Semiconductors & Op1cs: an introduc1on

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1 Semiconductors & Op1cs: an introduc1on Raphaël CLERC Université Jean Monnet & Ins1tut d'op1que Graduate School, Laboratoire Hubert Curien, Saint-E1enne, France 1

2 Semiconductors & Op1cs Introduc1on 2

3 Optoelectronics success story : LED for ligthing the efficacy of LED-based ligh6ng could reach 200 lm/w in 2020, crossing 100 lm/w in 2010 Haitz s law (2000) Cf Eva Monroy & Anne-Laure Bavencove s 3 Talk

Today D810 : 36.3 millions pixel, FX (35.

4 Optoelectronics success story : CMOS imagers Nikon D1, first commercial digital camera : 1999, CCD 2,7 millions pixel, DX (23,7 mm x 15,7 mm) Today D810 : 36.3 millions pixel, FX (35.9 mm x 24 mm) Pixel or photosite area is µm² Distance between pixels is 4.87 µm Cf Jerome Vaillant s Talk 4

Optoelectronics success story : Low cost IR detectors Low cost IR

detec6on, temperature measurement, people coun6ng, and fire & gas

pyroelectric detectors, thermopiles, microbolometers and thermodiodes

5 Optoelectronics success story : Low cost IR detectors Low cost IR detectors are today used for a wide range of func6ons including mo6on detec6on, temperature measurement, people coun6ng, and fire & gas detec6on. pyroelectric detectors, thermopiles, microbolometers and thermodiodes Résolu6on: 206 x 156 pixels Longueur d'onde µm Oyxde de Vanadium, microbolomètre non-refroidi Len6lle en verre de chalcogènure Cf Johan Rothman s Talk 5

radar systems, biosensing 100 Gb/s modulators and receivers : Mach-Zehnder modulators data rates up to

6 Optoelectronics success story : Silicon Photonics Paths for commercializa6on are now widely accessible Main applica6on : communica6on in data center Opportuni6es : medical diagnos6cs, LIDAR, spectroscopy, radar systems, biosensing 100 Gb/s modulators and receivers : Mach-Zehnder modulators data rates up to 50 Gbps Germanium photodiode at 120 GHz with 0.8 A/W responsivity at 1,550 nm 6 Cf Alexei Tchelnokov s Talk

7 Optoelectronics success story : and many others OLED display The raise of Thz imaging Single Photon Avalanche Detector New Solar Cells (Perovskite) Heterogenous integra6on & Packaging. 7

8 Dielectric Func1on, Op1cal index, Absorp1on Dielectric Func6on : ε r =1+χ= ε 1 +i ε 2 Refrac6ve index : ε r 2 =n+iκ ε 1 = n 2 + κ 2 ε 2 =2nκ Absorp6on coef (m-1) : α= 2π/λ κ n(ω)=1+ c/π VP 0 α(ω )/ ω 2 ω 2 dω Kramers Kronig rela6ons : 8

9 Op1cal proper1es of semiconductors Absorp6on : Photodiodes Emission : LED Modulator in Si and Ge 9

10 Introduc1on I Free carrier absorp1on Ex : Silicon High Speed Phase Modulator II Valence to Conduc1on Band Absorp1on Ex : Modula6on of absorp6on in tensile strained Ge III Valence to Conduc1on Band Spontaneous Emission Ex : LED spectrum & Resonant Cavity LED IV Compe11on with other recombina1on mechanisms Ex : LED efficiency droop 10

11 Bibliography 11

12 I Free carrier absorp1on G. Dresselhaus and M. S. Dresselhaus, The Optical properties of Solids, edited by J. Tauc (Academic, New York, 1966) 12

13 Free carriers absorp1on : Classical model Drude model for conduc6vity : m d v /dt + m eff v /τ = e E e iωt σ=env= n e 2 / m eff (1 iωτ) ε 1 = ε core (1 ω p 2 τ 2 /1+ (ωτ) 2 ) ε r = ε core (ω)+ iσ/ ε 0 ω ε 2 = ε core ω p 2 τ/ω(1+ (ωτ) 2 ) Screened plasma frequency : ω p = n e 2 /m ε 0 ε core 13

14 Free carriers absorp1on : Classical model Op6cal frequency : >> p, >> 1 ε 1 = ε core (1 ω p 2 τ 2 /1+ (ωτ) 2 )~ ε core ε 2 = ε core ω p 2 τ/ω(1+ (ωτ) 2 ) ~ ε core ω p 2 / ω 3 τ α= λ 2 e 3 /8 π 3 c 3 ε 0 ε core n/ m eff 2 µ 14

