9 & 10. Light-matter interactions - metals

Drude model and the wave equation

Complex dielectric function and complex

refractive index

Optical properties of metals

- skin depth-reflectivity

Plasma frequency of a metal

Drude model and the wave equation

2 2 2

022 2 20

1

E E P

z t tc

When we are considering metals, there is no “polarization,”but instead there is current – flowing electrons!

So what goes here?

We make a guess that can be related to

and we use Drude’s result which is equivalent to Ohm’s Law:

P N e x t J N e v t

0J t E t

Wave equation for an E-field in a metal2 2

0 022 20

1

E E E

z t tc

This wave equation is similar to what we encountered previously. So we solve it in a similar way.

Plug this into the above equation, and take the necessary t and zderivatives. The resulting equation is:

2 2

0 002 2

0 0

1

E zj E z

z c

Assume that E has a time dependence of ejt, and that the zdependence is a function E0(z) multiplied by ejkz:

0, j kz tE z t E z e

2 2

0 002 2

0 0

1

E zj E z

z c

Optical properties of metals

Thus, the solution is the same as the wave in empty space if we let the speed of light be modified:

0

0

0

1

c

j

modified speed: c

This looks just like . So we interpret the denominator as the (complex) refractive index.

0

ccn

0

0 0

1 n j

complex index:

This is the same as the usual wave equation in empty space, except for the factor in the parentheses.

Wait a sec… complexrefractive index??

Often denoted with the Greek letter kappa, , a dimensionless quantity!

Note: 0 always!

Complex refractive index

020, 0 jnk zz j tE z t E z e e e

Define the complex refractive index: 0

0

0

2

4

2

jn nk

n j

cn j

0

020 0

jj n k zk j tE z e e

k0 = k-vector in vacuum

Note the tilde!

So n n j

So a complex refractive index is just a convenient way of grouping n and together into one quantity.

Go back for a minute to the expression for waves in a dielectric medium (lecture 7):

Complex dielectric functionIf the refractive index is complex, what about ?

20

20

med n

n j

2 20 2R Ij n j n

Note: this sign convention is not universal - be careful!

0

0

12

12

R

R

n

This can also be inverted to give equations for n, :

what is this value?0

0 0

1 n j

Optical properties of metals

For a k-vector with a real and an imaginary part, the imaginary part gives rise to attenuation.

Absorption coefficient is: 02 Im 2 Imk k n

From this we can find the values of n and :

0

4Re Imn n n

which together tell us how waves propagate inside a metal:

0

0

0

200

0 00 0

,jj n k zkz j t

jnk zz j t z jkz j t

E z t E e e

E e e E e e

0k nk

For metals we found:

For real metals, is a big number, especially at lower frequencies

Example: Consider copper at a frequency of 100 MHz:

100

0

10

In that case, the complex refractive index is given by:

40 0 0 0

0 0 0 0

11

2

j jn j j e

in which case, the real and imaginary parts are equal:

0

02 n

47.32 10 for Cu at 100 MHz.

Skin depthIf the conductivity is high (i.e., >>1) then from we derive the absorption coefficient:

02

0 0 0

22

c 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 = 100 MHzskin depth is 3.2 m, about 0/900,000

Vacuum (or air)

dielectric medium

metallic medium

Medium

Attenuation of waves entering a medium

Real refractive index of metalsIf the conductivity is high (i.e., >>1) then the refractive index is also large:

0

02 n

47.32 10 for Cu at 100 MHz.

For Cu at 100 MHz, we find: 0.9999455R

metals make very good mirrors

We will soon learn that the reflectance of an object depends on its refractive index:

211

nRn

When is our guess likely to be wrong?Because 0 is REAL, our guess, J(t) = 0E(t), implies that the current is always in phase with the incident electromagnetic wave!

Recall: in the Drude model, the electrons are free to move - they are not bound to atoms by “springs”.

So, for low or moderate frequencies, this guess is ok.

But at a high enough frequency, it MUST fail.

2 2 2

022 2 20

1

E E P

z t tc

So, we are back to the inhomogeneous wave equation.

what goes here?

