Impedance of Soft Magnetic Multilayers:

Application to GHz Thin Film Inductors

ANDREY GROMOV

Department of Physics, Section of Nanostructure Physics

(Stockholm 2001)

Abstract

Transport and magnetic properties of magnetic multilayers have been a topic

of intensive research over the past 10-15 years, owing to such important discov-

eries as the oscillatory interlayer exchange interaction, giant magnetoresistance,

giant perpendicular anisotropy. These phenomena are behind the current un-

precedent growth rates in magnetic and magneto-optical data storage, and are

expected to result in new large scale applications, such as magnetic random ac-

cess memory and spin logic. Recently, the high frequency properties of magnetic

multilayers have been attracting an increasing attention, related to applications

in GHz inductors and sensors. This work is devoted to understanding the high

frequency response of ferromagnetic sandwiches and fabrication of an ecient

magnetic thin lm inductor.

A theoretical approach to calculating impedance of metallic magnetic/conductor

layered structures is developed. The frequency range considered extends to the

ferromagnetic resonance region of soft magnetic lms (of the order of 1 GHz).

The analysis includes the eects of screening of the high frequency elds by

eddy currents as well as the dynamics and relaxation of the magnetization of

the ferromagnetic sub-system. Analytical expressions for the impedance as a

function of frequency and material parameters and geometry of magnetic sand-

wich stripes are obtained. Two main cross-sectional layouts are considered: a

magnetic/conductor/magnetic sandwich stripe with and without ux closure

at the edges along the stripe length - with and without the magnetic lm en-

closing the conductor strip. The importance of good magnetic ux closure for

1

achieving large specic inductance gains and high eciency at GHz frequencies

is emphasized.

The theoretical results obtained were used to design and analyze magnetic

lm inductors produced using iron nitride alloy lms. Patterned sandwiches,

consisting of two Fe-N lms enclosing a conductor lm made of Cu, were fabri-

cated on oxidized Si substrates using lift-o lithography. The inductors exhib-

ited a 2-fold specic inductance enhancement at 1GHz. The magnetic contribu-

tion to the total ux in the narrow devices was less then predicted theoretically,

which was attributed to hardening of the magnetic material at the edges of the

strip leading to incomplete ux closure. Material and design issues important

for further improving the performance of the devices are discussed.

Contents

Outline 4

1 Impedance of ferromagnetic multilayers 4

1.1 Spin dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Phenomenological theory . . . . . . . . . . . . . . . . . . . . . 5

1.1.2 Spin relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 General electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Field and potential representation . . . . . . . . . . . . . . . . 8

1.2.2 Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . 9

1.2.3 Maxwell's equations in a steady state . . . . . . . . . . . . . . 10

1.2.4 Steady state potentials in conductive media . . . . . . . . . . . 12

1.2.5 Solutions for the scalar potential . . . . . . . . . . . . . . . . . 13

1.2.6 Expression for impedance . . . . . . . . . . . . . . . . . . . . . 15

1.3 Sandwich stripe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.1 Physical aspects of dimensional reduction . . . . . . . . . . . . 18

1.3.2 Approximation of the external ux . . . . . . . . . . . . . . . . 19

1.3.3 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Materials: soft ferromagnetic lms 24

2.1 Material requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Instrumentation for magnetic properties measurement . . . . . . . . . 28

2

2.4.1 Loop tracer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.2 Magnetometry -VSM . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.3 HF permeameter . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Fe-X-N lms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5.1 Magnetic properties of Fe-N and Fe-Ta-N . . . . . . . . . . . . 33

2.5.2 Anisotropy due to oblique deposition . . . . . . . . . . . . . . . 36

2.5.3 FMR susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Applications: GHz inductors 41

3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.1 Spirals and meanders. . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.2 Planar solenoids . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.3 Sandwiched strip . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Device fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.1 Lithography process . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.2 Film deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3.2.1 UHV system . . . . . . . . . . . . . . . . . . . . . . . 49

3.3.2.2 Reactive magnetron sputtering . . . . . . . . . . . . . 50

3.3.2.3 Electron beam evaporation . . . . . . . . . . . . . . . 51

3.4 GHz inductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4.1 Impedance measurement: calibration and deembedding . . . . 52

3.4.2 HF performance . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4.3 Flux closure at the edges . . . . . . . . . . . . . . . . . . . . . 59

3.5 Prospectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 Appendix 62

References 63

5 Acknowledgments 66

6 Appended papers 67

3

Outline

The thesis consists of three main parts. Part 1 contains an overview of theoretical

modeling of the impedance of soft ferromagnetic multilayers. Maxwell's equations

in the potential form combined with the Landau-Lifshitz equation for the dynamics

and dissipation of the magnetization of the ferromagnetic sub-system are analyzed.

The analysis is applied to a magnetic/conductor/magnetic sandwich, for which the

impedance is obtained analytically. The enhanced magnetic ux due to the mag-

netic lms in the sandwich yields high specic inductance, the property essential for

miniaturization of GHz inductors.

Part 2 contains a brief overview of magnetic materials suitable for use in thin lm

inductors. We discuss the properties required of the material, dictated primarily by

considerations of strong low-loss high frequency magnetic response and compatibility

of the material with integration into various layered structures. A number of recently

developed high moment, high resistivity soft magnetic alloy systems, suitable for use

at GHz frequencies, are briey reviewed. We conclude by describing the preparation

of reactively sputtered Fe(Ta)N lms and discuss the properties important for device

applications.

In Part 3 the status of research on magnetic lm inductors is reviewed. The fab-

rication of FeN/Cu/FeN/ sandwich strips using lift-o lithography is described. The

details of the high frequency impedance measurements are given. The performance

of the devices and the factors aecting it are discussed. We conclude with a view of

the prospects for magnetically enhanced GHz inductors.

This thesis is based on the research work that has been presented in the publica-

tions appended.

1 Impedance of ferromagnetic multilayers

This section is an introduction to the fundamentals of modeling the high frequency

response of metallic magnetic layered systems. The impedance of sandwiches, con-

sisting of soft ferromagnetic and nonmagnetic metal layers, is discussed for various

cross-sectional layouts. The frequency range of interest here extends to the ferromag-

netic resonance (FMR) range of the magnetic sub-system. Therefore, screening of

high frequency elds by eddy currents in the lms is to be considered in combination

with the intrinsic magnetization dynamics and relaxation.

We rst overview the theory of FMR [1], which is used universally to describe

4

the response of ferromagnetic materials at high frequencies. The magnetic response

is expressed through the permeability (or susceptibility) tensor. The dissipation is

treated phenomenologically by introducing a damping parameter, which determines

the rate of decay of the magnetization precession after the excitation is removed.

Next we pose the general electrodynamical problem and dene the impedance for

systems with alternating magnetic and non-magnetic layers, taking into account both

screening and magnetization dynamics.

Finally, we illustrate the theory by discussing the impedance of a thin and long

magnetic/conductor sandwich. This structure is often studied in experiments since it

nds applications in GHz inductors and sensors.

1.1 Spin dynamics

The dynamic magnetization and eddy currents form, in the most general case, a

coupled system described by a set of coupled dynamic equations for the spin preces-

sion/relaxation and Maxwell's equations for elds and currents in the material. Such

systems often are very complex and must be analyzed numerically. If the system

possesses a certain symmetry (excitation vs. the equilibrium magnetization), how-

ever, the spin dynamics and Maxwell's equations can be analyzed independently and

combined in the nal result for the impedance. Here we will deal with such systems

only, keeping the analysis in the analytical domain.

1.1.1 Phenomenological theory

A dynamic equation for the magnetization of a ferromagnetic material,!M , was rst

introduced by Landau and Lifshitz [1], in which they expanded the rate of change of

the magnetization along three orthogonal vectors:

@

@t

!M =

!M

!M !H eff L

!M !M !H eff : (1)

The magnitude of the magnetic moment is a function of temperature only and is

constant for motion at a xed temperature jM j = Ms. Therefore = 0 and the

resulting motion is precession with the gyromagnetic ration and relaxation constant

L. The nature of the relaxation term is clearly seen when (1) is rearranged into

Gilbert's form [2] and is a force proportional to the rate of change of the magnetic

5

moment (magnetization velocity) and opposing the excitation:

@

@t

!M = !M

!H eff

Ms

@

@t

!M

: (2)

The eective magnetic eld in (2) is, in general, a sum of the anisotropy eld, Ha,

externally applied uniform eld H , demagnetizing eld Hd, exchange eld Hex, and

excitation eld h:!H eff =

!Ha +

!H +

!Hd +

!Hex +

!h :

The demagnetizing eld reects the shape of the sample. We omit it for now and

will include it through boundary conditions for multilayer lms discussed later on.

The exchange eld is proportional to the degree of misalignment between neighboring

spins. It is eective for samples of size comparable to the exchange length, which

is typically 106cm. This is much smaller than the structures that we will be

discussing, so Hex is neglected here-forth. Finally, we assume that the anisotropy is

uniaxial in the plane of the lms with the easy axis taken to be along z, and the

external uniform eld (if any) is applied along z, so!Ha +

!H = H0

!ez .In the absence of excitation

!h the magnetic moment is the equilibrium magneti-

zation,!M =

!M0 = M0

!ez . For small excitations we can linearize (2) in the standard

way [3]:

!M =

!M0 +

!mei!t

!H eff =

!H0 +

!h ei!t

!m and!h are small compared to M0 and H0, and have zero z-components. Substi-

tuting these into (2) we obtain:

i!!m = !m !H0 !M0

!h i!

!M0 !m;

!mx + (i H0 + !)my = i hy

!my + (i H0 + !)mx = i hx:

In the matrix form, !m = b!h , where b is the susceptibility tensor. The permeability

6

tensor is

b = b1 + 4b =

0B@ ia 0

ia 0

0 0 1

1CA ; (3)

where b1 is the unit tensor and

=!H (!H + !M ) !2

!2H !2

(4)

a =!M!

!2H !2

(5)

!M = 4Ms ; !H = (H0 +Ha) i!: (6)

For homogeneous materials the components of the permeability tensor are coordinate-

independent. The above result can then be directly substituted into the Maxwell

equations for structures under study.

1.1.2 Spin relaxation

The phenomenological relaxation term introduced above (1)-(2) scaled by a constant

represents a number of fundamental dissipative mechanisms in a ferromagnetic ma-

terial. These mechanisms can be roughly divided into direct relaxation to the lattice

(phonons), and indirect relaxation through excitation of non-uniform magnetization

modes (also known as spin-waves or magnons), which in turn decay into the lattice. In

the case where direct ow of energy from the uniform excitation into lattice motions

mediated by magnetoelastic (spin-orbit) coupling dominates, the damping constant

can in general be expressed through the elastic constants of the material [4]. For

samples comparable or larger than the domain wall size (the limit of interest in this

work), decay of the uniform mode into spin-waves needs to be taken into account.

The latter depends on the size as well as the shape of the sample. In the case of

metals an additional loss mechanism is present, namely screening of high frequency

excitations by eddy currents of conduction electrons.

