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Physics World Archive

Single-electron transistorsMichel H Devoret, Christian Glattli

From

Physics WorldSeptember1998

IOP Publishing Ltd 2014

ISSN: 0953-8585

Downloaded on Wed Oct 15 16:32:56 GMT 2014 [131.193.117.248]

Institute of Physics Publishing

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FEATURES

While the e lectronics industry wonders w hat

will

happen when transistors becom e

so sma ll tha t quantum effects become im portan t, researchers are building

new transistors tha t actively exploit the quan tum properties of electrons

Sin g le-elec tro n tran s is to rs

M ichel Devoret and Christ ian Glatt l i

THE INVENTION of the transistor

b y

Joh n Bardeen

and William Shockley in 1948 triggered a new era in

electronics. Originally designed simply to emulate

the vacuum tube , scientists soon found tha t

this

solid-

state device could offer m uch m ore. T he great poten-

tial of the transistor for speed, miniaturization and

reliability has been fully exploited since well con-

trolled m aterials suchasp ure single-crystal silicon b e-

came available. According to the latest roa d-m ap

for the microelectronics industry, microchips con-

taining one billion transistors and operating with a

clock cycle of a billionth of a second will be on the

market jus t a few

years

into the new millennium.

As

transistors continue to shrink, a question natur-

ally arises: will the qu antu m nature of electrons an d

atoms become important in determining how the

devices are b uilt? In o ther words, what will hap pen

when a transistorisreducedtothe size of a few atom s

or a single molecule?

Researchers seeking

to

answ er these questions have

devised the so-called single-electron tunnelling tran-

sistor - a device that exploits die quan tum effect of

tunnelling to control and measure the movement of

single electrons. Experim ents

have

shown that charge

does not flow continuously in these devices but in a

quantizedway. Inde ed, single-electron transistors are

so sensitive to charge that they can be used as

extremely precise electrometers.

1 O pe r a t io n o f a c o n v e n t i o n a l t r a n s i s t o r

channel

energy

At the beginning

Th e m ost common transistor

in

today's microchips

is

the metal-o xide-s em icond uctor field-effect transis-

tor (MO SFE T). Its operation

is

surprisingly simple: not much

quantum mechanicsisrequired to unde rstand how it works,

even thoug h die size of a typical deviceisnow justafew thou -

sand atom s placed sidebyside.

Two con ducting electrodes, called the source and drain, are

connected together by a channel of material in which the

density of free electrons can b e varied

in practice a semi-

con duc tor (figure

1). Avoltageis

applied to the semiconduct-

ing chan nel through the gate , a third electrode that is

separa ted from the channel by a thin insulatinglayer.When

the gate voltage is zero, the channel does not contain any

cond uction electrons a nd is insulating. But as die voltage is

increased, the electricfieldat the gate attracts electrons from

the source and drain, and the channel b ecomes conducting.

P H Y S I C S W O R L D S E P T E M B E R 1 9 9 8

distance

The metal-cxide-sem iconduc tor field-effect transistor

MOSFET)is the

basic switching

and am plification device of

digital

e lectronics.

The

current between the source

and

drain

electrodes

is

controlled

by the

gate

voltage, (a)

Whenthe gatevoltage is

zero,

no

conduction electrons are presentin the

channel,

(b) Whenthe gateis at a positive

voltage,

electrons fromthe source and drain accumulate in the areaof the channel

closeto the gate, (c) As the gate

voltage is increased further, the num ber of electrons in

the channel increases until saturation is

reached. The

potential

seenby the

electrons is

alsoshown along

a

linegoing fromthe gate

to

the

channel.

With

no

gatevoltage,

electrons in the channelexperience a potential that is higherthanthe bias po tential,

shown by the dashed line (d). As thegatevoltage increases, the potential in the channel

gradually lowers and electrons accumulate there( e - f ) .

This field effect leads to an amplification mechanism in

which die gate voltage can control the current flowing

b etween die source and d rain when a bias voltage is applied

across these two electrodes (figure 2). The source-drain cur-

rentisdetermined b y the conductance of the channel, which

in turn depends on

two

factors: die density of the cond uction

electrons and their mobility. The mobility of electrons

depends on how often the electrons collide with crystal

imperfections, and

is

essentially inde pend ent of the gate volt-

age. In contrast, the density of electronsiscontrolled direcdy

by the gate voltage.