The “polarization” when there is currentLet’s go back to our forced oscillator model. Newton’s law F=ma gave us:

22 002 2 e e e j t

ee

d x t dx t eEx tdt dt m

force due to the incident light field

For a Drude metal, there is no spring holding the electrons. So what if we take 0 = 0?

2

2 20

/( )2

eNe mP t E tj

From this, we found the polarization:(see lecture 7)

resonant frequency of the spring

20

spring constant

em frictional

damping

The plasma frequency

2

2

/( )2

eNe mP t E tj

We now use this as a ‘new and improved’guess: plug this in for the polarization term in the wave equation.

Define a new constant, the “plasma frequency” P:2

2

0

Pe

Nem

2

0 2( )2

PP t E tj

Thus

Back to the wave equation22 2 2 2

0 0 022 2 2 2 20

12

PE E P E

z t t j tc

But this is the same as a problem we solved earlier:

22 2

22 2 20

1 1 02

PE E

z j tc

This is the wave equation for a wave propagating in a uniform medium, if we define the refractive index of the medium as:

2

221

2P

metalnj

So this must be the (complex) refractive index for a metal.

The new and improved result

Instead of making a ‘guess’ that

we make the better guess that 2

0 2( )2

PP t E tj

and then the Drude model (plus the wave equation) predict the optical properties of metals as:

2

212

Pmetaln

j

0 ,dP E tdt

2

0 212

Pmetal j

or

At low frequencies, << , this new result gives the same answer as our first guess, as long as we identify (the Drude scattering time) with 1/2.

High frequency dielectric of metalsHow does this dielectric function behave at higher frequencies, e.g., >> ?

2

0 21

P

In this limit, we find:

The dielectric function becomes purely a real number.And, it is negative below the plasma frequency and positive above the plasma frequency.

Some numbers:

Recall from Drude theory, that ~ 10-14 sec, so ~ 1/ ~ 1014 Hz.

For a typical metal, P is 100 or even 1000 times larger.

(corresponding to the frequency of infrared light)

(corresponding to the frequency of ultraviolet light)

2

0 212

P

j

A plot of Re() and Im() for illustrative values:

• imaginary part gets very small for high frequencies• real part has a zero crossing at the plasma frequency• real and imaginary parts are equal in magnitude at

Dielectric function of metals

0 2000 4000 6000 8000 10000

0-1-2-3-4-5

12345

Frequency (cm-1)

(

)/0 Re()Im()

linear scale

P = 4000 cm-1

= 20 cm-1

P

10-3

10-1

101

103

105

(

)/0

100 101 102 103 104

Frequency (cm-1)

log scale

P

Re()

Re()

Im()

105

10-5

2

0 212

P

j

High frequency optical properties

2

21 Pn

In the regime where >> , we find:

For frequencies below the plasma frequency, n is complex, so the wave is attenuated and does not propagate very far into the metal.

For high frequencies above the plasma frequency, n is real. The metal becomes transparent! It behaves like a non-absorbing dielectric medium.

reflectivity drops abruptly at the plasma frequency

This is why x-rays can pass through metal objects.

Another example: the ionosphere

the uppermost part of the atmosphere, where many of the atoms are ionized. There are a lot of free electrons floating around here…

2

0

2 9 MHz Pe

Nem

For N ~ 1012 m-3, the plasma frequency is:

Radiation above 9 MHz is transmitted, while radiation at lower frequencies is reflected back to earth.

That’s why AM radio broadcasts can be heard very far away.

Drude theory: it works pretty well

real / 0

imag / 0

large and negative (below P)

small and positive

… but not perfectly

Plasma frequency ~ 9 eV. So the stuff at ~4 eV is not due to P.

They are due to inter-band (valence-to-conduction band) transitions. This stuff is why gold is gold-colored and silver is silver-colored.

The full explanation of this requires the quantum theory of solids.

Drude model

dramatic departure from Drudemodel

“Optical dielectric function of silver,”Phys. Rev. B vol. 91, 235137 (2015)