A microscopic treatment, even qualitative, of spin relaxation in magnetically or-

dered media requires an extensive knowledge of the microstructure of the sample, and

is seldom possible for technologically interesting materials. In the case of a ferro-

magnetic metal one can list [5] a large number of elementary microscopic scattering

processes that can play a role in magnetization relaxation. These can be subdivided

into magnon-phonon, magnon-magnon, and magnon-conduction electron scattering.

7

A few examples of the latter would be in order: (i) eddy current damping through in-

duced emf, (ii) exchange in the strong screening regime, (iii) direct dipolar interaction

of a localized magnetic spin with either the spin or orbital moment of an itinerant

conduction electron, and exchange interaction between the same. Although the re-

laxation constant, , is only a qualitative phenomenological representation of many

physical processes, it is found in many cases to provide a satisfactory description of

the experiments.

In what follows we will account explicitly for dissipation due to non-uniform mag-

netization modes and eddy currents by solving the dynamic equations in conductive

multilayered samples in a given and xed geometry as to the interfaces and bound-

aries. The 'intrinsic' damping constant (in the absence of screening and spin waves)

will be assumed to have been determined experimentally (see section 2.5.3 on page 39).

1.2 General electrodynamics

1.2.1 Field and potential representation

The basic laws of electromagnetism in the dierential form are:

Gauss0s law div!D = 4 (7)

Ampere0s law (modified) curl!H =

4

c

!J +

1

c

@!D

@t(8)

Faraday0s law curl!E +

1

c

@!B

@t= 0 (9)

Absence of free magnetic poles div!B = 0: (10)

It required the genius of J. C. Maxwell to see the inconsistency in Amper's law,

and thus modify the set of equations now known as Maxwell's equations. They can

be solved for eld distributions in some simple situations. It is often convenient,

however, to introduce potentials in order to obtain a smaller number of second-order

equations, while satisfying the Maxwell equations identically (see, for example, [6]).

Since div!B = 0, we can dene

!B in terms of a vector potential:

!B = curl

!A: (11)

Then the other homogeneous equation (9), Faraday's law, can be written as

8

curl

!E +

1

c

@!A

@t

!= 0: (12)

This means that the quantity with vanishing curl in (12) can be written as a gradient

of some scalar function, namely, a scalar potential ':

grad' =!E +

1

c

@!A

@t

or

!E = grad'

1

c

@!A

@t: (13)

The denition of!B and

!E in terms of the potentials

!A and ' according to

(11) and (13) satises identically the two homogeneous Maxwell's equations. The

inhomogeneous equations (7-8) can then be written in terms of the potentials as

r2'+1

c

@

@tdiv

!A = 4; (14)

r2!A 1

c2@2!A

@t2 grad

div

!A +

1

c

@'

@t

=

4

c

!J : (15)

We have now reduced the set of four Maxwell's equations to two equations. But they

are still coupled equations. The uncoupling can be accomplished by exploiting the

arbitrariness involved in the denition of the potentials.

1.2.2 Gauge Transformations

When there is a simple or no charge distribution, a useful gauge for the potentials is

the so-called Coulomb, radiation or transverse gauge:

div!A = 0: (16)

From (14) we see that the scalar potential then satises the Poisson equation,

r2' = 4: (17)

The scalar potential is the instantaneous Coulomb potential due to the charge density

(!r ; t) or due to potentials on the boundary. The vector potential satises the

9

inhomogeneous wave equation:

r2!A 1

c2@2!A

@t2=

1

cgrad

@'

@t

4

c

!J : (18)

The term involving the scalar potential can, in principle, be calculated separately

from (17). It is interesting to note a peculiarity of the Coulomb gauge. Electromag-

netic disturbances are well known to propagate with a nite speed. Yet the solution

to (17) indicates that the scalar potential "propagates" instantaneously everywhere

in space. The vector potential, on the other hand, satises the wave equation (18),

with its implied nite speed of propagation, c. At rst glance it is puzzling to see how

this obviously unphysical behavior is avoided. A preliminary remark is that it is the

elds, not the potentials that concern us. Both potentials in this situation play the

role of auxiliary mathematical functions, derivatives of which in combinations return

the real physical quantities.

1.2.3 Maxwell's equations in a steady state

Let us consider a harmonic external power source. The components of the elds are

scalar quantities and therefore their sinusoidal time variations can be represented by

means of complex numbers. The time dependence of!E , for instance, is

!E (t) = Re

!Eei!t

=

1

2

!Eei!t +

!Eei!t

; (19)

where!E is a complex amplitude in the extended 6D space. All the other sinusoidal

eld vectors can be represented similarly in terms of corresponding complex vectors,

and all sinusoidal scalar quantities in terms of corresponding complex scalars.

We must note, however, that, in order for all of the electromagnetic quantities

to be sinusoidal functions of time, the entire electromagnetic system must be linear.

Thus, a sinusoidal steady state can exist only if

!B = b!H (20)!D = "

!E (21)

!J =

!E ; (22)

with the permeattivity, ", conductivity, , and permeability tensor components, ij

(see (3) in section 1.1.1 on page 5), that are independent of time at each point in

10

space. In this case the dierential eld laws (7-10) can be written as linear dierential

equations relating the eld vectors and the free-source densities:

curl(b0

!B ) =

!E + "

@!E

@t(23)

curl!E =

@!B

@t(24)

div!B = 0 (25)

div("!E ) = : (26)

Here 0 = 4 107 is the permeability of free space and b is the inverse tensor,0B@ ia 0

ia 0

0 0 1

1CAwhere =

22a

, a =a

22a

together with (4)-(6). SI units have been used here.

Substitution of the harmonic vectors dened above for!E (19) and

!B into Eq.24

yields

curl!Eei!t + curl

!Eei!t = i!!Bei!t + i!

!Bei!t:

This equation must be satised regardless of the time origin selected, i.e., it must be

satised when an arbitrary constant t0 is added to t [7]. Suppose, for instance, that

t0 =

2!, we obtain then

curl!Eei!t curl

!Eei!t = i!!Bei!t i!

!Bei!t:

Sum of the last two equations yields

2curl!Eei!t = 2i!!Bei!t:

Similarly, for all the equations (23-26) we obtain8>>>><>>>>:curl

!E = i!!B

curl(b!B ) = 0( + i!")!E

div!B = 0

div("!E ) = :

(27)

11

We can conclude then that Eq.27 is equivalent to Eq.23-26 in the sinusoidal steady

state. Clearly, all linear relations between sinusoidal time functions can be trans-

formed into equivalent complex relations by substituting for the time functions the

corresponding complex quantities, and for the dierential operator @

@tthe imaginary

quantity i!.

1.2.4 Steady state potentials in conductive media

We divide our system in to the regions where the media parameters " and can be

approximated by constants in each region, so one can take them outside the space

derivatives: 8>>>><>>>>:curl

!E + i!

!B = 0

curl(b!B ) = 0( + i!")!E

div!B = 0

div!E = 0;

(28)

Assuming no free electric charges.

The following procedure is a transition from unknown eld amplitudes to the

potential amplitudes, rst substituting!B = curl

!A into the rst equation of (28),

and next setting the expression obtained under curl equal to a gradient of some scalar

function. The expression (13) in the sinusoidal steady state becomes:

!E = grad' i!

!A: (29)

Using the dierential Ohm's law, the current density is obtained as

!J =

grad'+ i!

!A: (30)

Next, using (29) and the Coulomb gauge, div!A = 0, the second and forth equations

in (28) become

curlbcurl!A = 0( + i!")(grad'+ i!

!A ); (31)

r2' = 0: (32)

Examples below will be given for soft magnetic alloys, which are technologically

attractive for use in planar devices. The typical conductivity is = 5 105(1m1),

12

the maximum frequency to be considered 10 GHz, and the dielectric constant is

approximately that of vacuum, "0 = 8:854 1012

(faradm1). An estimate of the

complex constant + i!" in (31) gives

(1 + i 1:76 107

):

Obviously, the imaginary part is negligible. By expanding the curl, equation (31)

can be reduced to

r2!A + curl

"1

1

@Ay

@x@Ax

@y

!ez i

a

@!A

@z

#= 0

(grad'+ i!

!A ): (33)

1.2.5 Solutions for the scalar potential

The problem for the scalar potential amplitude can be considered as an electrostatic

problem, with no current living the surface of the structure. This corresponds to the

Neumann boundary condition, @'

@njs= 0, assuming an induced current component

normal to the surface is also zero. We will be considering structures with interfaces

(internal as well as external) that are parallel to the z-axis (Fig. 1), with uniform

potentials applied to the edges along z.

z

y

x

y

dmax

A

A

AA

xz

l

Fig. 1. A cylindrical inductor of arbitrary cross-section dmax `. The

cross-section is constant along z.

The solution to (32) is then gradz' = const = Ed in all regions with dierent

conductivity, and the potential dierence is U = Ed `, where ` is the conductors'

13

length. The projection of (33) onto the z-axis is then

r2Az ia

@

@z

@Ay

@x@Ax

@y

= 0

(grad'+ i!Az):

In the case where the potential dierence is applied to the conductor only, the elec-

tric eld becomes a function of all coordinates due to current redistribution. However,

a numerical solution of (32) shows that the redistribution region is comparable with

the layer thickness and the driving electric eld approaches a constant value expo-

nentially over that distance. The eld map for an axial cylinder with nonuniform

boundary conditions is depicted in Fig. 2. The section shown is along the cylinder

from its center to the outer surface. This short redistribution length allows us to ne-

glect the edge eects and take the driving electric eld to be constant in the volume

of the device.

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20Redistribution

conductor

magnetic

Dis

tanc

e al

ong

radi

us

Distance along z-axis

Fig. 2. Driving electric eld distribution along z-axis of a two-layer cylin-

der. A uniform potential is applied to the conductor region at the left edge.

Conductivity of the magnetic layer is 50 times lower then that of the conductor.

For conductors long compared to the transverse dimensions and short compared to

the signal wavelength,

14

dmax ` ; (34)

the translational symmetry in the z-direction can be used to reduce the problem for

Az to 2D:

r2Az(x; y) = ? [Ed i!Az(x; y) ] : (35)

where

?=

0

= 0

2a

(36)

is the eective transverse permeability in SI units.

For example, conductors of a few micrometers in transverse size and a millimeter

in length operating at 1GHz ( = 0:3m) will satisfy condition (34), as would a

0:1mm-diameter wire of 1 cm in length at 100MHz.

Since in (35) the driving eld amplitude, Ed, is constant, it is convenient to work

with a normalized vector potential, eA =Az

Ed, and transform (35) to

r2 eA(x; y) = ?h1 i! eA(x; y)i ; (37)

which has to be solved separately in each conductive region. eA describes the properties

of the device, independent of the external source.

1.2.6 Expression for impedance

Assume a potential dierence is applied to the conductor only. Performing averaging

in (30) over the cross-section of the conductor,

1

S

ZS

!J @!s =

1

S

ZS

!r' @!s i!1

S

ZS

!A @!s ;

and then integrating along the cylinder,Z`

0

SI @z =

Z`

0

@ h'is

@z@z i!

1

S

Z`

0

ZS

Az @s @z;

`

SI + i!

`

V

ZV

Az @V = h'(0)is h'(`)i

s; (38)

we obtain

15

R0I + i!` hAziV = U0;`; (39)

where R0 is the DC resistance and the angle brackets mean averaging over the volume.

