Dielectic function Phys 774: Ellipsomety Optical vibations (phonons) Fee electons (plasma) Electonic tansitions (valence conduction band) Dielectic function and efactive index ae geneally complex: ε ε ' + iε '' ; ; ε ' n R -n I ; ε '' n R n I absoption coefficient α k 0 n I n I - extinction coefficient Fall 007 1 ε( ω) 1 + χ ( ω) + χ ( ω) + χ ( ω) ε + χ ( ω) + χ ( ω) χ ( ω) χ ph FC ( ω) Ph FC E Ph FC S j j ωto ω + iωγ j j Dielectic function contibutions Ω p ω + i ωγ ne Ω p ε m 0 * Spectoscopic Measuements in IR Fouie Tansfom Infa-Red Spectoscopy ELLIPSOMETRY 30 10000 cm -1 χ ( ω) E P j j ω 0 j ω + iωγ j 3 Technique that detemines the change in polaization state of light eflected fom a sample. Typically used fo chaacteizing thin films, stongly coelated mateials. Used in a wide ange of industial applications. Non contact, non destuctive method. 4
Pinciples and goals of Ellipsomety Ellipsomety and Gating Spectometes ρ p φ vaiable 0 s PMT o Photodiode Disadvantages: Cannot use lenses due to Depolaization effects Nomalization is difficult due to complicated esponse of the gating Ellipsomety: Linea polaized light is incident on the sample unde an angle φ 0 and polaization of the eflected light is measued. 5 6 Ellipsomety and Gating Spectometes PMT o Photodiode Disadvantages hee? Resolution deceases if we cannot illuminate all goves of the gating Spectoscopic Ellipsomete SOPRA GESP5 SOPRA vaiable angle spectoscopic ellipsomete GESP5 7 8
FT-IR Ellipsomety Advantages: 1. Multiplexing. Thoughput (no slits needed) Types of Ellipsomete Ellipsometes may be single wavelength o spectoscopic They may opeate as otating element o as nulling ellipsometes They may be single point o imaging IR-bolomete 9 10 Plane of Incidence Plane of incidence of light is the plane containing the nomal to the sample suface and the incident diection of light. The s-plane is pependicula to the plane of incidence. The p-plane is paallel to the plane of incidence. 11 Fesnel s equations give the complex amplitude of the eflected light y x Polaization P I y + I x E Jones linea Fesnel s Equations I cosα E sinα 1 0 Jones E cicula E0 i I Jones Fomalism ncosφ ncosφ 1 i y y y e δ n1cosφ + ncosφ φ ncosφ ncosφ 1 i x x x e δ ncos φ + ncosφ 1 φ Jones J J J 11 1 J1 J y ϕ φ Optical elements (polaizes, mios, etc) E J J E x 11 1 0x E y J E 1 J 0 y 1 x
p i ρ tan Ψ e s Ellipsometic Equations ε sin φ sinφ tan φ 0 + 0 0 p φ0 vaiable, tan Ψ, δp δs s 1 ρ 1+ ρ ε is a function of fequency (enegy) in eal expeiment we need to connect ρ with the polaize angles of the instument using, e.g., Jones matices Pseudo-dielectic function, which coincides with dielectic function fo unifom and isotopic bulk mateials Anisotopy can be measued diectly by changing φ 0 13 What does an ellipsomete measue? The polaization change of light eflected fom a sample in tems of two paametes and Ψ These values ae elated to the atio of the Fesnel eflection coefficients fo p- and s- polaized light p s tan Ψ e This atio is complex tanψ measues the atio of the modulus of the amplitude eflection atio The phase diffeence between p- and s-polaised eflected light is given by i 14 What can ellipsomety tell us about samples? Taken on thee own values of and Ψ tell us little about a sample. We take the measued values of and Ψ, typically as a function of wavelength and angle of incidence (Vaiable Angle Spectoscopic Ellipsomety). An optical model is then built using as much infomation about the sample as possible What cannot be diectly detemined fom ellipsomety measuement? Select the model and Optical Functions The optical model will geneally consist of The thickness of a film o laye The complex efactive index N n in i Sometimes, the complex dielectic constant ε ε iε i is given, whee N ε Fo a Mixtue of mateials use the Effective Medium Appoximation Multilaye sample stuctue (it is not a foensic tool) 15 16
Extacting data in ellipsomety Acquie data of spectal ange and angle of incidence equied Build an optical model that descibes sample stuctue. Geneate theoetical data fom optical model. Compae expeimental and theoetical data Vay model paametes, such as film thickness, until a best fit to expeimental data is obtained. measue cos and tanψ constuct optical model compae data and model Global minimum. pint model dat a 17 Regession Regession algoithms ae used to vay unknown paametes and minimize the diffeence between model and expeimental data. The Levenbeg-Maquadt algoithm is the most widely used. The algoithm seeks to minimise the function χ 1 N M { ( tanψ measued tanψ model ) + ( cos measued cos model ) } N whee N is the numbe of data points at the diffeent measued wavelengths and M the numbe of model paametes. 