ANALYTICAL THEORY AND NUMERICAL SIMULATION OF NON-EQUILIBRIUM TRANSPORT EFFECTS IN LASER HETEROSTRUCTURES. N.A. Zakleniuk and M.J.

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1 NUSOD 005, Berli ANALYICAL HEORY AND NUMERICAL SIMULAION O NON-EQUILIBRIUM RANSPOR EECS IN LASER HEEROSRUCURES N.A. Zakleiuk ad M.. Adams Phtics Research Gru, ESE Deartmet Uiversity f Essex, Clchester, UK

2 CONENS Surface Ridge IP/IGaAsP Laser Gemetry is Csidered Outlie f the Mdel Key Prcesses Aalytical hery 1D DESSIS Simulati Belw ad Abve hreshld Cmaris f the Results Why 1D hery is i s Gd Agreemet with Numerical Mdel Why 1D hery is t Alicable fr Reliable Aalysis Cclusis

3 MOIVAION Multi-Secti uable Laser Dide: Light Outut I Gai I Phase I DBR Ades -IP Gai Regi uig Regis Sectis E g <E u -IP Substrate DBR Gratig Reflecti Catig Cathde AR Catig Wavelegth tuig is achieved via chage f the electric art f the dielectric fucti i the assive art f the waveguide. he mai gal is t btai maximum free-carrier desity fr a give ijecti currets I DBR /I Phase r bias.

4 GEOMERY AND PARAMEERS 0 Ade 0.4μm μm 3μm Y -IP Q-IGaAsP -IP X Active/ uig Regi 8 μm -IP E BI Δ 1 + Δ ev N A cm eV Δ eV Δ c 0.191eV Q-IGaAsP N A cm -3 E g Q 0.873eV Q 0.159eV Δ v 0.86eV Δ 0.953eV E g IP 1.35eV 0.048eV -IP N D cm -3 Cathde Mbilities: IP: μ -IP μ -IP 150 cm /V s; μ -IP μ -IP 100 cm /V s; Q-IGaAsP: μ Q 4000 cm /V s; μ Q 00, 100, 50 cm /V s; Recmbiati: IP: τ SRH s; Radiative: C cm 3 /s; Auger: C C cm 6 /s; Q-IGaAsP: τ SRH s; Radiative: C cm 3 /s; Auger: C 0, C cm 6 /s; Effective masses: IP: m 0.078m, m 0.7m ; Q-IGaAsP: m 0.05m, m 0.50m ; Ohmic ctacts: R0; Rm temerature: 300K; Abrut dig rfiles; Built-i vltage: V bi 1.35 V;

5 HEEROSRUCURE A HIGH-INECION REGIME: HE LA-BAND CONDIIONS Schematic f Bad Prfile ad e/h Quasi-ermi Levels Used i Aalytic hery at High-Ijecti Regime: Mst Geeral Case E c -IP E v y d 1 ev Q E cq Q E vq Q-IGaAsP Ey d d ev y -IP E c E v y d 1 d y y HE NECESSARY CONDIIONS A. Griberg, AP : 1 / ε Q k 1. L D << d ; Q 4 e r 10 π 18 cm -3 : L D 5 m. Q. Vltage Dr Acrss Q-Layer: eδv Q << k r I 50 ma, eδv Q 0.8 mev. 3. Quasi-Neutrality L D <<d Q ad Hmgeeity L Dif >>d Q, L Dif 1μm!.

6 BASIC RANSPOR PROCESSES IN LASER DIODE EL. LEAKAGE: DRI- DIUSION IN HE SCR ELECRONS: HERMIONIC EMISSION Ey ELECRONS: DRI- DIUSION IN HE SCR A N O D E EL. LEAKAGE: DI-RECOMB IN QNR E/H RECOMBINAION HOLE LEAKAGE: DI-RECOMB IN QNR C A H O D E P-IP Q-IGaAsP N-IP HOLES: DRI- DIUSION IN HE SCR HOLES: HERMIONIC EMISSION HOLE LEAKAGE: DRI- DIUSION IN HE SCR y