The transistor therefore works like a tap controlling the

flow of water between two tanks, where the opening of the

tap

is

set by the pressure of the w ater

in

a third tank. Th e dif-

2 9

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gate voltage

The

current in

a

field-effect transistor varies with gate

voltage

and

with the bias voltage between the source and

drain.

For

a

fixed bias voltage, the current

is

turned on when

the gate voltage is positive,and turned offwhen

th e

gate

voltage is negative.

ference is that electrons in the

channel behave as a compressible

fluid with a local density that

depends strongly on the electric

potential at that point. In other

words, the electric field produced

by the gate does not generate a

hard

wall

for electrons inside the

channel, but a smoothly varying

potential that is modified by the

presence of electrons (figure 1

d-f .

Note that

we

have ma de no refer-

ence to the wave-like properties of

electrons, nor to the fact that the

channel is made from individual

atoms.

Th e only quantum property

that comes into play is the Pauli

exclusion principle, which dictates

that each possible state in the chan -

nel can be occupied by only one

electron. This means that only a

certain num b er of electrons can accumulate in the channel,

and this sets a limit on the cur rent flow.

However, the quantum properties of electrons and atoms

willplayamore importantroleas transistors are made smaller.

For example, the wave-like nature of electrons will influence

the way in which they travel through the channel. When the

width of the channel b ecomes comparab le to the wavelength

of electrons (around 100 nm), electron propa gation b ecomes

more sensitive to the atom ic disorder in the device, which is

createdinthe fabrication

process.

This will poseamajor prob -

lem if the reduction insizeisnot accompaniedbyan improve-

men t in the atomic structure of the fabricated devices.

The technological constraints of moving towards the

atomic scale may force us to adopt a new physical principle

for achieving the transistor's function. Alternatively, a new

principle might be found that can provide functions that are

no t

possible

with cu rrent devices.

Towards single-electron devices

In 1985 DmitriAverinand Kon stantin Iikharev, then working

at the University of Moscow, proposed the idea of a new

three-te rminal device called a single-electron tunnelling (SET)

transistor.Two yearslater Theodore Fulton and Gerald Dolan

at Bell Labs in the US fabricated such a device and demon-

strated how it operates.

Unlike field-effect transistors, single-electron devices are

based on an intrinsically quantum phenomenon: the tunnel

effect. This is observed when two metallic electrodes are sep-

arated by an insulating barrier about 1nm thick - in other

words, just 10atomsin a row. Electrons atthe Fermi energy can

tun nel throu gh the insulator, even thoug h in classical terms

their energy would be too low toovercomethepotential barrier.

Th e electrical b ehaviour of the tunne l junctio n depe nds on

how effectively the barrier transmits electron waves, which

decreases exponentially withits thickness, and on the num b er

of electron-wave modes that impinge on the b arrier, which is

given by the area of the tunnel junc tion dividedbythe square

of the electron wavelength. A single-electron transistor exploits

the fact thatthe transfer of charge throughthe b arrier becomes

quan tized when t he junc tion

is

m ade sufficiently resistive.

This qu antization process

is

shown particularly clearly in a

2 Ampl ificat ion in a f ie ld-effec t t ran sisto r

simple system known as a single-

electron box (figure 3). If a voltage

source charges a capacitor, C

g

,

through an ordinary resistor, the

charge on the capacitor is strictly

proportional to the voltage and

shows no sign of charge quantiza-

tion. But if the resistance is re-

placed by a tunnel junction, the

metallic area b etween the capacitor

plate and one side of the junction

forms a conducting island sur-

rounded by insulating materials. In

this case the transfer of charge on to

the island becom es quantized

as

the

voltage increases, leading to the so-

called Cou lomb staircase (figure

3c).

This effect was first observed by

Philippe Lafarge and collaborators

in our lab oratory in 1991.

