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ECE507 - Plasma Physics and Applications

Lecture 4

Prof. Jorge Rocca and Dr. Fernando Tomasel

Department of Electrical and Computer Engineering

2

Constant, uniform B

)3(0

)2(

)1(

dt

dv

Bvm

q

dt

dv

Bvm

q

dt

dv

z

zx

y

zyx

Let’s align B with the z-axis, so B = Bzk. Then we can write F = q (v x B) = = i q vyBz – j q vxBz. The differential equations of motions are

Taking the time derivative of (1) and using (2) we obtain

xzx v

m

qB

dt

vd2

2

2

This is the diff eq for a harmonic oscillator with angular frequency c=|q|Bz/m (cyclotron or Larmor frequency)

ECE 507 – Lecture 4

3

Constant, uniform B

)3(

)2(

)1(]Re[

0

)(

)(

zz

ti

y

ti

x

vv

eivv

evv

c

c

A similar equation is found for vy. The solution of these equations can be written as

We drop the Re –we are dealing with real quantities

)/(tan 00

1

2

0

2

0

2

xy

yx

vv

vvv

Integration constants

vx and vy are /2 out of phase, we have circular motion in a plane perpendicular to B . Integrating once more we obtain

)'3(

)'2(

)'1(

00

)(

0

)(

0

tvzz

eev

yy

eev

ixx

z

iti

c

iti

c

c

c

v/c is called Larmor radius, or gyro-radius

Find these equations by yourself!


4

Constant, uniform BAveraging the motion during an orbit we find the equation for the trajectory of the guiding center

tvzz

ev

yy

ev

ixx

zcg

i

c

cg

i

c

cg

00

0

0

So in the case of a constant, uniform magnetic field charged particles spiral about the magnetic field lines. Note that positive ions and electrons rotate in opposite directions – the upper sign on the preceding equations corresponds to positively charged particles. The direction of rotation of both negative and positive particles is such that reduces the ambient magnetic field – plasmas are diamagnetic!The Larmor frequency and radius depend on the mass of the particles. For the same absolute value of charge, ions will have larger orbits and lower frequencies.


5

Constant, uniform B

Negatively charged particle moving in a region of constant, uniform B. The magnetic field faces into the page. For a hydrogen plasma immersed in a 1T magnetic field, ce = 28 GHz and ci = 15.2 MHz

A low energy beam is injected across a magnetic field in a chamber with argon at low pressure . Note the dark gap between the cathode and beam onset. In this sheath region the electrons still have insufficient energy for light excitation.


6

Constant, uniform B• The ion and electron Larmor radii and frequencies provide space- and

time-scales in a magnetized plasma.– Phenomena that occur on space-scales much smaller than the gyro-radius, or

on time-scales much shorter than one Larmor period can be described using equations for an unmagnetized plasma.

– For large space-scales and long time-scales, gyro motion is essential to the description of the plasma behavior.

– In some plasmas, the electrons may be magnetized, but the ions may not.

• Homework: Look through articles in Physical Review Letters, Physics of Plasmas, Plasma Physics, Plasma Sources Science and Technology or in other journals over recent years and find at least one article each about laboratory, solar or terrestrial, and astrophysical plasmas immersed in magnetic fields. Evaluate electron and ion gyro-radii and Debye radius. Compare to system sizes for each case. Calculate number of particles in a Debye sphere. Evaluate Larmor frequencies and compare to the time-scale evolution of the overall plasma. Which of these are really plasmas? Which of these are magnetized or unmagnetized?


7

Constant, uniform B and E

)3(

)2()(

)1()(

zz

zxy

y

zyxx

Em

q

dt

dv

BvEm

q

dt

dv

BvEm

q

dt

dv

Let’s again align B with the z-axis, so B = Bzk. From the force law F = q (E + v x B) the differential equations of motions are

Taking the time derivative of Eq 2 and replacing in Eq 1 we obtain

y

z

xzy

x

z

yzx vB

E

m

qB

dt

vdv

B

E

m

qB

dt

vd2

2

22

2

2


8

Constant, uniform B and E

)3('

)2(''

)1(''

0

)(

)(

tm

qEvv

eivB

Evv

evB

Evv

zzz

ti

z

xyy

ti

z

y

xx

c

c

These equations are similar to those we found before for constant B, but not quite the same. Looks like a transformation could help. Noting that (E x B)/B2 = i Ey/Bz – j Ex/Bz, we will rather analyze v’ = v - (E x B)/B2. The solutions for v’ are

)'/'(tan

'''

00

1

2

0

2

0

2

xy

yx

vv

vvv

Integration constants

tm

qE

B

BEvv z

zcg

20

So the guiding center is moving with velocity v’z in the reference frame moving with velocity (E x B)/B2, and is moving with velocity

with respect to the laboratory. Note that the drift is independent of q, m, and v!


9

ExB drift

Smaller Larmor radius here

Larger Larmor radius here

0.88 m in 88 sec

0.75 m in 75 sec

E

B M

m M/m = 3

Since the drift velocity is independent of the characteristics of the charged particles, the whole plasma will drift together across the electric and magnetic field lines.


10

Uniform E and B: drift and electric mirror

B

Ey

Ez


11

Constant, uniform B and a force FIt is simple to generalize to the case of any other simple force. We can simple replace the electric force qE with a general force F. The guiding center will now move with velocity vF = (F x B)/qB2 or, in the case of gravity, vF = m(g x B)/qB2.

