Complex Permittivity and Refractive Index for Metals

Aaron Webster

Last Update: November 9, 2012

The following figures plot the frequency dependent complex permittivity ε = ε′ + iε′′ and refractiveindex n = n′ + in′′ for silver, aluminum, gold, copper, chromium, nickel, tungsten, titanium, beryl-lium, palladium, and platinum using either the Drude (Equation 3), Lorentz-Drude (Equation 4), orBrendel-Bormann (Equation 5 models in the range 200 nm to 2000 nm. The material parameters andmathematical formalism detailed in [1]. These tables are generated programmatically. Three differentmodels for the complex permittivity are tabulated. The first two are the Drude and Lorentz-Drude (LD)models.

εD = εD (1)

εLD = εD + εL (2)

where εD is contribution from the Drude model, representing free electron effects

εD = 1−√f0 ω

′2p

ω(ω − iΓ′0)(3)

and εL is the Lorentz contribution, representing the bound electron effects

εL =

k∑j=0

fjω′2p

ω′2j − ω2 + iωΓ′j(4)

The third is the Brendel-Bormann model which is based instead on an infinite superposition of oscillators

εBB =1√

2π σn

∫ ∞−∞

exp

(− (x− ω′n)

2σ2n

)fjω

2p

(x2 − ω2) + iωΓ′ndx (5)

1

0.1 Ag

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

−150

−100

−50

0

wavelength (µm)

ε′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

5

10

15

wavelength (µm)

ε′′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

0.5

1

1.5

wavelength (µm)

n′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

5

10

15

20

wavelength (µm)

n′′

D LD BB

Figure 1: Material parameters for Ag based on the Drude, Lorentz-Drude, and Brendel-Bormann models.

2

0.2 Al

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

−400

−300

−200

−100

0

wavelength (µm)

ε′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

20

40

60

80

wavelength (µm)

ε′′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

1

2

3

wavelength (µm)

n′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

10

20

30

wavelength (µm)

n′′

D LD BB

Figure 2: Material parameters for Al based on the Drude, Lorentz-Drude, and Brendel-Bormann models.

3

0.3 Au

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

−150

−100

−50

0

wavelength (µm)

ε′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

5

10

15

20

wavelength (µm)

ε′′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

0.5

1

1.5

wavelength (µm)

n′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

5

10

15

wavelength (µm)

n′′

D LD BB

Figure 3: Material parameters for Au based on the Drude, Lorentz-Drude, and Brendel-Bormann models.

4

0.4 Cu

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

−200

−150

−100

−50

0

wavelength (µm)

ε′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

10

20

wavelength (µm)

ε′′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

0.5

1

1.5

wavelength (µm)

n′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

5

10

15

20

wavelength (µm)

n′′

D LD BB

Figure 4: Material parameters for Cu based on the Drude, Lorentz-Drude, and Brendel-Bormann models.

5

0.5 Cr

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

−40

−20

0

wavelength (µm)

ε′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

10

20

30

40

50

wavelength (µm)

ε′′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

1

2

3

4

wavelength (µm)

n′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

2

4

6

8

10

wavelength (µm)

n′′

D LD BB

Figure 5: Material parameters for Cr based on the Drude, Lorentz-Drude, and Brendel-Bormann models.

6

0.6 Ni

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

−60

−40

−20

0

wavelength (µm)

ε′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

20

40

60

wavelength (µm)

ε′′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

1

2

3

4

wavelength (µm)

n′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

2

4

6

8

10

12

wavelength (µm)

n′′

D LD BB

Figure 6: Material parameters for Ni based on the Drude, Lorentz-Drude, and Brendel-Bormann models.

7

0.7 W

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

−100

−80

−60

−40

−20

0

wavelength (µm)

ε′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

10

20

30

wavelength (µm)

ε′′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

1

2

3

4

wavelength (µm)

n′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

5

10

wavelength (µm)

n′′

D LD BB

Figure 7: Material parameters for W based on the Drude, Lorentz-Drude, and Brendel-Bormann models.

8

0.8 Ti

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

−20

−15

−10

−5

0

wavelength (µm)

ε′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

10

20

30

40

wavelength (µm)

ε′′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

1

2

3

4

wavelength (µm)

n′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

2

4

6

wavelength (µm)

n′′

D LD BB

Figure 8: Material parameters for Ti based on the Drude, Lorentz-Drude, and Brendel-Bormann models.

9

0.9 Be

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

−80

−60

−40

−20

0

wavelength (µm)

ε′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

10

20

30

40

wavelength (µm)

ε′′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

1

2

3

wavelength (µm)

n′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

5

10

wavelength (µm)

n′′

D LD BB

Figure 9: Material parameters for Be based on the Drude, Lorentz-Drude, and Brendel-Bormann models.

10

0.10 Pd

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

−80

−60

−40

−20

0

wavelength (µm)

ε′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

20

40

60

80

wavelength (µm)

ε′′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

1

2

3

4

wavelength (µm)

n′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

5

10

15

wavelength (µm)

n′′

D LD BB

Figure 10: Material parameters for Pd based on the Drude, Lorentz-Drude, and Brendel-Bormannmodels.

11

0.11 Pt

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

−80

−60

−40

−20

0

wavelength (µm)

ε′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

20

40

60

80

wavelength (µm)

ε′′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

2

4

6

wavelength (µm)

n′

D LD BB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

·10−6

0

5

10

wavelength (µm)

n′′

D LD BB

Figure 11: Material parameters for Pt based on the Drude, Lorentz-Drude, and Brendel-Bormann models.

12

References

[1] A.D. Rakic, A.B. Djurisic, J.M. Elazar, and M.L. Majewski. Optical properties of metallic films forvertical-cavity optoelectronic devices. Applied Optics, 37(22):5271–5283, 1998.

13