15 Free carriers absorp1on : Applica1on in Si modula1on Pockels effect, Kerr effect and the Franz Keldysh effect are too weak in Si 1.55 µm: 1.3 µm: 15

16 16

17 II Valence to Conduc1on Band Absorp1on 17

18 Form of the electrons Hamiltonian in a crystal H 0 = ( P ) 2 /2m + V crist ( r ) Independant electrons in a crystal : H 0 Ψ n k = E n ( k )Ψ n k Both the group IV (Si, Ge) elements (diamond structure) and many of the III V semiconductors such as GaAs (zinc blende structure) have the same shape of first Brillouin zone. Ge Si GaAs 18

19 Band to Band absorp6on : Form of the Hamiltonian in an Electromagne6c Field H 0 = ( P ) 2 /2m + V crist ( r ) Independant electrons in a crystal : H 0 Ψ n k = E n ( k )Ψ n k Independant electrons in a crystal + light wave : classical approxima6on = no photons, no spontanous emission Introducing the vector poten6al A : E = V A / t B = A Coulomb gauge condi6on : div A =0 H= ( P q A ) 2 /2m + V crist ( r ) (see R Loudon The Quantum Theory of Light (Oxford Science Publica1ons) for details) 19

20 Band to Band absorp6on : Form of the Hamiltonian in an Electromagne6c Field For low light levels : H= ( P q A ) 2 /2m + V crist ( r ) ~ P 2 /2m + V crist ( r ) q P A /m H= H 0 q P A /m Poten1al vector for a monochroma1c plane wave : E =Re( E 0 ε e j(ωt+ k p r ) H= H 0 ) + q E 0 ε P /mω ( e j(ωt k p r ) e j(ω A = E 0 ε /ω sin (ωt k p r ) Fermi Golden Rule : (1 rst order perturba6on theory) H= H 0 +M e jωt Transi6on probability by 6me unit : P i f = 2π/ħ Ψ f M Ψ i 2 δ( E f E i ħω) 20

21 Band to Band absorp6on : Energy conserva6on H= H 0 +M e jωt H= H 0 +M e +jωt P i f = 2π/ħ Ψ f M Ψ i 2 δ( E f E i ħω) Absorp6on P i f = 2π/ħ Ψ f M Ψ i 2 δ( E f E i +ħω) Emission Absorp6on E f = E i +ħω Emission E f = E i ħω 21

22 Band to Band absorp6on : Wave vector conserva6on Matrix element (V = Crystal volume, = primi6ve unit cell volume) Ψ f M Ψ i 2 = q E 0 /2mω 2 V u c k ( r ) e j k r e j k p r ( jħ) ε [ u v k ( r ) e j k r ]= [ u v k ( r )] e j k r +j k u v k ( r ) e j k r Ψ f M Ψ i 2 = q E 0 /2mω 2 R Ω u c k ( r ) e j( k + k p k ) r ε ( Ψ f M Ψ i 2 q E 0 /2mω 2 R e j( k + k p k ) r Ω u c k ( r ) ε ( R e j( k + k p k ) r δ( k + k p k ) k = k + k p k 22

23 Band to Band absorp6on : Absorp6on Probability P i f = 2π/ħ Ψ f M Ψ i 2 δ( E f E i ħω) 2π/ħ q E 0 /2mω 2 ε p cv 2 p cv = jħ Ω u c k ( r ) ( [ u v k ( r )]+j k u v k ( r ))= Ω u c k ( r ) [ u Absorp6on for all state k (/m 3 /s) Absorp6on E f = E i +ħω P Abs = P i f ( k )f( E v ( k ))[1 f( E c ( k ))] d 3 f = Distribu6on func6on (not necessary Fermi (equilibrium)) Effec6ve mass approxima6on : E c ( k )= E c0 + ħ 2 k 2 E v /2 m c ( k )= E v0 ħ 2 k 2 /2 m v 23