The impedance is the proportionality coecient between the voltage and the current,

and from (39) it is

Z = R0 + i!` hAziV

I: (40)

If we assume bias conditions such that the current amplitude I is xed, then the

z-component of the vector potential completely describes the impedance.

In the literature [8] one can nd reactance denition as a ratio of magnetic ux

linkage to the total current. Thus in (40) the ux linkage is

link = ` hAziV =1

S

Z`

0

ZS

Az @s @z =1

S

ZS

@s

Ic

!A @!l =

=1

S

ZS

@s

ZS0

!B @

!s0 = h(x; y)i

s: (41)

Where we have extended the integration path from line a to contour c closed through

innity, where!A1

= 0 (Fig. 3). c is chosen such that tangent vectors !n b, !n0

bare

normal to!A . Therefore the ux linkage is a ux averaged over all possible contours

crossing the volume of the conductor. Expression (39) represents a macroscopic Ohm's

law for harmonic signals taking into account a nite thickness of the device. For such

structures, circuit parameters must be derived directly from the eld distribution.

z

x

y

l/2

-l/2

A

A

c

a

b´

b

→nb

→nb´

Fig. 3. One of the integration contours, c, extended to innity. The magnetic

ux through this contour is equal to the vector potential integrated along the

line parallel to the z-axis.

16

If the bias conditions are such that a xed voltage amplitude is applied to areas with

dierent conductive properties, the simplest approach for the impedance calculation is

to nd the total current, and take the potential dierence averaged over the edges (as

in (38)). Then, the current density distribution completely describes the impedance.

Let us assume that the scalar potential gradient has only one component, gradz' =

Ed = Const, in conductive media. We can then drag the constant eld outside the

integral and normalize by it the current density:

Z =U

I=

Ed`RS

!J @!s

=`P

k

RSk

(Jzk=Ed) @s=

`PkkRSk

1 i! eAk

@s

; (42)

k denotes the regions connected to the voltage source and tilde means normalization

on driving eld. In order to get the impedance, one needs to solve only the equation

for the normalized vector potential.

1.3 Sandwich stripe

Here we illustrate the model with a generic structure of rectangular cross-section. A

schematic of the planar inductor structure is shown in Fig. 4. A voltage is applied to

a magnetic/conductor/magnetic strip along the z-direction, which is also taken to be

the easy axis direction for the magnetization.

y

x“0”

“1”

“2”

w/2z

M→l

Jd1=σ1E

Jd0=σ0E

t1

t0

Fig. 4. A schematic (x > 0) of a magnetically enclosed stripe inductor. Regions

0, 1 and 2 denote the conductor, magnetic lm and air, 0 and 1 are the

permeabilities of the conductor and the magnetic lm, respectively.

17

There are three regions 0, 1 and 2 denoting the conductor, the magnetic lm and

air. In what follows t0, 0 and 0 are the thickness, conductivity and permeability of

the conductor, and t1, 1 and 1 = r0 are those of the magnetic lm. The conductor

width is w, and the length of the structure is `. Equation (33) for the vector potential

has to be solved in every region and the solutions joined at the regions boundaries.

1.3.1 Physical aspects of dimensional reduction

In section 1.2.5 we have shown that accuracy is not compromised by limiting the

driving electric eld to one non-zero component, Ez(t), which results in one non-

zero component of the vector potential, Az(t). For long stripes, with ` w, we

can therefore use the two-dimensional equation (35). For the given geometry, with

t0; t1 w and ux closure at the edges of the stripe, the magnetic moment has a

preferred direction in the plain, which reduces the demagnetizing energy. Since in

practice r 1, the y-component of the magnetic induction is small, By = @A

@x

Bx, which further reduces the problem to 1D. For the sake of clarity, in what follows

A will denote the amplitude of the z-component of the vector potential, with the

subscript k denoting the region:

@2

@y2Ak(y) = 2

kAk(y) kkEd; (43)

where k =1+i

Æk, and Æk = (!kk)

12 is the skin depth in every region. The general

solutions to (43) are

Ak = Ck cosh(ky) +Dk sinh(ky) iEd

!: (44)

The symmetry along y, A0(y) = A0(y), yields D0 = 0.

Since we have By 0, the boundary condition on the normal component of the

induction is satised automatically. We have to require that the vector potential be

continuous across the boundary. The conditions at the two boundaries become8>>>><>>>>:A0(

t0

2) = A1(

t0

2)

1@

@yA0(

t0

2) = 0

@

@yA1(

t0

2)

A1(t0

2+ t1) = A2(

t0

2+ t1)

0@

@yA1(

t0

2+ t1) = 1

@

@yA2(

t0

2+ t1)

(45)

18

1.3.2 Approximation of the external ux

If the external ux constitutes an appreciable portion of the total ux, then A2 must

be taken into account. The diculty in calculating the external ux in idealized

systems (assumed innite in at least one dimension) is the divergence of the vector

potential at innity. The standard approach to avoid this problem is to use a cut-o

distance for the vector potential of the order of the characteristic size of the device,

the length of the stripe in our case.

z

x

y

y=t0/2+t1

l/2

-l/2

Isurf

Fig. 5. A stripe of length ` with surface current density Isurf .

We performed a direct calculation of the 3D vector potential of an innitesimally thin

stripe conductor (Fig. 5):

A3D(x; y; z) =

Z w

2

w

2

@z0Z1

1

@y0Z `

2

`

2

@z0Jsurf Æ(y

0

)p(x x0)2 + (y y0)2 + (z z0)2

:

We can use the 3D vector potential of a current carrying stripe at (x = 0; z = 0),

taking it to represent the functional form of A2(y).

A2 = Cf(y) (46)

= C

"`

2wln

R+ w

R w

2y

warctan

w`

2yR

+ ln

`+ Rpw2 + 4y2

!#;

R =

pw2 + 4y2 + `2;

19

with f yw!0 ! 1 + ln

2`

w

;@f

@y

y

w!0

!

w.

The error in the external ux introduced by this procedure is <10%, which results

in an even smaller error in the total ux. This expression correctly reproduces the

magnetic eld at innity and results in a non-diverging vector potential for a nite

length stripe.

1.3.3 Impedance

Using (44) and (46), the vector potential becomes

A0 = iE

![C0 cosh(0y) 1] ; (47)

A1 = iE

![C1 cosh(1r) +D1 sinh(1r) 1] ; (48)

A2 = iE

!C2f(y): (49)

After substituting (47-49) into the boundary conditions (45), together with approxi-

mation (46) we obtain for the coecients

C0 = D1;

C1 = D1

cosh(0

t0

2) cosh(1

t0

2)

r01

10sinh(0

t0

2) sinh(1

t0

2)

;

D1 = D1

r01

10sinh(0

t0

2) cosh(1

t0

2) cosh(0

t0

2) sinh(1

t0

2)

;

where

D =

coth

0

t0

2

coth (1t1) +

r01

10+w01

1

1 + ln

2`

w

coth

0

t0

2

+

r01

10coth (1t1)

sinh (1t1) sinh

0

t0

2

The coecients are numbered with respect to the region (C0; C1; D1; C2). Coecient

C2 is not given above, since it will not be used in the following discussion.

From (30) the complex amplitude of the total current density is

J0 = E0 cosh (0y) (50)

J1 = E1 [C1 cosh(1y) +D1 sinh(1y)] : (51)

20

Figure 6 shows the current density (50-51) normalized to the driving electric eld as

a function of y.

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

MagneticConductor

Distance from centre, y (µ m)

Nor

mal

ised

cu

rren

t d

ensi

ty |J

/E| (

S)

(a)

0.0 0.2 0.4 0.6 0.8 1.0-2.0

-1.5

-1.0

-0.5

0.0

MagneticConductor

Distance from centre, y (µ m)

Ph

ase

(rad

)

(b)

Fig. 6. Magnitude (a) and phase (b) of the normalized current density jJ=Ej

as a function of y at 500MHz (w = 10m, ` = 1mm, t0 = 1m, t1 = 0:5m,

0 = 5 107S, 1 = 5 10

6S, r = 100).

21

The current in the magnetic lm is a sum of the eddy currents and the driving current,

and is increasingly concentrated at the magnetic/air interface as the frequency is

increased. A signicant portion of this current is in phase with the driving electric

eld. The external magnetic ux can contribute appreciably to the total magnetic

ux in the structure.

Since the magnitude of the driving electric eld is constant, the impedance can

be obtained as the proportionality factor between the voltage and the total current

in the device:

Z =`1

2w1

1 +

q01

10tanh

0

t0

2

tanh (1t1)q

01

10tanh

0

t0

2

+ tanh (1t1)

+ i!`0

2

1 + ln

2`

w

: (52)

This expression includes the DC resistance, reactance, the skin eect and FMR

contributions. 1 = ?

is dened by (36) with Ms=21000 kG, Ha =50 Oe, and

=0.01. The inductance, L = Im(Z)=!, resistance, R = Re(Z), and quality fac-

tor, Q = Im(Z)=Re(Z), are plotted in Fig. 7 as a function of frequency for typical

parameters.

10M 100M 1G0.1

1

10

100

L

R

Q

Qu

ali

ty f

act

or,

Res

ista

nce

(Ω

)

Frequency (Hz)

-1

0

1

2

3

4

5

Ind

ucta

nce (n

H)

Fig. 7. Impedance characteristics of a magnetically enclosed inductor from

(52), with w = 20m, ` = 1mm, t0 = 1m, t1 = 0:1m, 0 = 5 107S,

1 = 106S, 4Ms = 21000G, Ha = 50Oe, = 0:01.

22

The impedance exhibits a resonance at approximately 3 GHz, the FMR frequency of

the ferromagnetic sub-system. The specic inductance is enhanced three-fold over the

air-core value of the bare conductor ( 1 nH/mm), a great advantage in miniaturizing

planar magnetic ux devices.

23

2 Materials: soft ferromagnetic lms

The impedance of magnetic sandwiches has recently become of interest because of

new applications in magnetic ux devices operating at frequencies up to the GHz

range. High frequency operation and compatibility issues impose certain requirements

on the magnetic materials to be used, which are summarized below. With these

requirements in mind, we briey discuss the two main material groups, magnetic

oxides and metallic lms. We conclude the section by describing the preparation and

magnetic characterization of our Fe-based nitrogen containing lms, which are then

used to fabricate planar devices.

2.1 Material requirements

There are two basic requirements to physical properties of the magnetic materials

for use in GHZ inductive devices: high loss-free permeability and compatibility with

under/over-layers in multi-layered device designs. For achieving a high permeability

at high frequencies the following properties are desirable:

high saturation magnetization,Ms. This is a precondition for high permeability.

The permeability employed in most of the designs is the transverse permeabil-

ity (?, hard axis response) in lms of uniaxial anisotropy, which is directly

proportional to Ms;

controllable anisotropy, Hk, in the range 10 50 Oe. At low frequencies ?

