9. Light-matter interactions - metals

Transcription

1 9. Light-matter interactions - metals Complex dielectric function and complex refractive index The Drude model Optical properties of metals - skin depth - reflectivity

2 A few announcements Reminder: no class on Monday (Winter recess) Problem set 3 due Friday. Problem set 4 will be posted on Friday afternoon. It will include a note reminding you to sign up for an exam time slot. Sign-up for Exam 1 will be posted some time on Friday. AFTER it is posted: - You need to Dr. Mittleman to sign up for a time slot. - only. No other method is acceptable. - Include a 1 st choice AND a 2 nd choice. - Time slots assigned on a first- , first-allocated basis. - Sign-up deadline is Friday Feb Exam dates: March 2-3. See sign-up page for more info. - IF YOU WILL BE UNAVAILABLE ON THOSE DATES: contact Dr. Mittleman IMMEDIATELY to arrange an alternative.

3 Electric field in a medium depends on n, z/2 Ezt (, ) E( z)e exp[ j( nkz t)] Absorption causes attenuation of the field with increasing propagation distance z Refractive index changes the k-vector and therefore the wavelength Frequency doesn t change! and the dielectric of the medium, med, is related to n: n c c med

4 Complex refractive index, z 2 jnk z E z t E z e e e jt E z e 2k e j j nk z j Define the complex refractive index: 2k Note the tilde! Complex index: this is just a convenient way to group n and in a single quantity. n n n n jt j 4 c j 2 k = k-vector in vacuum Often denoted with the Greek letter kappa,, a dimensionless quantity! Note: always! So n n j

5 Complex dielectric function If the refractive index is complex, what about? R I med 2 n n j j n j n Note: this sign convention is not universal - be careful! This can also be expressed as equations for n, : 1 n R 2 1 R 2

6 Example: complex refractive Index of water x-ray: n ~ 1 visible: n ~ 1.3 radio frequencies: n ~ 9 increasing frequency

7 Complex dielectric function vs complex index It is strictly correct that: Re Im Re Im n n It is NOT strictly correct that: is related only to the phase of the wave is related only to the attenuation of the wave is related only to the phase of the wave is related only to the attenuation of the wave But, if << n (as is often the case), then 2 Re n which is related only to the phase of the wave Note: it is NOT the case for metals

8 Does the forced oscillator model work for everything? We started with the assumption that the electrons are bound to nuclei. nucleus That s a good description of insulating materials. electron E applied (t) But not such a good description of metals, where at least some of each atom s electrons are free to move through the lattice.

9 Drude theory of metals The Drude Model of metallic conductivity (~19) Lattice of (immobile) atomic cores (+ charges) Electrons are free to roam throughout the lattice Every once in a while, each electron collides with something. This collision randomizes the electron s velocity. The model has one single free parameter: the average time between collisions, Typical value in a metal: ~ 1-14 seconds Paul Drude ( )

10 Drude theory of metals the movie

11 Drude conductivity of metals Suppose there is an applied electric field, E. Between each collision, the electron accelerates, due to F = ma = ee. average velocity v ave = acceleration time ee m e N electrons/m 3 The average current flowing through an area A in a time t: area A distance v ave t I ave number of electrons charge/electron N v t A ave time t e

12 Drude conductivity of metals Plug in v ave : I ave 2 Ne A Define average current density J = J ave Ne m 2 e E applied m Define: = Drude conductivity J E ave applied e E average current area Ne m 2 1 Units of m sec Cm volt 2 e This is Ohm s Law! me kg ohm meter Ne meter C sec 1 Note: 1 mho = 1 Siemens = (1 ohm) -1

13 Electric field in a metal What if the electric field is oscillating? E = E e j(kz t) Recall the inhomogeneous wave equation: E 1 E P z t t c In this case, we do not have a polarization P, but instead a flow of unbound electrons: a current! How do we handle this situation? What do we use instead of P(t)? Recall that P(t) is proportional to the electron s position: P(t) = N e x e (t)

14 Polarization in a metal P(t) = N e x e (t) But the current is proportional to the average velocity of the electrons: J(t) = N e v e (t) = dp/dt So, as a guess: Replace dp/dt with the current density J(t) = E(t): c E E J E z t t t (We will revisit this guess in our next lecture )

15 Wave equation for an E-field in a metal c E E E z t t This is almost the same situation we had when we first encountered the polarization (lecture #7). So solve it in a similar way. Assume that E has a time dependence of e jt, and that the z dependence is a slowly varying function multiplied by e jkz : E z, t E z e Plug this into the above equation, and take the necessary t and z derivatives. The resulting equation is: z 2 2 E j E z z c j kzt

16 Optical properties of metals z 2 2 E j E z z c This is the same as the usual wave equation in empty space, except for the factor in the parentheses. Thus, the solution is the same as the wave in empty space if we let the speed of light be modified: c modified speed: c 1 j c c n (complex) refractive index. complex index: n 1 j This looks just like. So we interpret the denominator as the But we have seen that a complex index is just a convenient way of grouping n and together.

17 Optical properties of metals n 1 j what is this value? From this we can find the values of n and : 4 nre n Im n which together tell us how waves propagate inside a metal: E zt, E z e 2k e j j nk z jt

18 For real metals, is a big number Example: Consider copper at a frequency of 1 MHz: 1 In that case, the complex refractive index is given by: j 4 1 j n 1 j j e 2 in which case, the real and imaginary parts are equal: 1 n for Cu at 1 MHz.

19 Skin depth If the conductivity is high (i.e., >>1) then from we derive the absorption coefficient: 2 c 2 2 c As we have seen, this is a very large number for metals. Skin depth or penetration depth : depth of propagation of light into a metallic surface = 1/ For metals, this depth is much less than the wavelength. Example: copper at = 1 MHz skin depth is 3.2 m, about /9,

20 Attenuation of waves entering a medium Vacuum (or air) Medium metallic medium dielectric medium

21 Real refractive index of metals If the conductivity is high (i.e., >>1) then the refractive index is also large: n for Cu at 1 MHz. We will soon learn that the reflectance of an object depends on its refractive index: n 1 R n 1 For Cu at 1 MHz, this is: R metals make very good mirrors 2

22 Failures of Drude theory There were several notable ways in which Drude theory fails. These failures could not be explained until quantum mechanics came along. Here s one: why is gold gold-colored and silver silver-colored? Here s another: Drude theory predicts So, a tri-valent atom (like titanium Ti +3 ) should be a much better conductor than a monovalent atom (like sodium, Na +1 ). Ne m That is, the conductivity should increase if the number density of electrons increases. 2 e But it s not: Ti mho/cm Na mho/cm