑟𝜌𝑗(ℏ𝜔)𝑓𝑐(ℏ𝜔)[1 − 𝑓𝑣(ℏ𝜔)] 𝑓𝐶(𝜀2) ≈ exp (− 𝜀₂− 𝜀_𝐹𝑐 𝑘𝐵𝑇 ) 1 − 𝑓_𝑉(𝜀1) ≈ 𝑒𝑥𝑝 (− 𝜀𝐹𝑣 − 𝜀1 𝑘_𝐵𝑇 ) 𝑅𝑠𝑝(ℏ𝜔) = 1 𝜏𝑟𝑎𝑑 𝜌𝑗(ℏ𝜔)exp (− ℏ𝜔 𝑘𝐵𝑇 ) 𝑒𝑥𝑝 (Δ𝜀𝐹 𝑘𝐵𝑇 ) Δ𝜀_𝐹 = 𝜀_𝐹𝑐 − 𝜀_𝐹𝑣 𝑅𝑠𝑝(ℏ𝜔) = 1 𝜏𝑟𝑎𝑑 1 2𝜋2( 2𝑚_𝑟 ℏ2 ) 3 2 √ℏ𝜔 − 𝜀𝐺 𝑒𝑥𝑝 (− ℏ𝜔 − 𝜀_𝐺 𝑘𝐵𝑇 ) 𝑒𝑥𝑝 (Δ𝜀𝐹− 𝜀𝐺 𝑘𝐵𝑇 ) 𝑅𝑠𝑝(ℏ𝜔) ∝ √ℏ𝜔 − 𝜀𝐺𝑒𝑥𝑝 (− ℏ𝜔 − 𝜀_𝐺 𝑘𝐵𝑇 )

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25 𝑅_{𝑠𝑝,𝑇} = 1 𝜏_𝑟𝑎𝑑∫ 𝜌_𝜀 𝑗(ℏ𝜔)𝑓𝑐(ℏ𝜔)[1 − 𝑓𝑣(ℏ𝜔)]𝑑(ℏ𝜔) 𝐺 𝑅_{𝑠𝑝,𝑇} = exp_𝑘Δ𝜀𝐹 𝑏𝑇 𝜏_𝑟𝑎𝑑 ∫ 𝜌𝑗(ℏ𝜔) exp (− ℏ𝜔 𝑘_𝐵𝑇) 𝑑(ℏ𝜔) ∞ εG ∫ √𝑥 exp 𝑥𝑑𝑥 = ∞ 0 √𝜋 2 𝑅𝑠𝑝,𝑇 = 1 𝜏𝑟𝑎𝑑 𝑁𝑟𝑒𝑥𝑝 ( Δ𝜀_𝐹− 𝜀_𝑔 𝑘𝐵𝑇 ) 𝑁_𝑟 𝑁_𝑟 = 1 4( 2𝑚𝑟∗𝑘𝐵𝑇 𝜋ℏ2 ) 3 2 ⁄ 𝑅_{𝑠𝑝,𝑇} = 1 𝜏_𝑟𝑎𝑑 𝑁𝑟 𝑁_𝑐𝑁_𝑣𝑛𝑝 ≡ 𝐵𝑛𝑝 𝐺₀ 𝑅₀ = 𝐺₀ = 𝐵𝑛₀𝑝₀ = 𝐵𝑛_𝑖2 𝑛₀ 𝑝₀ 𝑛₀𝑝₀ = 𝑛_𝑖2 Δ𝑛 = Δ𝑝

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26 𝑅_𝑟 = 𝐵𝑛𝑝 − 𝐵𝑛₀𝑝₀ = 𝐵(𝑛𝑝 − 𝑛₀𝑝₀) 𝑑𝑛 𝑑𝑡 = 𝑑𝑝 𝑑𝑡 = 𝑅0− 𝐵𝑛𝑝 = −𝐵(𝑛𝑝 − 𝑛0𝑝0) = −𝐵(𝑛𝑝 − 𝑛𝑖 2₎ 𝑑𝑛 𝑑𝑡 = 𝑑Δ𝑛 𝑑t⁄ ⁄ 𝑑𝑝 𝑑𝑡 = 𝑑Δ𝑝 𝑑𝑡⁄ ⁄ 𝑑Δ𝑛 𝑑𝑡 = −𝐵(𝑛0Δ𝑝 + 𝑝0Δ𝑛 + Δ𝑛Δ𝑝) Δ𝑛 = (𝑛0 + 𝑝0)Δ𝑛(0) [𝑛0+ 𝑝0+ Δ𝑛(0)] exp[𝐵(𝑛0+ 𝑝0)𝑡] − Δ𝑛(0) 𝑛 ≈ 𝑝 ≫ 𝑛0, 𝑝0 𝑑𝑛 𝑑𝑡 = 𝑑Δ𝑛 𝑑𝑡 = −𝐵(Δn) 2 _{≈ −𝐵𝑛}2 𝐸_𝑇 𝑓_𝑡(𝜀_𝑇) = 1 1 + exp (𝐸𝑇_𝐾− 𝐸𝑓 𝐵𝑇 )