Ms=Hk, so high Hk results in too low a permeability. For low Hk the FMR

frequency is decreased: fFMR ' p4MsHk. At FMR the permeability is

mostly imaginary producing high losses;

small FMR line-width. This is commonly dened as the full-width at half-

maximum in the bell-shaped imaginary part of the permeability. In a system

without dissipation the FMR line is innitely narrow. In a real system it can

be rather broad due to various dissipation processes (see section 1.1.2);

high resistivity. Eddy currents are one of the many dissipation channels. How-

ever, in widely used soft magnetic alloys screening often dominates as a magnetic

loss mechanism;

single domain state, for low magnetic loss as well as for reproducibility. Varia-

tions in inductance caused by changes in the domain pattern of the (soft) lms

will most likely be unacceptable for applications;

24

low magnetostriction is preferable, since the fabrication process may result in

stress in the lms leading to stress-induced anisotropy, which limits the perme-

ability.

It is important that the magnetic material should be compatible with the under/over-

layers in multi-layered device designs. The highest impact for integrated inductors, for

example, is seen in scaling down CMOS or BICMOS RF ICs. Process compatibility

in this respect would mean the ability to fabricate the material on various 'imperfect'

substrates (polycrystalline or amorphous insulation or metallization layers, SiO, Al,

etc.), and a restricted process temperature required for other on-chip components.

2.2 Oxides

Ferrites and other oxides are attractive for one reason, their high resistivity. Other

properties of ferrites are disadvantageous compared with those of soft magnetic al-

loys. The saturation magnetization is a factor of 5 lower. Generally, the anisotropy

in oxides is much more sensitive to growth conditions and is harder to control. There

is no simple and robust method of growing soft uniaxial ferrite lms in a biasing eld.

Epitaxial garnets are known to have the smallest FMR linewidths among the magnetic

materials. However, their low saturation magnetization (10 times lower than that of

alloys) and especially the need for single-crystalline lattice-matched substrates, make

garnet lms uninteresting for use in integrated inductors. Polycrystalline ferrite lms

prepared at relatively low temperature have relatively large FMR linewidths. Ex-

change biasing of ferrite lms to achieve a single-domain remnant state has been

demonstrated, but is less straightforward than with alloy lms. However, the high

resistivity of ferrite lms warrants research on integrating them into Si-based tech-

nology.

A common method to prepare ferrite cores for power inductors and transformers

is to use a prefabricated ferrite powder, mix it with a polymer by ball mill rotation,

spin cast or screen print it on a substrate, and nally cure the resulting paste at a

few hundred ÆC [9] up to 800ÆC [10]. The typical relative permeability obtained by

this method is r = 20 30, which is sustainable up to 10MHz [9].

NiZn-ferrite lms produced on glass substrates by a low-temperature (T < 100ÆC)

plating technique, having 4Ms 6 kG and a large anisotropy perpendicular to the

plane, for use in microwave non-reciprocal devices have been reported [11]. This

technique is very promising and it would be interesting to see if the method can be

25

used to produce ferrite lms with properties required in a GHz inductor (see previous

subsection).

Sputtering and pulsed laser deposition have been used to fabricate ferrite lms.

Typically, sputtered as-deposited lms are highly disordered or amorphous [12], and

have undesirable magnetic properties, even spin-glass behavior [13]. The lms often

require a post-deposition annealing at up to 1000ÆC. Pulsed laser deposition of high-

quality (Mn,Zn)-ferrite lms, having bulk magnetic properties, was achieved at 400600

ÆC by a careful selection of buer layers [14]. Also exchange biasing has been

demonstrated in ferrite bilayers [15].

At present, however, there appears to be no readily available techniques for fabrica-

tion of ferrites having the desired properties for use in GHz inductors and compatible

with the Si-based integration. In this respect the situation is quite dierent with soft

magnetic alloys.

2.3 Alloys

Soft magnetic alloy lms can be prepared by a variety of deposition techniques practi-

cally on any surface in various multilayer congurations at low temperature. Permal-

loy as the core material has dominated the eld of planar inductive devices for decades.

Its relatively low saturation magnetization ( 10 kG) and resistivity ( 20 cm),

however, make the material too lossy to be used at above 100MHz. Recent suc-

cess in fabrication of a number of alloy systems with high saturation magnetization

( 20 kG) and high resistivity ( 100 cm and higher) gives a large choice in

designing HF devices. A rigorous classication or a complete survey of the enormous

eld of soft magnetic alloys is beyond the scope of this introduction. We will limit

ourselves to mentioning a few illustrative examples of soft alloy lms developed for

high frequency applications.

The general approach is to start with an intrinsically high moment material, such

as Fe, Co or FeCo and modify, by alloying in additional elements, the structure on the

atomic or nanometer scale in order to achieve the desired soft magnetic properties,

high resistivity, low magnetostriction, high thermal stability etc. Amorphous alloys

have been particularly popular as materials for high frequency inductive applications

([16, 17] and references therein). In these Si, B, C, P, Zr, Nb, etc., are added (at 10-30

atomic %) to promote a glass-like structure with no long range order. The result is

a magnetically soft material (Hc; Hk 1 Oe, Ms 10 15 kG), with a vanishing

crystal anisotropy and magnetostriction and increased resistivity (100 200 cm).

26

Amorphous lms are typically prepared by 'quenching' the material on cold substrates.

They are often annealed at 400600ÆC to ne tune the magnetic properties (see [18],

and references therein). The annealing induces crystallization of nano-sized Fe(Co)-

rich grains embedded in an amorphous ferromagnetic matrix - a morphology known

as nanocrystalline.

The combination of high magnetization( 20 kG), high resistivity (50 100 cm), and excellent soft magnetic properties (Hc; Hk 1Oe) has made nanocrystalline

Fe-X-N compositions (with the alloying element X=Ta, Hf, Si, Zr, Al, Cr, Ti, etc.)

one of the most promising systems for high frequency inductive applications (see, e.

g., [16, 19, 20] and the extensive literature cited therein). Nitrogen atoms intersti-

tially incorporated into the iron lattice promote ne crystalline Fe-rich grains, 10nm in diameter, embedded in a disordered N-rich phase. The size of the grains is

smaller than the exchange length in the material, which can explain the soft magnetic

properties as resulting from vanishing (averaged over the grains) magnetocrystalline

anisotropy. It is essential for achieving soft magnetic properties that the inter-grain

material is ferromagnetic and provides a suciently strong exchange coupling be-

tween the grains. For this reason incorporation of large amounts of nitrogen, leading

to weakening of the inter-grain couplings, is not desirable. Adding various alloying

elements into FeN can improve magnetic softness, thermal stability, etc.

Another member of the nanocrystalline family of magnetic lms is known as gran-

ular lms. These consist of ne Fe(Co)-rich grains mixed with an insulating non-

magnetic phase, typically Al2O3 (see, e. g., [21]). The relative volume of the insulat-

ing phase is critical since Al2O3 breaks the magnetic coupling at the grain boundaries.

When the volume fraction of the magnetic metallic phase is bellow the percolation

threshold, the grains couple through dipolar interactions only and the lms exhibit

hard magnetic properties or even a super-paramagnetic behavior (if the magnetic en-

ergy of a single grain, proportional to the volume, is smaller than kT). Good soft mag-

netic properties in combination with very high resistivity values (500 1000 cm)

can be obtained.

Finally, we would like to mention two more systems, ultra-high moment Fe16N2

lms and high moment high resistivity electrodeposited lms.

High moment materials, being intrinsically the best magnetic ux ampliers, have

traditionally been of great interest. Among these, a delicate (metastable) phase

Fe16N2, rst found in 1972 by Kim and Takahashi [22], has been attracting an increas-

ing attention. Epitaxial Fe16N2 lms obtained using ultra-high vacuum evaporation

have been reported [23]. Single crystal substrates of In0:2Ga0:8As (100) were used

27

to provide a good lattice match. A thin underlayer of pure Fe was rst deposited,

on which Fe16N2 was grown at an extremely low rate, 0.006-0.03 Å/s. Record high

saturation magnetization values were reported, 4Ms=28-29 kG, with the resistivity

30 40cm at room temperature [24]. Deposition of the same phase (4Ms=26-

27 kG) by sputtering under carefully optimized deposition conditions has recently

been reported [25], [26]. The high sensitivity of this phase to preparation conditions

is due to the presence of a number of phases with competing formation energies. The

high magnetic moment is a great advantage in inductive applications. It remains to

be seen, however, whether Fe16N2 can be produced in a way that is technologically

compatible with incorporation in planar device structures.

Electro-deposition is often preferred to sputtering for soft lms because of its low

capital cost, good control of the lms' properties, and most importantly because in

many cases it oers high exibility in fabrication of various patterned structures.

Recently, soft Co-Fe-Ni alloy lms with very high saturation magnetization (20-21

kG) and resistivity (>100 cm) have been obtained by using electrolytes containing

various additives, such as S, C, Mo (see the recent review by T. Osaka, [27]). These

results are very promising and are likely to warrant a conversion from sputter-based

fabrication processes to electro-deposition.

In this work FeN and (Fe90Ta10)N are used as the magnetic layers in mag-

netic/conductor/magnetic sandwiches. Our interest was in the material aspects of

incorporating the lms in a multilayer device rather than in optimizing the properties

of single layer lms. These aspects include variations in the magnetic anisotropy for

deposition on various surfaces, such as SiO and Cu, with various hight gradients.

2.4 Instrumentation for magnetic properties measurement

The experimental techniques and instrumentation used to magnetically characterize

lms in this study are described in this section. These instruments were designed and

built from scratch at Nanostructure Physics (KTH). Setting up and maintaining the

instrumentation was part of this Ph. D. project.

2.4.1 Loop tracer

The function of this instrument is based on Faraday's eect, where an EMF is in-

duced by a magnetic ux change (see dierential form in (9) on page 8). A low

frequency magnetic eld changes the magnetization of the specimen periodically; the

corresponding ux change is picked up by a coil, the output voltage of which is inte-

28

grated and displayed as a function of the driving eld. We have developed a thin lm

loop tracer and optimized it specically for studies of in-plane anisotropy.

Our implementation was inspired by the single-wire-pickup loop tracer with near

monolayer sensitivity described by H. Oguey in 1960 [28]. The changes introduced

in our design exploit advances since the 1960's in low noise amplier electronics and

signal enhancement by a lithographic patterning of the planar pick-up loop. The block

diagram of the instrument is shown in Fig. 8. A sinusoidal driving eld is supplied

by a Helmholtz coil. The pick-up voltage is amplied, integrated and applied to the

Y-input of a real time digital oscilloscope. The signal for the phase compensation

and X-input of the oscilloscope is taken directly from 1 resistor in series with the

driving coil.

Amplifier

Phase compensation

Integrator

R61Ω

Audio amplifier

Function generator

dM/dt

M

H

Scope

Film

Fig. 8. Block diagram of at-coil-pickup M-H loop tracer.

The central part of the instrument is a at two-section pickup coil (g. 9). The

coil plane is oriented along the external magnetic eld, which signicantly reduces

the direct pickup. The two coil sections, with windings in opposition, compensate

any remaining o-axis eld, while summing up the fringing ux from the sample.

When produced lithogracally, this sensitive part of the instrument has two main

advantages. First is the increased sensitivity due to numerous nely dened turns

(2x10 in our case), which simplies design of the amplication circuit and removes

the need for a voltage boosting transformer used in the original design [28]. Secondly,

the ability to use high precision lithography to dene two copper spirals of essentially

29

identical geometry greatly reduces the unwanted pickup.