This Coulomb staircase is only

seen under certain con ditions. Firstly, the en ergy of the elec-

trons due to therma lfluctuationsmustbe significantly sm aller

than the Coulomb energy, which is the energy needed to

transferasingleelectron onto the island when the applied volt-

age iszero.This Coulomb energyisgivenby e

2

/2C,where

e

is

the charge of an electron and Cis the total capacitance of the

gate capacitor,C

g

,and the tunneljunc tions. Secondly, the tun-

nel effect itself should be weak enoughtopreventthe charge of

the tunnelling electrons from b ecom ing delocalized over the

two electrodes of the junctio n, as happ ens in chemical bon ds.

According to recent work by theorists at the universities of

Freiburg and Karlsruhe in Germany , this means that the con-

ductance of the tunnel junction should b e much

less

than the

quantum of conductance,

2e

2

/h ,

where

h

is

Planck's constant.

Wh en b oth these conditions are met, the steps observed in

the charge are somewhat analogous to the quantization of

charge on oil droplets observed by Millikan in 1911. In a

single-electron b ox, however, the ch arge on the island is not

random but

is

controlled by the applied voltage. As the tem-

perature or the conductance of the ba rrier is increased, the

steps become rounded and eventually merge into the straight

line typical of an ord inary resistor.

Asingle-electron trans istor

Th e S ET transistor can b e viewedasan electron box that has

two separate junctions for the entrance and exit of single elec-

trons (figure

4).

It can also b e viewed

as

afield-effect ransistor

in which the channel is replaced by two tunnel junctions

forming a metallic island. The voltage applied to the gate

electrode affects the amount of energy needed to change the

num b er of electrons on the island.

Th e SE T transistor comes in two versions that have been

nicknamed metallic and semiconducting . These names

are slightly m isleading, however, since the principle of b oth

devicesisb ased on the use of insulating tunnel b arriers to sep-

arate cond ucting electrodes.

In the original metallic version fabricated by Fulton and

Dolan, a metallic material such as a thin aluminium film is

used to make all of the electrodes. Th e m etal isfirstevapor-

ated through a shadow mask to form the source, drain and

gate electrodes. The tunnel junctions are then formed by

30

P H Y S I C S W O R L D S E PT E M B E R 1 9 9 8

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3 A n e l e c t r o n in a bo x

capacitor resistor

/

\

capacitor

/

tunnel

/ junction

E 0

ra o

3

1

0

-

, j

ft

1

e

O

3e

voltage

5e

2 0 ,

a) Whena cap acitor is charged through a resistor, the charge onthe

capacitor is proportional to the app lied voltage and shows no sign of

quan tization, (b) When a tunnel junction replaces the resistor, a

condu cting island is formed between the junction and the cap acitor plate.

Inthis case the average charge on the island increases in steps as the

voltage is increased (c). The steps are sharper for m ore resistive barriers

an d

at lower temperatu res.

introducing oxygen into the chamber so that the metal

b ecomes coated by a thin layer of its natural

oxide.

Finally , a

second layer of the metal - shifted from the first by rotating

the sample - isevaporated to form the island.

In the semiconducting versions, the source, drain and

island are usually ob tained by cutting regions in a two-

dimensional electrongasformed at the interface b etween two

layersof semiconductors suchasgallium aluminium arsenide

and gallium arse nide. In this case the condu cting regions are

definedbymetallic electrodes pattern ed on the top semicon-

ducting layer. Negative voltages applied to these electrodes

deplete the electron gas just b eneath them, and the depleted

regions can b e ma de sufficiently narrow to allow tunnelling

between the source, island and drain . Moreover, the electrode

that shapes the island can b e usedasthe gate electrode.

In this semiconducting version of the SET, the island is

often referred to as a qua ntu m dot, since the electrons in the

dot are confined in all three directions. In the last few years

researchers at the Delft University of Technology in the

Netherlands and at NT TinJapa n have shown that quantum

dots can b ehavelikeartificial

atoms.

Indeed, it has b een poss-

ible to construct a new periodic table tha t describ es dots con-

taining different num b ers of electrons (see Qu an tum d ots

by LeoKouwenhoven and Charles Marcus

hysics Worldjune

pp35-39).

Operat ion ofa S Tt rans is tor

So how does a SET transistor work? The key point is that

charge passes through the island in quantized units. For an

electron to hop onto the island, its energy must equal the

Coulomb energy

e

2

/2C.