Although somewhat similar, note that in this case the drift velocity does depend on m and q. Consequently, the presence of gravity will result into a net current density in the plasma. The gravitational drift is horizontal, not vertical!

B

g


12

Constant, nonuniform B

iti

c

cg

ti

zxy eev

BeqvyBqvF cc )()()(

How much can we tell about the movement of charged in nonuniform magnetic fields without knowing the exact form of B(x,y,z)? Assuming that the variation of the magnetic field within the Larmor radius is small, we can Taylor expand B

and then replace into Eqs (1) and (2), page 10

y

ByyBB cgcg

)(

B

Smaller Larmor radius here

Larger Larmor radius here

Show it!


13

Constant, nonuniform B

y

BqvF

c

y

2

2_

Now remembering that we are interested on the real part of these exponentials, and taking the initial phase equal to zero, we average in a gyration to obtain

so the guiding center drift velocity for perpendicular gradients can be written as

32

2

2 2

1

B

BB

q

W

B

BBv

B

BF

qv

c

grad

Note that this drift velocity, like in the case of gravitational drift, depends on the sign of the charged particle, and so it results in a net current and a volumetric electric field.

Show it! What happens with Fx?


14

Curved B: Curvature drift

• What is the magnetic field is curved? As particles move along the line fields, they will feel a centrifugal force given by

2

2

//

2

// ˆc

c

c

cR

Rmvr

R

mvF

Using the equations defined before, we can directly deduce the curvature drift

22

//

22

2

//

2

21

c

c

c

cccurv

R

BR

qB

W

R

BR

qB

mv

B

BF

qv

Note that the constant, “curved” magnetic field does not satisfy Maxwell’s equations, so the gradient drift needs to be added.


15

Curvature drift - Complete

For the magnetic field to have zero curl in all directions perpendicular to B, the magnetic field strength must fall off as

]0)(1

)([2

rB

rrBscoordinatelcylindricain

R

RBB z

c

The total drift (curvature + gradient) can then be written as

22

22

//32

2

22

2

// vv

B

BB

q

m

B

BB

qB

mv

R

BR

qB

mvvv

c

cgradcurv

Show it!

For an isotropic, Maxwellian plasma the average curved-field drift can be written as

3//3

22

1

B

BB

q

TWW

B

BB

qvv gradcurv

Show it!


16

Other drifts…


17

A summary of guiding center motion

2:

2:

:

:

:

1:

22

//3

22

//

3

2

2

2

vv

B

BB

q

mvvvacuuminfieldBCurved

R

BR

qB

WvdriftCurvature

B

BB

q

WvdriftGradientB

B

Bg

q

mvfieldnalGravitatio

B

BEvfieldElectric

B

BF

qvFforceGeneral

curvgrad

c

ccurv

grad

g

E

F


18

Example: Sputtering and reactive sputtering• In concept, sputtering and reactive sputtering are simple

processes

Sputtering gas (Argon)

Reactive gas (Oxygen, Nitrogen, etc)

(-)

(+)

MetalTarget (cathode)

Power Supply

Metallic or dielectric coating

Substrate

• Metal sputtered from a target is deposited onto the substrate.

• If the metal is sputtered in the presence of a reactive gas, it will readily form a compound deposit.

• DC or pulsed-dc power offers the most straightforward and cost effective option for such a process…


19

Issues with glow discharge sputtering sources

• High ion current densities (> 1 mA/cm2), necessary to achieve acceptable deposition rates, force to operate the discharge with a high voltage (~ 2-5 kV).

• However, the sputtering efficiency is relatively low at these energies, and decreases with increasing energy.

• Discharge is maintained by secondary electron emission from the cathode. Pressures must be high enough (>30 mTorr) so the secondary electrons are not immediately lost to the walls.

• These pressures, however, are higher than optimum for deposition of sputtered atoms onto the substrates (sputtered atoms are scattered by argon atoms)


20

A solution: planar magnetron discharges

Magnetron flange mount

Front View Side View

Race Track

AluminumTarget

Backing plate

Magnetic trapping field

Argon In

Ar+

Ar+

Al

Al

Al

Substrate

Aluminum Deposit


21

Magnetron discharge

Map of the magnetic field measured above the surface of the

target. The magnetron is cylindrically symmetric. The solid

line is a cross section along a radius of the intensity of the radiation emitted by the plasma (200-800

nm).

2-D image of the plasma seen with a narrow pass filter centered at

396.1 nm (Al I 3s2 ( 1S)3p -3s2 ( 1S)4s).


22

Typical high-speed end-on picture of a microarc. The gate width is 100 ns, and the delay from the beginning of the current pulse is 350 ns. The arc is used to inject a stream of electrons that evidence the E

x B drift, resembling the use of dyes in the study of fluid motion


23

A different view of the arc. The image is the result of applying a transformation that straightens the centerline of the etch track

(indicated by the solid line).

Cross sectional view of the luminous streak along the

centerline.


24

Cross sections of the streak for different delays (indicated in ns). Note that the intensity profile does not change appreciably with the delay. The extent of the

disturbance at the earliest delay at which we were able to acquire images (170 ns) indicates that the perturbation introduced by the arc travels with a minimum speed

of approximately 7.4x105 m/s. [Tomasel et al., Plasma Sources Sci. Technol. 12, 1 (2003)].

Exposure: 100 ns


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