H

Iind

Fig. 9. Layout of pickup coil.

The phase compensation unit has no mechanical parts or varyometers found in [28].

0Æ 180

Æ phase adjustment is done electronically. No DC restorers have been used

since the integrator, built on an active band-pass lter, is very ecient with respect

to icker noise. Any remaining ham (50 or 60 Hz) was removed using the digital

oscilloscope averaging. The schematic and description of the amplier/integrator are

given in the Appendix.

An additional major advantage of the loop tracer developed here is the free access

to the back surface of the sample. The back side of the substrate is placed on a non-

magnetic rod using a double-sided tape. The rod is then placed through an opening

transverse to the Helmholtz axis with the lm facing the at pickup coil. Thus, a

free in-plane rotation is achieved, allowing a 'real time' evaluation of the angular

dependence of the magnetization loop. With driving frequencies of typically 200-400

Hz, digital trace averaging for less than one second was sucient for good signal

to noise ratio, and the entire in-plane magnetization map could be obtained within

seconds.

2.4.2 Magnetometry -VSM

A home built Vibrating Sample Magnetometer (VSM) was used for characterization of

magnetic lms (Fig. 10). The VSM consist of room temperature eld coils(Hmax<220

Oe), a pair of counter-wound pick-up coils, and a vibrator mechanically connected to

the sample. Following Foner, the inventor of the VSM, we have used a loud speaker

30

for producing the vibrations. A lock-in amplier, a function generator and a DC

power supply complete a magnetometer.

PC

Lock-in Amplifier Vibrator

Pick-up coilsDC field coil

IDC

Power Supply

osc.

Fig. 10. Block diagram of the VSM.

The function generator output is used to drive the loud speaker. A magnetic sample

xed on a non-magnetic vibrating rod induces voltage in a pair of pick-up coils. The

induced voltage depends on the geometry of the coils, amplitude and frequency of

the vibration, and is directly proportional to the magnetic moment of the sample.

This voltage is then phase-sensitively measured using the lock-in, and read in to a

computer along with the DC-eld (solenoid current) value. The instrument is well

suited for studies of soft magnetic lms, and allows sensitive measurements down to

10nm in lm thickness.

2.4.3 HF permeameter

This apparatus is based on a Vector Network Analyzer and a strip-loop xture, and

is used to measure the complex magnetic permeability of lms [29]. The complex per-

meability data contain information about the inductive properties of the lms as well

as dissipation (also known as magnetization relaxation or Ferromagnetic Resonance

loss).

Fig. 11 shows a schematic of the measurement setup. A computer-controlled

3GHz impedance analyzer HP8714C is used to measure the complex reection coef-

cient of the xture containing the sample under test. The xture consists of a copper

strip loop mounted directly on an SMA connector and a standard N to SMA

31

adapter, which is then connected to the impedance analyzer reection port. A

Helmholtz coil is used to apply an external DC magnetic eld. The same loop is

used both to generate the excitation eld and to detect the sample's response. The

loop impedance, obtained from the reection coecient, is recorded with and without

the sample in the loop as a function of frequency. The recorded change in the loop

impedance is then converted into complex susceptibility or permeability spectra.

HP8714C

N-JACK SMA-PLUGN-PLUG toSMA-JACK

adapter

Helmholtz coil

Sample

Striploop

Hdc

I

Film

w

h

Loop section

Hdc

Fig. 11. Schematic of the setup.

The magnetic eld in the center of a strip loop of width w carrying current I (see

inset to Fig. 11) is H = kHI

w; where a geometrical factor, kH =

2

arctan

w

h

for

innitely long loop of width w and height h (w = 6:5mm; h = 1:2mm; kH = 0:88 in

our case). The total length of the measuring loop was kept below 10 mm to ensure

that the loop is a lumped element even at the highest frequencies used.

The principle of the method can be understood by considering the following. The

ux due to a magnetic lm enclosed by a strip loop is [28]

= 0HV

w;

where 0 is the permeability of free space, is the susceptibility of the lm, V is the

volume of the lm and 1 is a geometrical correction term, which, in general, is

32

a function of the lm-coil geometry. Once the value is computed for given loop

dimensions, it does not vary appreciably (within 3%) with the size of the lm. We

can write for the impedance induced by the ux :

Z =V

I=

1

I

@

@t= i!0

V

w2kH

or separating real and imaginary parts:

L = 00tmk

R = !000tmk

where k =Sm

w2 kH is a geometrical factor of the order of unity, Sm and tm are the

area and thickness of the magnetic lm, respectively. Thus, the change in inductance

and resistance of the loop when the sample is inserted is proportional to the real

and imaginary parts of susceptibility, respectively, scaled by the lm thickness. Since

= 1 and 1, the real and imaginary parts of the impedance are similarly

related to the permeability, Z = R+ i!L = k!0tm(00+ 0).

2.5 Fe-X-N lms

In this section we discuss the preparation and properties of Fe-based nitrogen con-

taining lms, illustrating the issues relevant for their incorporation in planar devices,

such as anisotropy and dissipation.

2.5.1 Magnetic properties of Fe-N and Fe-Ta-N

As was discussed above the magnetic properties of nanocrystalline Fe N (Fe X N) are sensitive to the shape and size of the grains, and to the nature of the

grain boundary material. The microstructure is inuenced by the reactive sputtering

process parameters, such as the substrate temperature (T ), nitrogen partial pressure

(pN2), total gas pressure (p), output power of the magnetron (P ), as evidenced by

many detailed studies on the subject (see section 2.3). Fig. 12 illustrates the variation

of the coercivity and resistivity of our reactively sputtered FeN lms with the nitrogen

partial pressure in an Ar +N2 gas mixture, pN =pN2

pAr+N2

100%. The coercivity along

two orthogonal in-plane directions are denoted Hce (easy-axis) and Hch (hard-axis),

and both show a rapid decrease already at pN 1% and a steady rise at high nitrogen

concentrations.

33

0 2 4 6 8 10 12 140

20

40

60

80

ρ

ρ (

µΩ

cm

)

pN (% )

0

10

20

30

40

50

Hc

Hce

Hch

Fig. 12. The coercivity and resistivity of FeN lms versus nitrogen partial

pressure.

The incorporation of nitrogen into iron lms can generally be divided in two stages.

Below the interstitial nitrogen solubility limit of the single Fe(Ta) crystalline phase

(pN ' 1%), N acts as a 'grain rener'. The softening of the lms in this regime can

be explained in terms of a vanishing local magnetocrystalline anisotropy. At high

nitrogen concentrations the soft magnetic behavior is lost progressively in spite of the

ne nanostructure. This is due to weakening of the inter-granular exchange coupling

often attributed to a transformation of the amorphous matrix into a lower Tc phase

with addition of excess N [19].

The resistivity of the lms increases almost linearly with increasing pN . Such an

increase may reect two main indirect contributions to the scattering of the conduction

electrons: one being grain boundary scattering due to a decrease in the grain size and

increase of the volume fraction of the inter-granular amorphous-like phase; the other

being a lattice distortion scattering in the grains, due to the nitrogen incorporation.

The coercivity of FeTaN (Fig. 13) has a similar broad minimum. However, the

absolute minimum is shifted to higher pN .

34

0 2 4 6 8 10 12 140

20

40

60

80

ρ

ρ (

µΩ

cm

)

pN (% )

0

10

20

30

40

50

Hc

Hce

Hch

Fig. 13. The coercivity and resistivity of FeTaN lms with 10w=o of Ta versus

nitrogen partial pressure.

FeN lms sputtered at pN = 7% exhibit similar properties to FeTaN lms prepared

at pN = 11%. Both systems show a weak in-plane anisotropy in this range of pN .

The resistivity of FeTaN is larger than that of FeN for low pN (due to the Ta

alloying), but the two become comparable by pN 5%. Thus, we nd no signicant

dierences in properties between the two systems in the range pN = 4 12%. The

choice of high N contents (10 12%) may seem obvious in view of high frequency

applications. We nd, however, that high pN makes the material more sensitive to

deposition conditions, such as the angle of incidence (see next section).

Both FeN and FeTaN are found to become softer with lowering the total sputtering

gas pressure and with increasing the magnetron output power. The lowest pressure

needed to maintain a stable plasma in our deposition system was 2:5 103Torr.

High magnetron power can lead to an increase of the substrate temperature [30] during

deposition, sometimes causing hardening of the material or melting of the mask when

devices are made using the lift-o technique. We therefore limited the magnetron

power to 300W (15W=cm2).

35

2.5.2 Anisotropy due to oblique deposition

Magnetic lms deposited on tilted substrates possess properties dierent from those

of normally (on-axis) deposited lms. The incidence angle, , at a given point is

dened as the angle between the normal to the substrate and the line connecting

the point with the center of the target (g. 14). Films thus deposited are found to

posses a uniaxial in-plane anisotropy with the easy axis directed perpendicular to the

deposition plane (containing the substrate normal and the target center).

Averageatom fluxdirection

Ha

θ

Fig. 14. The angle of incident atom ux to the normal and corresponding

anisotropy direction.

M H traces for magnetic lms sputtered at pN = 1% are shown in Fig. 15a for

three dierent incidence angles. The normal incidence lms ( = 0) show no preferred

in-plane orientation of the magnetization. The lms deposited at 20Æ develop a

weak in-plane anisotropy. The easy-axis coercivity, Hce, increases while that for the

hard axis, Hch, decreases, and a uniaxial anisotropy with Hk 10Oe develops. The

lms sputtered at = 60Æ show Hce 200Oe and Hk 400Oe (the hard axis loop

for 21Æ incidence is shown for comparison, Fig. 15b). This growth induced anisotropy

leads to more than an order of magnitude reduction in the transverse permeability,

which is responsible for the inductive response of the lm.

36

-50 -40 -30 -20 -10 0 10 20 30 40 50

-1.0

-0.5

0.0

0.5

1.0

M/M

S

H (Oe)

e h

00

210

(a)

-400 -300 -200 -100 0 100 200 300 400

-1.0

-0.5

0.0

0.5

1.0

M/M

S

H (Oe)

e h

210

600

(b)

Fig. 15. M-H curves for Fe-N (1%) lms deposited at (a) = 0Æ, 21Æ, and (b)

21Æ, 60Æ incidence angles.

Two potential mechanisms of this anisotropy are discussed in the literature (see, e.g.,

[17]). The rst is the shape anisotropy of magnetic columns, which are typically

37

present in lms deposited at oblique angles due to the self-shadowing eect [31, 32].

This eect has recently been analyzed in detail for Co lms on underlayers deposited

at oblique angles [33] and was found to result in up to 1 kOe of anisotropy. Another

possible origin of the oblique-growth induced anisotropy in magnetic lms discussed

in the literature is due to magnetostriction. The later is less likely to result in such

high anisotropy values (Hk 1 kOe).

Films containing dierent amounts of nitrogen have dierent sensitivity to oblique

deposition. This is illustrated in Fig. 16 by xing at 21Æ and varying the nitrogen

partial pressure. The hard axis loops, measured for lms corresponding to pN =

1; 2; 4; 8% demonstrate an increase in anisotropy from 10Oe to 100Oe. The easy

axis coercivity, Hce, showed a slight increase from 5Oe to 15Oe on increasing pN .