When b oth the gate and

bias

voltages

are zero, electrons do not have enough energy to enter the

island and curre nt does not flow. As the b ias voltage b etween

the source and drainisincreased, an electron canpassthrough

the island when the energy inthe system reaches the Co ulomb

energy.This effectisknownasthe Coulomb blockade, and the

critical voltage needed to transfer an electron on to the island,

equal to

e/C

iscalled the Cou lomb gap voltage.

Now imagine diat the bias voltage is kept below die

Coulomb gap voltage. If the gate voltage is increased, the

energy of the initial system (with no electrons on the island)

gradually increases, while the energy of the system with one

excess electron on die island gradually decreases. At die gate

voltage corresponding

to

die point of m aximum slope on the

Coulom b staircase, b om of these configurations equally qual-

ify as me lowest energy states of the system. This lifts the

Coulom b b lockade, allowing electrons to tunnel into and out

of die island.

The C oulomb b lockadeislifted w hen die gate capacitance

ischarged widi exacdy m inus half an electron, whichisnot as

surprisingasit may seem. T he islandissurroundedbyinsula-

tors,which mean s that the charge on it must b e quandze d in

units ofe but the gate is a metallic electrode connected to a

plentiful supply of electrons. Th e charge on die gate capaci-

tor merely represents a displacement of electrons relative to a

background of positive ions.

If

we

furmer increase the gate voltage

so

that die gate capa-

citor b ecomes charged w idi

-e ,

die island again has only one

stable configuration separated from die next-lowest-energy

states by the Coulom b energy. Th e Co ulomb b lockade is set

up a gain, b ut the island now contains a single excess electron.

The conductance of die SET transistor dierefore oscillates

b etween m inima for gate charges ma t are integer multiples of

e

and maxima for half-integer multiples ofe(figure 5 ).

Accurate measures of cha rge

Such a rapid variation in cond uctanc e m akes die single-elec-

tron transistor an ideal device for high-precision electrometry.

In this type of application the SET has two gate electrodes,

and the b ias voltage is kept close to the Co ulom b b lockade

voltage to enh ance the sensitivity of the curre nt to changes in

die gate voltage.

The voltage of the first gate is initially tuned to a point

where the variation in current reaches a maximum. By

adjusting the gate voltage around this point, the device can

measure die charge of a capacitor-like system connected to

die second gate electrode.Afraction of this measured cha rge

is shared by the second gate capacitor, and a variation in

charge of Vieisenough to change die current by ab out half

the maximum current diat canflowhrough the transistor at

the Coulom b blockade voltage. Th e variation in current can

be as large as 10 billion electrons per second, which means

that tiiese devices can achieve a charge sensitivity that out-

performs other instruments by several orders of m agnitude.

Indeed, a collaboration between researchers at Yale Uni-

versity in the US and Chalmers University in Gothenburg,

Sweden, recendy showed diat charge variations smaller tiian

10~

5

e

can be detected in a measurement period of just one

second andwidia b andwiddi of several hundred megahertz.

SET transistors have already been used in mesoscopic

physics experiments that have required extreme charge sen-

sitivity. For example, earlier this year R ob ert W estervelt and

co-workers at Harva rd University in the US used this type of

device to measure the rounding of steps observed in a

Coulom b staircase.

Electrometers b ased on S ET transistors could also be used

to measure die delicate quantum superpositions of charge

states in a supercond ucting island connectedbya tunnel

junc-

P H V S I C S W O R L D S L P I I M B E R 1 9 9 8

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4 Pr inc i pl e of th e SET t ran s i s to r

ion to a superconductor. Such an

island can accommodate not only

several charge states corresponding

to different numbers of Cooper

pairs,

but also coherent quantum

superpositions of these

states.

Super-

conducting islands could therefore

provide a means for implementing

the quantum bits needed for a quan -

tum compu ter (see Fundam entals

of quantum information

by

Anton

Zeilinger hysics M^rWMarchpp35-

40).

The feasibility of the idea has

been shown b y experiments at Delft

University, the State University of

New York at Stony Brook, the PTB

Laboratory in Germany, the NEC

Lab oratory in Japan and in our lab.