18 Fit the data Regession Analysis The data ae then compaed with the data geneated fom the theoetical model. Unknown paametes in the optical model, such as film thickness o optical constants, ae vaied to ty to poduce a best fit to expeimental data. Regession algoithms, such as Levenbeg-Maquadt, ae used to vay unknown paametes and minimize the diffeence between expeimental and modelgeneated data. Physical paametes ae obtained once a good fit is achieved. Advantages of Ellipsomety It measues the atio of two values so is highly accuate and epoducible, does not need a efeence sample, and is not so susceptible to light souce fluctuation Since it measues phase, it is highly sensitive to the pesence of ulta-thin films (down to submonolaye coveage). It povides two pieces of data at each wavelength. Moe film popeties can be detemined. 19 0
Some applications of Ellipsomety: Data stoage: Chaacteization of thin films used the manufactuing of stoage media Display technology: Chaacteisation of mateials used in flat panel displays Optical coatings: Study of coatings on filtes, beam splittes and othe optical components. Biology / Chemisty: Study of oganic semiconductos, and tissue samples. In-situ measuement: Allows eal time monitoing of etching and thin film deposition pocesses. 1 What is fa-ir Ellipsomety? Fa-Infaed Ellipsomety is a technique which allows one to measue vey accuately and with high epoducibility the complex dielectic function ε(ω) ε(ω) 1 + i ε(ω) of oxide thin films and single cystals. It measues the change in polaization of Infaed light upon non-nomal eflection on the suface of a sample to be studied. To extend the Ellipsomety technique to the Fa-Infaed pat of the electomagnetic spectum, we ae going to cay out these expeiments at Bookhaven National Laboatoy, National Synchoton Light Souce. Synchoton light povides thee odes of magnitude moe billiant light in the Fa-Infaed as compaed to conventionally available light souces, like mecuy ac lamps. Example of fa-ir spectoscopic Ellipsomety: Hadening of the soft modes in STiO 3 thin films Anothe example: Sensitivity to thin film Ellipsomety of STiO 3 thin films Bae Si Bae Si Si + 1nm oxide A.A. Sienko, et al., Natue 404, 373 (000) 3 4
Poblems with tanspaent samples Poblems with tanspaent samples The eo in is popotional to cosec Fo tanspaent samples, ~ 180 0 This can be ovecome by placing a etade in font of the light souce to intoduce a phase shift into the light. Altenatively, a otating compensato ellipsomete could be used 5 6 Poblems with tanspaent samples Hafnium oxide on silicon Poo fit 7 8
Hafnium oxide on silicon Patially oxidised sacificial a-si on SiC Excellent fit with paameteisation 9 30 Patially oxidised sacificial a-si on 4H-SiC Model (SiO database asi Foouhi) Ellipsomety of conventionally gown oxide on 4H-SiC 10 8 ε i unoxidised, etched ε i oxidised, etched ε unoxidised, etched ε oxidised, etched SiO (1.41176 µm) 6 ε 4 asi + SiO with density gadient (7.9 nm) asi (64.7 nm) SiC substate 31 0 1 3 4 5 Photon Enegy ev 3
Vey thin films Fo films of thickness about 10 nm o less, thee exists a coelation between thickness and efactive index Complementay techniques must be used to measue some paametes, usually thickness. GIXR employed 33 Gazing Incidence X-ay Reflectomety X-ays used in XRR have wavelengths of the ode of 1Å. Since the efactive index of mateials at these shot wavelengths is slightly less than unity, x-ays ae totally extenally eflected at angles of incidence typically up to a few tenths of a degee. As the incidence angle inceases above the citical angle fo total eflection, x-ays penetate the layes of the mateial. 34 Gazing Incidence X-ay Reflectomety vs. Ellipsomety At the inteface between each laye, some of the adiation is eflected to the laye above, while some is tansmitted (efacted) to the laye below. The wave intefeence poduces a seies of finges in the eflectivity cuve. Laye densities, thickness and inteface oughness can be detemined fom the eflectivity cuve. Ellipsomety of conventionally gown oxide on 4H-SiC tan (ψ) 0.50 0.45 0.40 0.35 Best fit using non-oxidised substate tan (ψ) data tan (ψ) best fit cos ( ) data cos ( ) best fit 0.9 0.8 0.7 cos ( ) 0.30 0.5 0.6 0.0 0. 0.3 0.4 0.5 0.6 0.7 0.8 Wavelength (µm) 0.5 35 36
Ellipsomety of conventionally gown oxide on 4H-SiC 0.50 0.45 Best fit using etched themal oxide as substate 0.9 Anothe example: Compaison of XTEM and Spectoscopic Ellipsomety tan (ψ) 0.40 0.35 0.30 tan (ψ) data tan (ψ) best fit cos ( ) data cos ( ) best fit 0.8 0.7 cos ( ) 0.5 0.6 0.0 0. 0.3 0.4 0.5 0.6 0.7 0.8 Wavelength (µm) 0.5 37 38