-150 -100 -50 0 50 100 150-1.5

-1.0

-0.5

0.0

0.5

1.0p

N = 1% 2% 4% 8%

Incident angle 21o

M/M

S

H (Oe)

Fig. 16. Hard axis loops for FeN lms, sputtered at 21Æ and dierent pN .

The increase in Hce and Hk with pN is dramatic for large incidence angles. The data

for = 60Æ are shown in Fig. 17.

38

0 1 2 3 4 5 6 7 8 9 10 11 120

100

200

300

400

500

600

700

800

PN / P

tot (% )

HC

Hk Hce

Fig. 17. Hce and Ha for FeN deposited at = 60Æ versus nitrogen partial

pressure.

Here we see a rapid hardening of the lms for nitrogen partial pressure 0-2%, with

the coercivity approaching 200Oe and the anisotropy eld 600Oe. Further increase

of pN does not inuence Hce and leads to a gradual increase in Hk. This trend is

opposite to that of the normal incidence lm. Thus, the soft properties of normal

incidence lms is in no way a guarantee of magnetic eciency of the material used in

a device, where deposition on tilted surfaces (microstrip edges) is required.

2.5.3 FMR susceptibility

We have measured the high frequency magnetic response of the lms using the tech-

niques described above (see section 2.4.3). To illustrate the variation in the response

of the lms, we show in Fig. 18 the complex susceptibility of two FeN lms deposited

at approximately 10Æ and 15Æ, represented with open and closed symbols, respectively

(squares - real component, circles - imaginary component).

39

0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4-600

-300

0

300

600

900

1200

1500

Fe-N(4.5%) 4πMS=21kG

Su

scep

tib

ilit

y

frequency (GHz)

Fig. 18. Complex susceptibility versus frequency for FeN(4.5%), =10Æ (open

symbols) and 15Æ (closed symbols).

The measured values of the initial permeability are 600 (10Æ) and 360 (15Æ). The real

part of the permeability goes through zero at the FMR frequency, 2.2 GHz and 2.8

GHz, respectively. Theoretical tting of the susceptibility spectra using (36) yields

the following parameters of the material: Hk=33 Oe, 4Ms=21 kG and =0.018 for

=10Æ and Hk = 55Oe, 4Ms = 20 kG and = 0:012 for =15Æ.

40

3 Applications: GHz inductors

Electronic products continue to undergo a rapid reduction in size and weight. An in-

creasing demand for communication products has been motivating research on mono-

lithic integration of radio components and systems. The fundamental electronic com-

ponent least compatible with silicon integration is the inductor, which is required for

implementation of lters, oscillators and matching networks. The dominating inte-

grated inductor design is an air-core spiral. Spirals are inecient magnetically, rela-

tively large, and often perform poorly in silicon integrated circuits. Use of magnetic

lms as ux-amplifying components allows for smaller inductors. Magnetic inductor

designs exist with most of the ux contained within the magnetic lms, which reduces

stray elds and the associated losses. After summarizing the current state of research

in the eld, we discuss the material and design issues involved in developing ecient

magnetic lm inductors. We then continue by analyzing one specic implementation

- a magnetic sandwich strip inductor.

3.1 Background

Saleh and Qureshi [34] proposed a magnetic thin lm inductor consisting of a square

spiral deposited between two Permalloy lms, 0.3 m thick on a glass substrate.

The inductor operated at 10 MHz and had a quality factor Q '18 with a 15%inductance enhancement over the free space value. In spite of the small gain in the

specic inductance, this work is still relevant for high-frequency magnetic inductor

design because of the use of the transverse permeability (AC eld transverse to the

equilibrium magnetization), as well as the segmentation of the magnetic lm to reduce

displacement currents due to the distributed conductor-to-lm capacitance. Soohoo

[35] gave a basic magnetic analysis for a magnetically sandwiched spiral and a mag-

netic lm core solenoid. He also presented a prototype inductor with copper lm

winding 'wrapped' around a Permalloy lm/glass(Si) substrate, which showed a 700-

fold enhancement in the specic inductance (no frequency or quality factor data were

provided).

Over the last 10-15 years there have been numerous eorts to fabricate an ecient

IC-compatible magnetic inductor and extend its operating frequency range from 1-

10 MHz to 100-1000 MHz. Shirae and coworkers have implemented a number of

structures, starting with planar coils embedded in SiO and sandwiched between two

Permalloy lms [36]. This design did not yield an ecient inductor (Q 1) and

41

the resonances at a few tens of MHz were attributed to the distributed coil/SiO/lm

capacitance in the structure. Going from magnetically sandwiched planar coils to

magnetically sandwiched conductor strips (Py/Cu/Py trilayers) patterned into planar

coils was found to improve the high-frequency characteristics of the devices: Q 3

at 100 MHz. The gain in the specic inductance due to the magnetic cladding was

however 'too small' [37]. In this regard the importance of magnetic material at the

edges of the conductor (anges) to achieve ux closure was pointed out [38]. A

gain in inductance up to a factor of 4 with Q2-3 at 100 MHz was obtained by

Yamaguchi et al. [39],[40] for Permalloy-coated conductor strips, with and without

magnetic closures at the edges, formed into meanders and spirals. Korenivski and

van Dover [41] have studied Cu strips sandwiched with Permalloy and Co-Nb-Zr,

with and without anges at the edges. Up to 7-fold inductance enhancements over

the air-core value (100 nH/cm linear inductance density) with Q 2 at f 250 MHz

were observed for 10-50 m strips. Another geometry, a solenoid, was studied by

Shirakawa and coworkers, who have demonstrated 10-fold inductance gains with Q

= 10-15 at f = 10-100 MHz for planar solenoids with laminated amorphous magnetic

cores [42],[43]. Two recent papers discuss the use of high moment, high resistivity Fe-

Ta-N and Fe-Al-0 lms (rather than Permalloy) in planar inductors, the trend that

will certainly continue. The design, however, was chosen such that in one case the

frequency range was limited to about 20 MHz [44] and in the other case the gain in

inductance was below 15% [45, 46].

3.2 Design

There is a number of ways one can incorporate a magnetic lm into a magnetic

ux device. These will vary in eciency, frequency range of operation, and ease of

fabrication. Here we briey argue advantages and disadvantages with the common

inductor designs for the GHz frequency range. We conclude the section by discussing

why a magnetic/conductor sandwich is a promising system for this application.

3.2.1 Spirals and meanders.

As mentioned above, there have been numerous attempts to minituarize a spiral

inductor by sandwiching it with magnetic lms. The main advantage of this structure

is that a well developed integrated fabrication process exists. However, there are

numerous disadvantages with this design:

42

the spiral is a compromise between the need for a planar layout and a solenoid,

and it is not very ecient magnetically. An extra metallization level is required;

insulation is required when coils are sandwiched between metallic magnetic

lms. This introduces a distributed coil-to-lm capacitance, which is strongly

geometry dependent and is recognized to be a limitation for operation at 1GHz;

the structure contains relatively large air gaps, which is undesirable from the

basic magnetic analysis point of view;

planar coils produce large out-of-plane elds, which in turn produce in-plane

eddy currents in the magnetic lms and Si substrate (Fig. 19). In-plane sec-

tioning of the magnetic lms can be used to reduce eddy currents in the magnetic

lms, but this may cause additional demagnetizing problems.

hard-axis (rotational, fast) permeability is preferred over easy-axis (switching,

domain wall motion, slow) permeability. In this regard biasing the magnetic

lms is not straightforward since the eld of a spiral is biaxial. A solution to

magnetically cover one-half of the coil [34] is available, though at the expense

of reducing the magnetic volume by 50%.

H

J

J

Jind

Si

Fig. 19. Large z-component of the magnetic eld of a spiral induces in-plane

eddy currents in metallic ferromagnetic lms.

43

3.2.2 Planar solenoids

A planar solenoid can be formed by plating the patterned lower conductor layer,

covering it with polyamide or SiO, depositing the magnetic core of desired geometry

with another layer of polyamide/SiO, arranging for metal via contacts and nally

plating the patterned top conductor layer. The advantages with this geometry are:

a well-developed fabrication process;

magnetically most ecient;

biasing the magnetic core is straightforward;

most of the eld produced by the solenoid is in the plane of the core, so eddy

currents can be controlled by varying the thickness of the core (lamination if

necessary).

The disadvantages are:

large distributed capacitance in the structure;

relatively complex structure with multiple via contacts, which may add up to

a high resistance of the coil and lead to low Q. It is important to note that

the inductors used at 1GHz are often of only several nH, a value that can

be realized with a magnetic core solenoid having a single turn. A single-turn

solenoid is essentially equivalent to a magnetic sandwich strip inductor.

3.2.3 Sandwiched strip

A sandwich strip inductor is a thin lm multilayer having a conductor strip, typically

made of Al or Cu, 0:1 1m thick, 10 20m wide and 1mm long, sandwiched

between two magnetic lms, with optional insulation layers between the conductor

and the magnetic lms. In addition, ux closure can be achieved by incorporating

magnetic anges at the edges of the strip, ux-linking the magnetic lms. A magnetic

sandwich strip with a 5 10-fold inductance enhancement over the air-core value can

substitute a typical RF spiral, and thus make free a lot of chip area. If needed, longer

strips can assume a spiral or a meander layout to achieve large inductances. The

structure has the following advantages:

simple in fabrication;

44

biasing the magnetic lms is straightforward;

the external magnetic ux is a small part of the total ux. The external eld is

in the plane of the substrate, therefore a reduced dissipation in Si is expected;

the excitation eld produced by a current in the conductor is mostly in the

plane of the magnetic lms, so eddy currents can be controlled by varying the

thickness of the lms;

no insulation is required, since the dierence in the conductivity between the

conductor (Al, Cu) and currently available high-resistivity soft alloys is 100.

This eliminates the unwanted capacitance.

Disadvantages:

anges are desirable for magnetic eciency. Incorporating anges adds addi-

tional fabrication steps and is typically done by making the top magnetic lm

somewhat wider to cover the conductor strip. The process can result in some

additional stress in the magnetic lm at the edges and, hence, anisotropy. This

problem, however, is routinely dealt with in making writing heads in magnetic

recording. The latest advances in electrodeposition of high moment high re-

sistivity lms (see section 2.3) are promising, since the technique is known to

be eective in achieving magnetic ux closures in micro-devices (such as write

heads in magnetic recording).

3.3 Device fabrication

In this section we describe the fabrication and characterization of magnetic sandwich

inductors. We conclude by discussing the circuit characteristics of the devices and

the issues limiting their high frequency performance.

3.3.1 Lithography process

The inductor is a strip of width 2-100 m and length 1250 m (center-pad separation)

xed at the pitch of the RF probe. A scanning electron microscope (SEM) image in

Fig. 20 shows a set of stripe inductors with widths 5-20 m.

45

Fig. 20. SEM image of an inductor set.

The sandwich consists of three layers. The bottom and top magnetic lms have the

same thickness (typically 100-200 nm). The conductor layer is 400-1000 nm thick.

The substrate is silicon with a 1 m thermal oxide layer.