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27 • 𝑅𝑐𝑛 = 𝐶𝑛𝑛𝑁𝑇[1 − 𝑓(𝜀𝑇)] = 𝑣𝑡ℎ𝜎𝑛𝑛𝑁𝑇[1 − 𝑓(𝜀𝑇)] 𝑁𝑇 1 − 𝑓(𝜀𝑇) 𝐶_𝑛 = 𝑣_𝑡ℎ𝜎_𝑛 𝑣_𝑡ℎ= (3𝑘_𝐵𝑇 𝑚_⁄ _𝑛∗₎1/2 _𝜎 𝑛 • 𝑅_𝑒𝑛 = 𝑒_𝑛𝑁_𝑇𝑓(𝜀_𝑇) 𝑒_𝑛 𝑅_𝑐𝑛0 = 𝑅_𝑒𝑛0 𝑒𝑛 = 𝐶𝑛𝑛0 1 − 𝑓₀(𝜀_𝑇) 𝑓0(𝜀𝑇) 𝑓₀ 𝑛₀

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28 𝑒𝑛 = 𝐶𝑛𝑛0𝑒𝑥𝑝 ( 𝜀_𝑇− 𝜀_𝐹 𝑘𝐵𝑇 ) 𝜀𝑖 𝑒_𝑛 = 𝐶_𝑛𝑛_𝑖𝑒𝑥𝑝 (𝜀𝑇− 𝜀𝑖 𝑘_𝐵𝑇 ) = 𝐶𝑛𝑁𝑐𝑒𝑥𝑝 ( 𝜀_𝑇− 𝜀_𝑐 𝑘_𝐵𝑇 ) ≡ 𝐶𝑛𝑛1 𝑅_𝑛 = 𝑛𝑒𝑡 𝑐𝑎𝑝𝑡𝑢𝑟𝑒 𝑟𝑎𝑡𝑒 = 𝑅_𝑐𝑛− 𝑅_𝑒𝑛 𝑅_𝑛 = 𝐶_𝑛𝑁_𝑇{𝑛[1 − 𝑓_𝑡(𝜀_𝑇)] − 𝑛₁𝑓(𝜀_𝑇)} 𝑓(𝜀_𝑇) • 𝑅_𝑐𝑝= 𝐶_𝑝𝑝𝑁_𝑡𝑓(𝜀_𝑇) = 𝑣𝑡ℎ𝜎𝑝𝑝𝑁𝑇𝑓(𝜀𝑇), 𝑁_𝑡𝑓(𝜀_𝑇) 𝐶_𝑝 = 𝑣_𝑡ℎ𝜎_𝑝 𝑣_𝑡ℎ = (3𝑘_𝐵𝑇 𝑚⁄ _ℎ∗)1/2 _𝜎 𝑝 • 𝑅𝑒𝑝 = 𝑒𝑝𝑁𝑡(1 − 𝑓𝑡) 𝑒𝑝 = 𝐶𝑝𝑝1

(29)

29 𝑝1 = 𝑛𝑖𝑒𝑥𝑝 (− 𝜀_𝑇− 𝜀_𝑖 𝑘𝐵𝑇 ) = 𝑁𝑣𝑒𝑥𝑝 (− 𝜀_𝑇− 𝜀_𝑣 𝑘𝐵𝑇 ) 𝑅_𝑝 𝑅_𝑝 = 𝑅_𝑐𝑝− 𝑅_𝑒𝑝 = 𝐶_𝑝𝑁_𝑇{𝑝𝑓𝑡(𝜀𝑇) − 𝑝1[1 − 𝑓(𝜀𝑇)]} 𝑅_𝑛 = 𝑅_𝑝 𝑓(𝜀_𝑇) 𝑓(𝜀𝑇) = 𝐶𝑛𝑛 + 𝐶𝑝𝑝 𝐶𝑛(𝑛 + 𝑛1) + 𝐶𝑝(𝑝 + 𝑝1) 𝑛1𝑝1 = 𝑛𝑖2 𝑅_𝑛 = 𝑅_𝑝 = 𝐶𝑛𝐶𝑝𝑁𝑇(𝑛𝑝 − 𝑛𝑖 2₎ 𝐶_𝑛(𝑛 + 𝑛1) + 𝐶𝑝(𝑝 + 𝑝1) 𝜏𝑛 = 1 𝐶𝑛𝑁𝑇 , 𝜏𝑝 = 1 𝐶𝑝𝑁𝑇 𝑅_𝑛 = 𝑅_𝑝 = (𝑛𝑝 − 𝑛𝑖 2₎ 𝜏_𝑛(𝑛 + 𝑛1) + 𝜏𝑝(𝑝 + 𝑝1) 𝑛, 𝑝 ≫ 𝑛𝑖 𝑛, 𝑝 ≫ 𝑛1, 𝑝1 𝜏_𝑛 = 𝜏_𝑝 𝑅𝑆𝑅𝐻 ≈ 𝑛2 2𝜏𝑛𝑛 = 𝑛 2𝜏𝑛