As we have seen, the charge sen-

sitivity of the SET is ultimately

linked to the fact that electrons tra-

verse the island one at a time. In

1990 Bart Geerligs, Valerie Ande-

regg, Peter Holweg and H ans Mooij

at Delft, together with Hugues

Pothier, Daniel Esteve, Cristian U r-

bina and one of us (MD) at CEA

Saclay showed that electrons can

be counted one b y one by creating

devices that combine several SET

transistors. An d in

1996

John Mar-

tinis and colleagues at the National

Institute for Standards and Tech nol-

ogy in Boulder, Colorado, showed

that a device called the electron

pump can count electrons with an

accuracy of 15parts in a billion. Th e

same group is now attempting to

measure the charge of the electron

with an accuracy better than 1 part

pe r1 millionbycomb ining an elec-

tron pum p with a specially calibra-

ted capacitor. Oth er m etrology labs

are aiming to use arrays of single-

electron transistors to establish a

standa rd for electric curren t.

The precision with which elec-

trons can be counted is ultimately

limited by the quantum delocal-

ization of charge that occurs when

the tunnel-junction conductance be-

comes comparable with the conductance quantum,

2e

2

/h .

However, the curren t through a SE T transistor increases with

the conductance of the junctions,soitisimportant to under-

stand how the single-electron effects a nd C oulom b b lockade

disappear when the tunnel co nductance is increased b eyond

2e

2

/h .In 1991Ko nstantin Matveev, nowatDukeUniversityin

North Carolina, drew a parallel between the suppression of

the Coulomb b lockade and the Kond o effect, in which mag-

netic impurities in metals are screened by conduction elec-

trons.

Experiments b y Philippe Joyez and co-workers in our

lab, and by Leonid Kuzmin and collaborators at Chalmers

tunnel

y

junctions

Like

a

MOSFET. the single-electron tunn elling

(SET)

transistor consistsof a gate electrode that electrostatically

influences electrons travelling between the source

and

drain electrodes. H owever, the elec trons in the SET

transistor need to cross two tunnel junctions that form

an

isolated conducting electrode called the island. E lectrons

passing through the island charge and discharge it, and

the

relative energies of systems containin g 0

or 1

extra

electrons depends on the gate voltage. At

a

low

sou rce-d rain vo ltage, a current will only flow through the

SET transistor if these two charge configurations

have

the

same energy.

5 C o u n ti n g e le c t r o n s w i th a s i n g l e -

e l e c tr o n t r a n s i s t o r

current

i

bias

~ = v o l ta g e

voltage

2C

g

2C

g

2C

g

2C

g

The current flowing in a single-electron transistor increases

with the bias voltage between the source and

drain,

an d

varies periodically with th e gate voltage. For low bias

voltages, current flows when th e charge on the gate

capacitoris ahalf-integer multipleo fe , but issuppressed

for integer mu ltiples of e.Eachtime an electron isaddedto

the

gate, an electron

tunnels

intothe island,

which

sets

the

field

in

the gatecapacitor back

to

its initial

value.

Peaks in

the conductance are observed for half-integer multiples of

e,and minima are seenat integer m ultiples of e. For bias

voltages largerthan e/C, conduction occurs independently

of

the

gatevoltage.

University have confirmed that

quantum fluctuations in the charge

of the tunnelling electrons reduce

the Coulomb energy.

Towardsroomtemperature

Until recently single-electron tran-

sistors had to be kept at tempera-

tures of a few hundred millikelvin

to maintain the thermal energy of

the electrons below the Coulomb

energy of the device. Most early

devices had Cou lomb energies of a

few hundred microelectronvolts

b ecause diey were fabricated using

conventional electron-beam litho-

graphy, an d

thesize

and capacitance

of the island were relatively large.

For

a SET transistor

to

work at room

tempera ture, the capacitance of the

island must b e less than 10~

17

F and

therefore its size must be smaller

than 10 nm .

This year two experiments have

demonstrated that SET transistors

can work at room temperature.

Lei Zhuang and Lingjie Guo of

the University of Minnesota, and

Stephen Chou of Princeton Uni-

versity in the US fabricated a SET

transistor in a similar way to a field-

effect transistor with a channel just

16 nm

wide.