Normally, a sandwich with ux closure (with an overlap of the magnetic lms at

the edges) is produced using three mask steps as illustrated in Fig. 21 (left to right).

Mask N1 Mask N2 Mask N1

Fig. 21. Fabrication of a sandwich strip with ux closure using three mask

steps.

We have fabricated inductors with ux closure using only one lift-o mask.A two-layer

resist system is used, with a PMGI low contrast resist as the bottom layer and a high

contrast resist, ZEP, as the top layer. The mask after development had an undercut

46

of 0.5-1.5 m as schematically shown in Fig. 22. We made use of the dierence in the

shadow depth for evaporation and sputtering. An e-beam evaporation source placed

at a distance of 20 cm from the substrate was used to produce a sharp edge mapping

of the top resist layer (solid arrows in Fig. 22). Sputtering, on the other hand, results

in material deposition in the open as well as the undercut regions (dashed arrows in

Fig. 22).

Fig. 22. Schematic of the lift-o process.

The following sequence was used:

1. spin the two-layer resist (PMGI SF71, ZEP5202 diluted 1:2), expose3 and de-

velop4 the pattern. The development time for the bottom resist layer5 is slightly

longer, in order to create 1 m undercut;

2. sputter 100-200 nm thick FeN or FeTaN lm, evaporate 400-500 nm thick Cu

lm, sputter FeN or FeTaN (see section on page 51 for details);

3. lift-o6.

The pattern was dened using an e-beam writer. Test patterns written with the dose

of 100 C=cm2 and dose modulation 50-150% were used to determine the proximity

correction. An optical image of a typical mask with a large undercut is shown in Fig.

23.

1At 1000 rpm it gives 600nm thickness; (soft bake 200ÆC 10 min on hot plate is needed).2At 3000 rpm it forms 90 nm thickness; (soft bake is 160ÆC 10 min on hot plate).3At 30 kV dose is in the range 80-100 C=cm2; 60 m and 120 m e-beam apertures give

reasonable exposure time.4Develop top layer 35s in P-xylene; blow dry.5Develop bottom layer 4.5min in 60%MF322 + 40%H2O; rinse with Distill water; blow dry.6with the Shipley 1165 Remover in a water bath of 55-60ÆC; rinse with Distill water; blow dry.

47

Fig. 23. Optical microscope image (top view) of a two-layer mask for 2m

width inductor. The dark area is SiO, the top resist layer is transparent (red

lter used).

An optical image in Fig. 24a shows a Cu/FeTaN sandwich with a FeTaN ange

produced using the above mask. The cross-section of a FeTaN/Cu/FeTaN sandwich

produced using a mask with a small (0.5 m) undercut is shown in Fig. 24b. Ap-

preciable sputter deposition on the bottom resist wall in this case can result in the

ange wrapping up during the lift-o. Fig. 24b demonstrates that a thin magnetic

lm can wrap up on the edges. This can be as a result of either too small undercut

and remaining resist on the substrate or strong penetration of the sputtering.

In order to avoid edge curling deeper undercuts have to be used. These, however,

should not be too deep as that would lead to weak top-layer resist bridges (see section

3.3.2).

48

(a)

(b)

Fig. 24. (a) Optical microscope image (top view) of Cu/FeTaN; (b) SEM cross-

section of FeTaN/Cu/FeTaN produced with 0:5 m undercut depth.

3.3.2 Film deposition

In this section our deposition system is described, in setting up and maintaining/upgrading

which I took an active part. Evaporation and sputtering steps in the lift-o process

are discussed.

3.3.2.1 UHV system The system we use for making lms has a UHV design

with a load lock, bakable to 200ÆC (Fig. 25). It is turbo/TSP pumped and has a

built in manipulator for doing angle depositions. The pressure is measured at dierent

locations: in the load lock (thermocouple gauge), at the chamber bottom (ion gauge)

and top (combined ion/thermocouple gauge). A pirani gauge is used for oxidation

inside the chamber.

49

MDX 500HVpowersupply

O2

-+

-

+

N2

Ar

~

DepositionmonitorXTM2

SKY945

Load lock

Piranigauge

Thermo-couplegauge

Combigauge

Iongauge

Air inlet

Fig. 25. Schematic of UHV system with standard graphic symbols used in

vacuum technology.

There is one ve-pocket e-Gun and two 2 inch sputter sources (one regular magnetron

source and one specialized for magnetic targets). The system is set up for doing in-situ

oxidation and reactive sputtering (O2; Ar;N2). The gas ow is controlled manually

with leakage valves and a turbo pump shutter.

3.3.2.2 Reactive magnetron sputtering Reactive sputtering is a method of

depositing lms of composition dierent from that of the target, and consists of adding

to the sputter gas (typically Ar) a reactive gas (typicallyN2 or O2). Nitrogen (oxygen)

in the plasma reacts with the target material (say Fe) and is incorporated in the

50

deposited lm. The amount of nitrogen in the lm is controlled by varying the partial

pressure of N2 in N2Ar plasma. We have used AJA A300 series magnetron sources

with Advanced Energy MDX500 magnetron drives.

Fe and Fe9Ta1 targets have been used. The total pressure was kept at 3mTorr

with 0 12% partial pressure of nitrogen. Depositions were made at ambient tem-

perature. During sputtering (typically 7 min for a 200 nm lm) the temperature is a

function of magnetron power and can increase to 100ÆC, which can result in melting

of the mask. Therefore, the power was restricted to 300W , corresponding to maxi-

mum deposition rate of 5 Å/s. Sputter deposition in the (large) undercut region is

demonstrated in Fig. 26 for a thicker (500 nm) sputtered Al lm (Al allowed for

clean cross sections, sputtered iron lms had a similar prole in the undercut region).

Fig. 26. Cross-section of a mask with an Al lm sputtered at 300W .

3.3.2.3 Electron beam evaporation Electron beam heated sources dier from

resistance heated sources in two ways: the heating energy is supplied to the top

of the evaporant by the kinetic energy of a high current electron beam, and the

evaporant is contained in a water cooled cavity or hearth. As a result the source

of evaporated material can be conned to a small spot. The small spot combined

with the line-of-sight directional deposition makes e-beam evaporation suitable for

producing patterned structures with boundaries well dened by the edges of the lift-

o mask.

We have used a Thermionics ve pocket e-Gun system (Model 100-0050) for

depositing (through mask) the conductor lm (Cu, typically 3 108 Torr base

pressure, 7-10 Å/s rate). The Cu layer thus deposited maps exactly the mask layout

(Fig. 27).

51

Fig. 27. Cross-section of a mask with evaporated Cu. Thickness of the bridge

is 90 nm, the undercut is 0:5 m.

A gradual buildup of the material at the edge of the mask leads to a gradual reduction

in width of the lm during deposition, resulting in an edge slope of about 80Æ.

A 5nm thick capping layer of gold was evaporated on the top of the magnetic

layer to ensure a good electrical contact for impedance measurements.

3.4 GHz inductors

In this section we describe the measurements and discuss the performance of magnetic

sandwich inductors.

3.4.1 Impedance measurement: calibration and deembedding

A four-point technique was used for DC resistivity measurements of the lms. The

resistivity of FeN lms was in the range 20 80 cm, that of Cu 2.4 cm.

For impedance measurements we have set up a xture with an XYZmicro-positioning

stage, a Cascade 12GHz xed-pitch probe, a stereo-scope for sample positioning

and an HP8714C network analyzer (Fig. 28). The real and imaginary parts of the

impedance were obtained from measurements of the reection coecient, = x+ iy,

in the 30MHz 3GHz frequency range:

Re (Z) =100y

(1 x)2+ y2

52

Im (Z) =50

1 x2 y2

(1 x)

2+ y2

:

Fig. 28. The experimental setup for HF impedance measurements: XYZ micro-

positioning stage with the sample holder, 12GHz probe, stereo-scope, HP8714C

network analyzer, PC.

The measurements are controlled by a PC using LabView. The external magnetic

eld is applied using a Helmholtz coil positioned around the xture (not shown).

The following deembedding procedure was used. The network analyzer was cali-

brated to the probe plane (PP in g. 28). A correction was made for the electrical

53

delay corresponding to the probe length. The measured impedance was modeled as a

sum of the probe, contact (probe to pads), xture (coupling to Si),and Device Under

Test (DUT) impedances (see Fig. 29).

ZDUTRcontZprobe

Zfixture

Fig. 29. Schematic of the xture impedance.

The contact resistance was evaluated from the low frequency limit using a set of

strips of varying width. The DC-limit resistance scaled linearly vs. the inverse strip

width with a 0:2 0:5 oset, which was attributed to the contact resistance of the

probe/pads (Fig. 30).

0.05 0.10 0.15 0.200

2

4

6

8

10

12

14

Rcont

= 0.24ρ l/t =59

Rexp

Rcont

+ρ l/tw

1/w (µ m-1)

R (

Ω)

Fig. 30. DC resistance of a set of strips.

54

The Cascade probe introduces a series impedance, which was determined using Cu

test strips. The impedance of the strips of varying width could be accurately modeled

and compared with the measurements. The correction for parallel xture impedance,

Zfixture, due to the measurement conguration was found to be negligible for strips

5m and wider (see Fig. 31).

0.1 1-6

-4

-2

0

2

0.1 1

100µm

5µm

5µm

w=2µm

f (GHz)

∆R

(Ω

)

-1.0

-0.5

0.0

0.5

1.0

100µm

w=2µm

∆L

(n

H)

Fig. 31. The measured impedance of Cu test stripes with the series impedance

and the calculated impedance of the stripes subtracted.

3.4.2 HF performance

The measured inductance, L = Im(Z)=!, is plotted in Fig. 32 as a function of

frequency for dierent inductor widths, unbiased and in an externally applied eld of

35 Oe.

55

0.1 10

1

2

3

4

5 µ m

10 µ m

20 µ m

50 µ m

100 µ m

L (

nH

)

f (GHz)

H ~ 35 Oe H = 0

Fig. 32. Inductance as a function of frequency for 5, 10, 20, 50, and 100 m

wide strips, unbiased (dashed lines) and biased in 35Oe (solid lines).

The contribution of the anges to the magnetic inductance becomes smaller as the

width of the strip is increased. The measured inductance of the 50m strip is essen-

tially the value expected for a sandwich with perfect ux closure, hence the model for

the impedance of a closed magnetic structure should apply (see (52) on page 22 and

equivalent circuit approximation in appended paper V). The inductance and quality

factor for the 50m width strip measured in three biasing elds are shown as func-

tions of frequency in Fig. 33a and 33b, respectively, along with the theoretically tted

curves. Fixing the geometrical parameters at the measured values, a good t can be

obtained by varying the permeability and damping parameter.

If we use the same parameters to t the impedance of the 10m strip, the deviation

from the predicted performance is signicant (see Fig. 34). In order to obtain a good

t not only the permeability has to be signicantly reduced (by roughly a factor of

two) but the dissipation constant has to be signicantly increased: from =0.028

(consistent with values 0.020-0.025 measured on similar test lms using the technique

56

of 2.5.3) to 0.06.