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30 𝑑𝑛 𝑑𝑡 = 𝑅𝐺− 𝛽𝑛𝑁𝑡𝑛. 𝛽𝑛 𝑁𝑡 𝑅_𝑒𝑐 = 𝐶_𝑒𝑐𝑛2_𝑝 𝑅_ℎ𝑐 = 𝐶_ℎ𝑐𝑛𝑝2

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31 𝑅_𝑒𝑒 = 𝐶_𝑒𝑒𝑛 𝑅_ℎ𝑒 = 𝐶_ℎ𝑒𝑝 𝑅_𝑒𝑐0 = 𝑅_𝑒𝑚0 ; 𝑅_ℎ𝑐0 = 𝑅_ℎ𝑒0 𝐶_𝑒𝑐𝑛_𝑖2 = 𝐶_𝑒𝑒; 𝐶_ℎ𝑐𝑛_𝑖2 = 𝐶_ℎ𝑒 𝑅𝐴𝑢𝑔 = (𝑅𝑒𝑐 − 𝑅𝑒𝑒) + (𝑅ℎ𝑐− 𝑅ℎ𝑒) = (𝐶𝑒𝑐𝑛 + 𝐶ℎ𝑐𝑝)(𝑛𝑝 − 𝑛𝑖2) 𝐸 𝑘⃗⃗ 𝐸′ 𝑘′⃗⃗⃗⃗ 𝐸 + 𝐸′ 𝑘⃗⃗ + 𝑘′⃗⃗⃗⃗ 𝑛 = 𝑝 𝑛, 𝑝 ≫ 𝑛_𝑖 𝐶_𝑒𝑐 = 𝐶_ℎ𝑐 𝑅_𝐴𝑢𝑔 = 𝛾_𝐴𝑢𝑔𝑛3

(32)

(33)

(34)

34 𝐸𝑛 =

ℏ2𝜋2 2𝑚∗_𝐿2𝑛2

(35)

(36)

(37)

37 - - - - -

(38)

(39)

(40)

40 -

(41)

(42)

(43)

(44)

(45)

45

Δ𝑡

𝑃𝐿 𝑄𝑌 = 𝑒𝑚𝑖𝑡𝑡𝑒𝑑 𝑝ℎ𝑜𝑡𝑜𝑛𝑠 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑝ℎ𝑜𝑡𝑜𝑛𝑠

(46)

(47)

47 𝐿𝐴, 𝐿𝐵 𝐿_𝐶 𝑃𝐵 𝑃𝐶 𝜇 𝐿_𝐵 = 𝐿_𝐴(1 − 𝜇)

(48)

48 𝜇 𝐿_𝐶 = 𝐿_𝐴(1 − 𝐴)(1 − 𝜇) 𝐴 = 1 −𝐿𝐶 𝐿_𝐵 𝐿_𝐵+ 𝑃_𝐵 (1 − 𝐴)(𝐿𝐵+ 𝑃𝐵) 𝜂 η𝐿_𝐴𝐴 𝐿𝐶+ 𝑃𝐶 = (1 − 𝐴)(𝐿𝐵+ 𝑃𝐵) + 𝜂𝐿𝐴𝐴 𝐿_𝐶 𝐿_𝐵 𝑃𝐿 𝑄𝑌 = 𝜂 = 𝑃𝐶− (1 − 𝐴)𝑃𝐵 𝐿_𝐴𝐴

(49)

(50)

(51)

51 -

(52)

(53)

53 - - - - ∼

(54)

54 -

-

(55)

55 - - - - - -

(56)

56 -

(57)

(58)

(59)

(60)

(61)

61 - - - - - -

(62)

(63)

(64)

(65)

(66)

(67)

(68)

(69)

(70)

(71)

71 𝑛2

(72)

(73)

73 𝜇𝑠

(74)

(75)

(76)

(77)

(78)