24 Band to Band absorp6on : Absorp6on Probability P Abs = P i f ( k )f( E v ( k ))[1 f( E c ( k ))] d 3 k /4 π 3 P Abs = 2π/ħ q E 0 /2mω 2 ε p cv 2 δ( E c ( k ) E v ( k ) ħω) f( E v ( k ))[1 f( ε=e c ( k ) E v ( k ) ħω= E g ħω+ ħ 2 k 2 /2 ( 1/ m c + 1/ m v )= E g ħω+ ħ 2 k 2 / dε= ħ 2 / m r kdk d 3 k /4 π 3 = 4π k 2 dk/4 π 3 E c = E c0 + m r / m c P Abs = 1/ πħ 4 q E 0 /2mω 2 ε p cv 2 (2 m r ) 3/2 ħω E g f( E v )[1 f( E c E v = E v0 m r / m P EStim = 1/ πħ 4 q E 0 /2mω 2 ε p cv 2 (2 m r ) 3/2 ħω E g f( E c )[1 f( E v 24

25 Band to Band absorp6on : Absorp6on Probability Photon concentra6on /m 3 : np (app. npv >> 1) S = E H = 1/2 εc n o E 0 2 Photon Flux /m 2 /s : Φ= S /ħω = εc n o /2ħω E 0 2 Absorp6on coefficient (/m) α= P Abs P EStim /Φ α= 1/ πħ 2 q 2 /4m 1/εc n o 2 ε p cv 2 /mħω (2 m r ) 3/2 ħω E g [f( E Oscillator strength f= 2 ε p cv 2 /mħω P Abs =A n p ħω E g f( E v )[1 f( E c )] P EStim =A n p ħω E g f( E c )[1 f( E v )] 25

26 Band to Band absorp6on : Absorp6on Coefficient 26

27 Band to Band absorp6on : Formalism for full band transi6ons (direct and phonon assisted indirect transi6ons ) : 27

28 Band to Band absorp6on : The role of band engineering : Example of strain Ge Phonon-assisted op6cal interband transi6on based on the Green s func6on formalism second-order perturba6on theory (SOPT) Band structures from empirical pseudopoten6al method 28

29 Band to Band absorp6on : The role of band engineering : Example of strain Ge 29

30 Band to Band absorp6on : The role of band engineering : Example of strain Ge 30

31 III Valence to Conduc1on Band Spontaneous Emission 31

32 Spontaneous emission : Einstein theory Cf Einstein coefficient theory (1916) Star6ng from : P Abs =A n p ħω E g f( E v )[1 f( E c )] P EStim =A n p ħω E g f( E c )[1 f( E v )] We postulate : P ESpon =B ħω E g f( E c )[1 f( E v )] To find B, we consider the equlibrium condi6on : n p = 1/ V p 1/ e ħω/kt 1 f(e)= 1/ 1+e E E F /kt P ESpon = A/ V p ħω E g f( E c )[1 f( E v )] 32

33 Spontaneous emission : LED emission spectum (3D) P EmSpon = 0 c/ n opt α(ω)g(ω)dω/ e (ħω Δ E F )/kt 1 0 c/ n opt α(ω Op6cal Mode Density Boltzmann approxima6on : P EmSpon B n p OSRAM Golden DRAGON Plus (blue) 33

34 Spontaneous emission : LED emission spectum (3D) Current dependency : n varia1on : BLUE SHIFT n Wavelength (nm) n Energy (ev) Temperature dependency : gap varia1on : RED SHIFT T Wavelength (nm) Energy (ev) T

35 Spontaneous emission : Junc6on temperature P EmSpon = 0 c/ n opt α(ω)g(ω) e (ħω Δ E F )/kt dω 35

36 Spontaneous emission : Spontaneous emission in a resonant cavity 36

37 37

38 IV Compe11on with other recombina1on mechanisms 38

39 Other recombina1on mechanisms : Trap assisted : Defects (contamina6on), interface, metal contacts In photodiode : it penalizes illuminated and dark currents!! Auger : High injec6on 39

40 Other recombina1on mechanisms : InGaN based LED efficicency droop 40

41 Other recombina1on mechanisms : InGaN based LED efficicency droop IQE 0 W R rad (x)dx / 0 W ( R rad (x)+ R trap (x)+ R Auger (x))dx IQE B n 2 /An+B n 2 +C n 3 41

42 42

43 Conclusions Optoelectronics is a fascina6ng domain coupling many fields of physics and engineering : Op6cs, Solid state physics, electronics, material sciences III V materials have excellent intrinsic materials proper6es, but materials proper6es can be tuned Silicon and organic material are gaining in interest There is s6ll plenty of scien6fic discovery to be made Keep an eye open for innova6on : Technology may seem conserva6ve, but there is s6ll room for innova6on 43

44 Thank you for your ayen6on 44

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