(a)

0.1 10.8

1.0

1.2

1.4

1.6

H3=160 Oe

H2=84 Oe

H1=42 Oe

L (

nH

)

f (GHz)

experiment theory

(b)

0.1 1.0

1

10

H1

H2

H3

Q

f (GHz)

experiment theory

Fig. 33. L (a) and Q (b) versus frequency for a 50 m wide inductor in three

biasing elds. Solid lines are theoretical ts using (52).

The reduction in the eective permeability for narrow strips can be attributed to

the increasing inuence of the incomplete ux closures at the edges. However, the

57

increase in the damping constant is not expected to directly depend on the width of

the strip.

(a)

0.1 10

1

2

3

4

5

H1=42 Oe

H3=160 Oe

72 Oe

260 Oe

L (

nH

)

f (GHz)

experiment theory theory

(b)

0.1 1.00.1

1

H3

H1

260 Oe

72 Oe

Q

f (GHz)

experiment theory theory

Fig. 34. L (a) and Q (b) versus frequency for a 10m wide inductor in two

biasing elds. Solid lines are theoretical ts using (52) with the parameters

from Fig. 8. Dashed lines are ts with assuming a reduced permeability and

increased damping constant (see text).

58

A probable cause of the increased dissipation is that an increasing portion of the

magnetic ux leaks through the conductor near the ange, causing screening currents

in the conductor. This dissipation mechanism is not accounted for in the 'closed-ux'

model (52) but known to be the dominating loss mechanism in sandwiches without

ux closure [appended paper IV]. The performance of the inductors at 1 GHz is

summarized in Fig. 35. A 2-fold inductance gain with Q 3 was obtained with the

20m width inductor. By varying the width of the device the inductance gain can

be compromised for higher quality factor.

0 20 40 60 80 1000

1

2

3

4

5

6

7

L (

nH

), Q

w idth (µ m )

L Q

Fig. 35. Inductance and quality factor at 1 GHz versus inductor width.

3.4.3 Flux closure at the edges

From the low frequency limit of the inductance (Fig. 32, solid lines) one can determine

the magnetic lm contribution, which excludes any dissipation or resonance eects

at high frequencies. The total inductance in this limit, L = L0 + Lm, is the sum of

the air-core inductance of the strip, L0, and the contribution due to the magnetic

lm, Lm. In Fig. 36, Lm is plotted versus the strip width. The experimental data

deviate from the ideal 1=width (dash-dotted line) behavior expected when there is

perfect magnetic ux closure at the edges. The observed magnetic contribution to the

59

inductance is below the theoretical value for narrow strips, while it approaches the

expected values for wide strips. This behavior indicates an incomplete ux closure at

the edges, which plays a progressively larger role as the width of the strip is reduced

[41].

10 100

0.1

0.2

0.3

0.4

0.5

0.60.70.80.9

1

2

3

4

5

100

50

20

105

2

L

m (

nH

)

w idth (µ m )

FeN(4%) exp. models, µ

r=390:

g=44nm, wf=0.1µ m

g=0 g=t

c

Fig. 36. The magnetic contribution to inductance, Lm. The expected in-

ductance of a 'closed' magnetic structure (dash-dot line) and 'open' structure

(dashed line).

3.5 Prospectives

We have demonstrated a signicant inductance gain at 1 GHz by using magnetic/conductor

sandwiches. Several issues should be addressed in the future. The use of soft lms

with resistivities of 100 -cm and higher should allow thicker magnetic layers in

the sandwich, hence larger magnetic ux and inductance. Use of thicker conductor

layers, provided the edge ux closures are not thereby degraded, should yield higher

quality factors. Control of the properties of the magnetic lms at the strip edges

needed for ecient ux closure is crucial for both improving the inductance gain and

performance in the GHz range.

60

We see no principal limit on the inductance gain one can achieve in a magnetic

sandwich, even with the currently manufacturable materials. The magnetic volume

can be increased, and screening controlled, by using laminated magnetic lms. E-

cient magnetic closure must of course be maintained.

When the 'extrinsic' dissipation issues of screening and incomplete ux closure

are solved by using optimum device parameters, the 'intrinsic' dissipation in the fer-

romagnetic material (usually described by a phenomenological damping constant) is

seen as the main factor limiting the device performance. Although magnetic relax-

ation is a long established area of solid state physics, the understanding of relaxation

in the technologically attractive lms we have discussed is in its early stages. Ulti-

mately, one would like to know the microscopic and microstructural mechanisms of

the magnetization dissipation and, more importantly, be able to control it.

61

4 Appendix

Combined preampleer and integrator circuit

The two stage preampleer is built on very low noise, low oset, low drift Precision

Dual Operational Amplier, OPA2107. It has a xed gain of 1000.

In the compensation block we have an instrumentation amplier, which compares

the voltage between two middle points of R3=R4 and C2=P1 circuits. Varying the

resistance of P1(P2) in 0 1 range, one can shift input phase in 0 range,

while keeping the constant amplitude. The input voltage, taken over the 1 resistor,

is equal to the driving current in Helmholtz coil. In the next stage we subtract the

compensation signal (divided with P3 and P4) from amplied signal. The second

instrumentation amplier, INA111, is AC coupled (U4B) and has two additional

gains for dierent volumes of the samples.

The nal stage is the band-pass lter, optimized for attenuation of the low fre-

quency noise below 100Hz and possessing the dispersion function f(!) = 1

!above

0:5 kHz.

+12V

R4 4.43k2

3

6

74

1

8

5

U2

INA111

2

3

6

74

1

8

5

U3

INA111

11

3

A1 A2 A3

50 k50 k

50 k

50 k

1000pF 1000pF

11478

12

139 10

¾ U4A

UAF42

2

3

4

1

½ U1A

OPA21075

6

8

7

½ U1B

OPA2107

VSS

5

46

¼ U4B

UAF42

VCC

R9 33k

R2

10 k

R1

10

R8 6.8k

C1 100pF

IN

VCC

VCC

VCC

P3

1 k

P4

100

R7

1.5 k

R5

50k

VSS

R3

4.43 k

R61 Ω

C2

0.22

P1

10 k

P2

1 k& COIL

R101 M

R11

22 k

C3

0.1

VSS

VSS

R121 k

C4

0.01

C5

0.01

R13

150 k

R14

220 k

M

OUT

dM/dt

OUT

-12V

HOUT

VSS

VCC

Ω

Ω

S1

"GAIN"

from

SINsource

pick-up coil

"PHASE"

"FINE"

"AMPLITUDE"

"FINE"

× 8.3

× 2.5

× 1000

S2

Preamplifier

Compensation block

Integrator

Fig. 37. Schematic of analog integrator based on band-pass lter.

62

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65

5 Acknowledgments

I gratefully acknowledge nancial support from the Swedish funding agencies SI and TFR,

and Bell Laboratories, Lucent Technologies.

During my work at the department of Nanostructure Physics and, previously, at the Con-

densed Matter Physics department I have had the pleasure to work with several people,

whom I wish to acknowledge.

Professor David Haviland, head of Nanostructure Physics department, for giving me the op-

portunity to work in his group and for excellently organizing the work in a newly established

department.

Docent Vladislav Korenivski, my supervisor and friend, for motivating me to work on mag-

netic devices and for the stimulating discussions in a broad area of magnetism. Tatjana

Korenivski (absolute record in friendship: almost 20 years) for her life guidance.

Docent Anders Liljeborg for careful management of Nanofabrication Laboratory, so that we

PhD students have scientic results in spite of capricious E-beam lithography system.

Professor Alexander Sukstanskii, who has shown me the real meaning of theoretician and

teacher.

Karin Andersson and Mattias Urech, my colleagues and friends, for shearing our oce on a

high spiritual level. Special credit goes to Mattias for ideas on perfect phase compensation

in Loop-Tracer electronics.

Peter Ågren, Jan Johansson, Jonas Rundqvist and Jochen Walter, Doctor Michio Watanabe

and Doctor Volker Schöllmann, my colleagues and friends, for the shearing knowledge of

dierent elds in physics and for the humorous atmosphere in the nano-group.

Professor K. V. Rao, for inviting me to Sweden and introducing me to experimental physics.

Professor Alex Grishin for discussion of many subtle points, which lead me to the deeper

understanding the problems in physics.

Doctor Sergey Khartsev for much useful advice in experimental work.

Björn Rodell, Jesper Wittborn, Valter Ström, my colleagues at CMP-KTH for being such a

nice company when I rst arrived in Stockholm.

Andrius and Egle Miniotas, my close friends who made me feel at home in our student

housing in Kungshamra.

My family, Olga and Arina for their love and encouragement.

66

6 Appended papers

I. A Model for Impedance of Planar RF Inductors Based on Magnetic Films, A.

Gromov, V. Korenivski, K. V. Rao, R. B. van Dover, P. M. Mankiewich, IEEE

Transactions on Magnetics, 34 (1998) 1246.

Maxwell's equations are solved analytically for a magnetic/conductor/magnetic sand-

wich strip with ux closure at the edges and no driving current in the magnetic lm. The

nal result is the impedance of the sandwich as a function of frequency and geometrical

parameters. My contribution was all of the calculations and the rst draft.

II. Analysis of Current Distribution in Magnetic Film Inductors, A. Gromov, V.

Korenivski, D. Haviland, and R. B. van Dover, Journal of Applied Physics, 85 (1999)

5202.

This paper is an extension of the above model to account for driving current redistribution

across the interface between the conductor and the magnetic lm. The obtained impedance

is valid for long and thin strips (large length-to-width and width-to-thickness ratios). My

contribution was all of the calculations and the rst draft.

III. Electromagnetic Analysis of Layered Magnetic/Conductor Structures, A. Gro-

mov and V. Korenivski, Journal of Physics D: Applied Physics, 33 (2000) 773.

A method for calculating the impedance of layered magnetic/conductor structures of

arbitrary cross section is presented. The method is exemplied on a conductor of axial

symmetry (wire) coated with a high permeability lm. The dependence of the impedance

on an external magnetic eld is analyzed in relation to the eect known as Giant Magneto-

Impedance, which is promising for sensor applications. My contribution is the calculations

and rst draft.

IV. Impedance of a Ferromagnetic Sandwich, A. Sukstanskii, V. Korenivski, and

A. Gromov, Journal of Applied Physics, 89 (2001) 775.

Maxwell's equations coupled with the Landau-Lifshitz equation for the magnetization

dynamics are solved for a three-layer sandwich, consisting of two ferromagnetic layers sep-

arated by a non-magnetic conductive layer. A 2D problem is analyzed, with and without

magnetic ux closure at the edges of the stripe. The impedance of the magnetically closed

structure is demonstrated to be more ecient then open structure. My contribution was in

discussing the boundary conditions and solutions for the external region, and commenting

on the manuscript.

V. GHz sandwich strip inductors based on FeN lms, A. Gromov, V. Korenivski

and D. Haviland, submitted to Journal of Physics D: Applied Physics (2001).

67

Planar strip inductors based on Fe-N and Cu lms have been fabricated on oxidized Si

substrates. The impedance measurements were carried at frequencies up to 3 GHz, and

revealed a substantial inductance enhancement compared to the air-core values of the strips.

Magnetic characterization of the structures was used to analyze the factors determining their

impedance. My contribution was all experimental aspects as well as the rst draft.

68