Th e fabrication process

generated variations in the chann el

that act as tunne l junc tions defining

several different islands, and the

behaviour of the device is domin-

ated b y the smallest island.

As expected from theory, the

conductance of the device shows a

series of peaks as a function of g ate

voltage. The Coulomb energy of

the device is aroun d 100 meV,

which is large enough to reveal

single-electron effects at room tem-

perature. However, the Coulomb

energy istoo smalltoprovide a large

Coulomb gap, so the transistor is

not sensitive enough to be used as

an electrometer. Interestingly, the

wavelength of electrons is com par-

able with the size of the dot, which means that their con-

finement energy makes a significant contribution to the

Coulomb energy.

Meanwhile, Jun-ichi Shirakashi and colleagues at die

Electrotechnical Laboratory and the Tokyo Institute of

Technology fabricated a metallic SE T from a very thin layer

of niob ium. Tunnel junctionswerecreated b y oxidizing areas

of the niobium with die tip of a scanning electron micro-

scope. The tip had almost atomic resolution, allowing the

researchers to make very n arrow oxide barriers b etween the

island and the source and drain, and a wider b arrier b etween

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PHYSICS WORLD S E P T E M I E I 1998

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the island and gate.

The team measured the current through the device asa

function of b oth b ias and gate voltages, and the results closely

match theoretical predictions. The Coulomb energy ofthe

device is 25 0 meV, which mean s that single-electron effects

can readily be seen at room temperature. However, tunnel

barriers fabricated inthis wayare highly resistive, which

means that the current isab out 100 times smaller th anin

devices operating atlowtemperatures. This problemalsolim-

its die usefulness of th e device for electrom etry

Perspectives onthe future

Researchers have long considered whether SET transistors

couldbeusedfordigital electronics. Although the current

varies

periodically with

gate

voltage in contrast

to

the thresh-

old behaviour of the

field effect

ransistor a S ET could still

form a co mp act an d efficient mem ory device. However, even

the latest SET transistors suffer from offset charges , which

means that the gate voltage needed

to

achieve maxim um c ur-

rent

varies

random ly from device to device. Such fluctuatio ns

make it impossible to build complex circuits.

One way to overcome this problem might be to combine

the island, two tunnel junctions and the gate capacitor that

comprisea single-electron transistor inasingle m olecule

after

all,

the intrinsically q uantum b ehaviour of a SE T tran-

sistor should not b e affected at the molecu lar

scale.

In prin-

ciple, the reproduc ibility ofsuch futuristic transistors would

be determ ined b y chemistry, and not by the accuracy ofthe

fabrication process. Last yearateam led by Cees Dekker at

Delft and Richa rd SmaJley at Rice Universityinthe US m ade

a crucial step in this direction b y ob serving Coulom b block-

ade in an island consisting of a single carb on nano tub e.

It is not yet clear whether electronics based on individual

molecules andsingle-electron effects will replace c onven-

tional circuits based on scaled-down versionsof field-effect

transistors. Only one thing is certain:ifthe paceofminia-

turization continues unabated, the quantum propertiesof

electrons will become crucialindetermining the designof

electronic devices before the end of the next decade.

Further read ing

M HDevoret, DEsteve and CUrbina 1 992 Single-electron transfer inmetallic

nanostructures Nature36 0547

D CGlattli,M

Sanquer

and J Tran Thanh Van

(ed) 1994

Coulomb and

Interference Effects in Small Electronic Structures(Moriond

series,

Editions

Frontieres, Gif-sur-Yvette)

HGrabert andM H Devoret ed)1992 Single Charge Tunnelling{Nat oseries,

Plenum,

New York)

JShirakashi eta/. 19 98 Single-electron charging effectsinNb/Nb oxide-based

single-electron transistors atroomtemperature Appl

Phys Lett

721893

L

LSohn,

L PKouwenhoven and GSchoen 1997 M esoscopic Electron

Transport Natoseries, Kluwer, Dordrecht)

MTinkham 1996Introduction to Superconductivity(McGraw-Hill, New York)

L

Zhuang,

L Guo and S Y Chou1998 Siliconsingle-electron quantum-dot

transistor

switch operating at room

temperature

Appl Phys Lett

721205

MichelHtk and are at theService de Physique del E tat

Condense, CEA-Saclay,

F-91191

Gif-sur-Yvette, France

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for

semiconductor characterisation, laboratory

analysis and high energy physics. Call

us

and see how we can contribute

to

your

magnet system needs.