7 Drift-Diffusi i the SCR i -IP d y y :. 0 1 DD R G dy y dj e, dy y d y y j DD μ k y y E N y c c ex Budary Cditis at -IP/Q Iterface:. E DD j d y j hermiic Emissi Curret at -I/Q I:, + E E E j j j. ex - ex ex + Δ + e Q e Q c c E k k d k E A j α SE O EQUAIONS OR ELECRON INECION CURREN: BASIC EQUAIONS Electrs

8 ERMI LEVEL DISCONINUIY A HE INERACE Budary cditi at yd leads t the fllwig exressi: d k j l DD ex k + j E e Δ ex DD E j + j c + E k Q c e Q ex E c d k, where j DD thermiic ad j E emissi are sme characteri stic currets defied the drift ext - diffusi age. ad

9 SOLUION O BASIC EQUAIONS Electrs ELECRON INECION CURRENS: DRI-DIUSION + E EMISSION: + Δ + e Q Q c c e c k E k ev E ex ex E DD j j, k ev L N ed Deb c DD / 1 4 D Deb N e k L π ε, E m m A A α α / 10 K cm A A - Richards s cstat he smallest curret defies the ijecti bttleeck! - Debye s legth Icrrati f ijecti heatig it the mdel: Ijecti f e/h results i suly f the excess eergy it the Q-layer: ΔE e ev/carrier. his i tur leads t icrease f the e/h temerature i Q-layer ad icrease f thermiic emissi frm the Q-layer. ΔE e ΔE Q Q

10 DESCRIPION O VERICAL LEAKAGE CURRENS Drift-Diffusi i the SCR i -IP y y d 1 : j 1 e DD l dj y dy DD l G R 0. d y y E y y μ y, dy c y N c ex k DD E Budary Cditis at Q/-IP iterface: j y d j. l 1 E l + hermiic Emissi Leakage Curret: j j j. Diffusi-Recmbiati Leakage Curret i the QNR y y : E l l E l E y DR DR c j j ex l l, k DD DR Budary cditi at the QNR edge: j y y j y y. l l

11 EXPRESSIONS OR ELECRON LEAKAGE CURRENS k ev R L N ed Deb c DD l DR l DD l E l l he Smallest Curret Defies the Leakage! + Δ k ev k E e Q Q c c e l l ex ex k ev A E l ex α cth c DR l L d N L D e j - hermiic Emissi Leakage Curret. - Drift-Diffusi Leakage Curret. - Diffusi-Recmbiati Leakage Curret.

12 SE O EQUAION OR ACIVE LAYER Ctiuity Equati i Q-layer: 1 d Q e dy R, l Electr Recmbiati Curret: r l tal Recmbiati Curret: Q r Q Q Q Q r Q R, ed Charge Quasi-Neutrality: Q Q Q Q d 1 d l y he set f similar equatis/exressis are derived fr the ijecti ad leakage currets f the hles. he btaied equatis ctai three ukw variables: he height f the barriers V ad V, ad the electr ermi level Q.

13 SE O EQUAION OR V, V, Q he btaied equatis ctai three ukws: ev, ev, ad Q. herefre there shuld be three equatis:,, Q Q r Q l Q V V,, Q Q r Q l Q V V v c IP g ev ev E E E ev + + ] [ ] [ he third equati was btaied usig the flat-bad cditis see eergy bad diagram.

14 INCORPORAION O LASING REGIME [ ] [ ] [ ].,, 1,, 0, /,,,,,,, k g th L g th th k s k s P c G G c G P G R ed P G R ed + ω ω ω α ω ω ω ω ω α ω ω ω A. Griberg, 1994

15 NUMERICAL VALUES O CHARACERISIC CURRENS OR IP/IGaAsP LASER DR l DD l E l l / cm A DR l E DD j j ; / k ev cm A DD ; A/cm E. / k ev cm A DD. A/cm E ; / k ev R cm A DD l. / k ev R cm A DD l ; A/cm E l. A/cm E l / cm A DR l [ ], ~ ~ ~ B ~ ~ A ~ cm A Rad SRH C Aug Q r ~ cm. / 10 ~ ; / 10 ~ ; 10 ~ s cm C C cm s B B s A A. 16 ~, ~ 1, ~ C B A

16 SIMLIIED ANALYICAL IV CHARACERISICS O DH LASER DIODE APPROXIMAIONS: Neglectig Leakage Currets Neglectig Ijecti Heatig Assumig hermiic Emissi is a bttleeck A A e E k IP g 1 e Δ c + k Q ev 1/ e ev k.

17 RESULS ROM ANALYICAL HEORY: I-V CHARACERISICS I-V Characteristics Vertical Leakage Curret tal Curret ka/cm K 350K Electr Leakage Curret ka/cm Ade Bias V Ade Bias V

18 RESULS ROM ANALYICAL HEORY: CARRIER DENSIY Carrier Desity cm K 350K Carrier Desity cm Ade Bias V Curret Desity ka/cm

19 RESULS ROM ANALYICAL HEORY: V, V BARIERS K: V V 0.5 Barrier Height ev K: V V Barrier Height ev Ade Bias V Curret Desity ka/cm

20 DESSIS SIMULAION IDEAL 1D CASE here is a very gd agreemet with aalytical results if vltage dr i IP is take it accut Leakage Curret 1e005 talcurret Ade Curret A/mkm e006 6e006 4e006 e006 Electr Leakage Curret A/mkm Ade Bias V

21 DESSIS D SIMULAION Belw hreshld Results fr three differet gemetries with the same material arameters: D as i the igure; 1D hythetical device f cstat width w3μm; Quasi-1D Q-IGaAsP width is w3μm P-ctact N-ctact 0 X Y Axis f Symmetry 1.5μm -IP Q-IGaAsP -IP μm 0.4μm 0 μm 8 μm Ade Curret A/mkm Laser1_D Laser_1D Laser3_Q1D he I-V characteristics are almst Idetical fr biases belw 1.5 V Ade Bias V

22 EEC O ELECRON/HOLE MOBILIIES IN Q-LAYER ON I-Vs Laser1 mb_h100, mb_e4000 Laser1a mb_h00, mb_e4000 Laser1b mb_h50, mb_e4000 Ade Curret A/mkm Laser1c mb_h10, mb_e4000 Laser1d mb_hmb_e4000 Laser1e mb_h4000, mb_e100 Laser1f mb_hmb_e Ade Bias V

23 EEC O ELECRON/HOLE MOBILIIES IN Q-LAYER CARRIER DISRIBUION Distributi f the carriers alg Y-directi i the crss-secti lae at X0.075μm. Q1D Laser1 mb_h100, mb_e4000 Laser1a mb_h00, mb_e4000 Laser1b mb_h50, mb_e4000 Laser1c mb_h10, mb_e4000 Laser1d mb_hmb_e4000 Laser1e mb_h4000, mb_e100 Laser1f mb_hmb_e100

24 EEC O ELECRON/HOLE MOBILIIES IN Q-LAYER ON CARRIER DISRIBUION Distributi f the carriers alg X-directi i the crss-secti lae at Y.μm middle f Q. Laser1 mb_h100, mb_e4000 Laser1a mb_h00, mb_e4000 Laser1b mb_h50, mb_e4000 Laser1c mb_h10, mb_e4000 Laser1d mb_hmb_e4000 Laser1e mb_h4000, mb_e100 Laser1f mb_hmb_e100 Edge f Q-Layer

25 CONDUCION BAND PROILE ALONG Q-LAYER Cducti bad rfile alg X-directi i the crss-secti lae at Y.μm middle f Q. Laser1 mb_h100, mb_e4000 Laser1a mb_h00, mb_e4000 Laser1b mb_h50, mb_e4000 Laser1c mb_h10, mb_e4000 Laser1d mb_hmb_e4000 Laser1e mb_h4000, mb_e100 Laser1f mb_hmb_e100

26 BUIL-IN ELECRIC IELD IN Q-LAYER: X-COMPONEN Built-i electric field E x alg X-directi i the crss-secti lae at Y.μm middle f Q. Laser1 mb_h100, mb_e4000 Laser1a mb_h00, mb_e4000 Laser1b mb_h50, mb_e4000 Laser1c mb_h10, mb_e4000 Laser1d mb_hmb_e4000 Laser1e mb_h4000, mb_e100 Laser1f mb_hmb_e100

27 BUIL-IN ELECRIC IELD IN Q-LAYER: Y-COMPONEN Built-i electric field E y alg X-directi i the crss-secti lae at Y.μm middle f Q. Laser1 mb_h100, mb_e4000 Laser1a mb_h00, mb_e4000 Laser1b mb_h50, mb_e4000 Laser1c mb_h10, mb_e4000 Laser1d mb_hmb_e4000 Laser1e mb_h4000, mb_e100 Laser1f mb_hmb_e100

28 REDISRIBUION O AUGER RECOMBINAION RAE Auger recmbiati rate alg X- directi i the crss-secti lae at Y.μm middle f Q. Laser1 mb_h100, mb_e4000 Laser1a mb_h00, mb_e4000 Laser1b mb_h50, mb_e4000 Laser1c mb_h10, mb_e4000 Laser1d mb_hmb_e4000 Laser1e mb_h4000, mb_e100 Laser1f mb_hmb_e100

29 DISRIBUION O ELECRON CURREN DENSIY IN Q-LAYER: ABSOLUE VALUE tal electr curret decity alg X- directi i the crss-secti lae at Y.μm middle f Q. Laser1 mb_h100, mb_e4000 Laser1a mb_h00, mb_e4000 Laser1b mb_h50, mb_e4000 Laser1c mb_h10, mb_e4000 Laser1d mb_hmb_e4000 Laser1e mb_h4000, mb_e100 Laser1f mb_hmb_e100

30 DISRIBUION O ELECRON CURREN DENSIY IN Q-LAYER: LONGIUDINAL COMPONEN X-cmet f ecurret desity alg X- directi i the crss-secti lae at Y.μm middle f Q. Laser1 mb_h100, mb_e4000 Laser1a mb_h00, mb_e4000 Laser1b mb_h50, mb_e4000 Laser1c mb_h10, mb_e4000 Laser1d mb_hmb_e4000 Laser1e mb_h4000, mb_e100 Laser1f mb_hmb_e100

31 DISRIBUION O ELECRON CURREN DENSIY IN Q-LAYER: NORMAL COMPONEN Y-cmet f ecurret desity alg X- directi i the crss-secti lae at Y.μm middle f Q. Laser1 mb_h100, mb_e4000 Laser1a mb_h00, mb_e4000 Laser1b mb_h50, mb_e4000 Laser1c mb_h10, mb_e4000 Laser1d mb_hmb_e4000 Laser1e mb_h4000, mb_e100 Laser1f mb_hmb_e100

32 DISRIBUION O ELECRON CURREN DENSIY IN Q-LAYER: ABSOLUE VALUE tal hle curret desity alg X- directi i the crss-secti lae at Y.μm middle f Q. Laser1 mb_h100, mb_e4000 Laser1a mb_h00, mb_e4000 Laser1b mb_h50, mb_e4000 Laser1c mb_h10, mb_e4000 Laser1d mb_hmb_e4000 Laser1e mb_h4000, mb_e100 Laser1f mb_hmb_e100

33 DISRIBUION O ELECRON CURREN DENSIY IN Q-LAYER: LONGIUDINAL COMPONEN X-cmet f hle curret desity alg X- directi i the crss-secti lae at Y.μm middle f Q. Laser1 mb_h100, mb_e4000 Laser1a mb_h00, mb_e4000 Laser1b mb_h50, mb_e4000 Laser1c mb_h10, mb_e4000 Laser1d mb_hmb_e4000 Laser1e mb_h4000, mb_e100 Laser1f mb_hmb_e100

34 DISRIBUION O ELECRON CURREN DENSIY IN Q-LAYER: NORMAL COMPONEN Y-cmet f hle curret desity alg X- directi i the crss-secti lae at Y.μm middle f Q. Laser1 mb_h100, mb_e4000 Laser1a mb_h00, mb_e4000 Laser1b mb_h50, mb_e4000 Laser1c mb_h10, mb_e4000 Laser1d mb_hmb_e4000 Laser1e mb_h4000, mb_e100 Laser1f mb_hmb_e100

35 ELECRON CURREN DISRIBUION IN -IP tal electr curret decity alg X- directi i the crss-secti lae at Y.405μm -ip/q I. Laser1 mb_h100, mb_e4000 Laser1a mb_h00, mb_e4000 Laser1b mb_h50, mb_e4000 Laser1c mb_h10, mb_e4000 Laser1d mb_hmb_e4000 Laser1e mb_h4000, mb_e100 Laser1f mb_hmb_e100

36 CALCULAION O VERICAL LEAKAGE CURRENS HOLES IN 0; 0;.4 Q-LAYER Active Part Uder the Ridge HOLES OU 1.5; ELECRONS IN 1.5;.4 ELECRONS IN

37 LONGIUDINAL ELECRON AND HOLE CURRENS A HE RIGH EDGE O Q-LAYER Laser1: μ 4000 cm /Vs, μ 100 cm /Vs Distributi f X-cmets f e/h curret desity alg i the crss-secti lae at X1.501 μm.

38 NORMAL ELECRON AND HOLE CURRENS A HE OP AND BOOM O Q-LAYER Laser1: Y-cmet f the hle curret desity i the crss-secti lae Y1.999 μm i -IP just abve Q, ad Y-cmet f the electr curret desity i the crss-secti lae Y.399 i -IP just belw Q: HOLES ELECRONS

39 ESIMAED VALUES O ALL CURRENS Cavity Legth 500 μm. I y 17 ma HOLES IN 0; 0;.4 Q-LAYER Active Part Uder Ridge I x 4 ma HOLES OU 1.5; I x 45 ma ELECRONS IN 1.5;.4 ELECRONS IN I y 85 ma he currets ad their ratis strgly deed the hle mbility i Q-Layer. Whe hle mbility decreases the rati f I x electr/i x hle icreases ad the rati f I x hle/i y hle decreases.

40 EEC O LASING We csider tw gemetries: Laser1 with the active layer badga f ev ad with the thickess 0.4 μm. Laser has the same material arameters as Laser1, but the active layer badga is 0.8 ev ad the thickess is 0.14 μm. he rest f Q-layer is a symmetric SCH. he ttal thickess f Q-Layer is als 0.4 μm. Laser1 Laser

41 EEC O LASING ON ELECRON-HOLE PLASMA Crss-secti at Y. μm Laser1 W/ Lasig Laser1 With Lasig Laser W/ Lasig Laser With Lasig

42 HE SAME Larger Scale Crss-secti at Y. μm Laser1 W/ Lasig Laser1 With Lasig Laser W/ Lasig Laser With Lasig

43 EEC O LASING ON BAND EDGES Crss-secti at Y. μm Laser1 W/ Lasig Laser1 With Lasig Laser W/ Lasig Laser With Lasig

44 EEC O LASING ON OAL ELECRON CURREN DENSIY IN Q-LAYER Crss-secti at Y. μm Laser1 W/ Lasig Laser1 With Lasig Laser W/ Lasig Laser With Lasig

45 EEC O LASING ON X- AND Y-COMPONENS O ELECRON CURREN DENSIY IN Q-LAYER Electrs: Crss-secti at Y. μm X-cmet Y-cmet Laser1 W/ Lasig Laser1 With Lasig Laser W/ Lasig Laser With Lasig

46 EEC O LASING ON OAL HOLE CURREN DENSIY IN Q-LAYER Crss-secti at Y. μm Laser1 W/ Lasig Laser1 With Lasig Laser W/ Lasig Laser With Lasig

47 EEC O LASING ON X- AND Y-COMPONENS O HOLE CURREN DENSIY IN Q-LAYER Hles: Crss-secti at Y. μm X-cmet Y-cmet Laser1 W/ Lasig Laser1 With Lasig Laser W/ Lasig Laser With Lasig

48 EEC O LASING ON CARRIER DISRIBUION ACROSS Q-LAYER Crss-secti at X0.001 Laser1 W/ Lasig Laser1 With Lasig Laser W/ Lasig Laser With Lasig

49 SUMMARY Aalytical 1D macrscic thery f carrier trasrt i lasers rvides very gd arximati fr calculati f I-V characteristics. he 1D thery is helful i evaluati f the bttle-ecks f the electr ijecti ad vertical leakage currets. he thery has gd agreemet with crresdig results frm umerical mdellig here we used DESSIS Sysys/ISE. Hwever, i site f gd agreemet, the 1D thery is abslutely ialicable fr reliable aalysis f carrier distributis, eergy bad ad electric field rfiles. he alicability f 1D arximati becmes eve mre rblematic abve threshld whe lasig has very strg effect.

50 HANK YOU VERY MUCH!