Unit 13 , IDA Centre , Pearse St. Dublin2,I reland Tel: 353 1 670 4046 Fax: 3531670 4047

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contact@magnet ic-solutions.com Websi te: ht tp: / /www .magn et ic-solu t ions.co m

P H Y S I C S W O R L D S E P T E M B E R

1998

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Books from Springer

M.S.

Longair

Galaxy Formation

1998.XVIII, 510 pp. 142 figs.

(Astronomy and Astrophysics l ib rary )

Hardcover 37.50

ISBN .1-540-63785-0 7 ^ ,

Written by a well-known astrophysicist

who is also a superbly talented writer, this

work deals with the matter and radiation

content of the universe and the formation

of galaxies, providing a comprehensive

introduction into relativistic astrophysics

as needed for the clarification of cosmolo-

gical ideas.

R.J. leVeque,D .

Mihalas

E.A.

Dorfi

E.Muller

Co mputational Me thods

fo r

Astrophysical Fluid Flow

Saas-Fee Advances Course 17.

Lecture Notes 1997 Swiss Society for

Astrophysics and Astronomy

1998.

XIV, 508 pp. 124 figs., 6 in color.

Hardcover 46.00

ISBN 3-540-64448-2

This book leads directly to the most mod-

ern numerical techniques for compressible

fluid flow, with special consideration given

to astrophysical ap plications. Emphasis is

put on high-resolution shock-capturing

finite-volume schemes based on Riemann

solvers. The applications of such schemes,

in particular the PPM method, are given

and include large-scale simulations of

supernova explosions by core collapse and

thermonuclear burning and astrophysical

jets.

Parts two and three treat radiation

hydrodynamics. The power of adaptive

(moving) grids is demonstrated with a

number of stellar-physical simulations

showing very crispy shock-front structures.

F.Pobell

M a t t e r

a n d

Meth od s

a t

Low

Tempera tures

2nd ed. 1996. XIII,

371

pp . 197 figs., 28 tab s.,

77 problems.

Softcover 34.00

ISBN 3-540-58572-9

This textbook contains a wealth of infor-

mation essential for successful experiments

at low temperatures. The major part is

devoted to refrigeration techniques and the

physics on which they rely, the definition of

temperature, thermometry , and a variety of

design and construction techniques.

Bo-run Jiang

The

Least-Squares

Finite Element

lod

B .-

Jiang

TheLeast-Squares Finite

Element Meth od

Theory and Applications in

Computational Fluid Dynamics

a nd

Electromagnetics

1998. XVI, 418 pp. 130 figs.

(Scientific Computation)

Hardcover 157.00

ISBN 3-540-63934-9 M-

Here is a comprehensive introduction to

the least-squares finite element method

(LSFEM) for numerical solution of PDEs.

It covers the theory for first order systems,

particularly the div-curl and the div-curl-

grad sy stem. Then LSFEM is applied sy stem-

atically to permissible b oundary conditions

for the incompressible Navier-Stokes

equations, to show that the divergence

equations in the Maxwell equations are

not redun dant, and to derive equivalent

second-order versions of the Navier-Stokes

equations and the Maxwell equations.

P lease o rder f rom

S pri nger -V er l ag Lond on Ltd .

Fax:

+ 4 4 / 1 4 8 3 / 4 1 5 1 5 1

e-m ai l : a [email protected]

o r t h r o u g h y o u r b o o k s e l le r

lectron

Microscopy

Physics of

Image Formation

and

Microanalysis

L.Reimer

Scanning Electron Microscopy

P h y s i c s o f Im ag e F o rm a t i o n a n d

M i c r o a n a l y s i s

2nd completely rev. and upd ated ed.

1998. Approx. 510 pp. 260 figs.

(Springer Series in Optical Sciences,

Vol.

45 )

Hardcover 57.50

ISBN 3-540-63976-4

A description of the phy sics of electron-

probe formation and of electron-specimen

interactions. The different imaging and

analytical modes using secondary and

backscattered electrons, electron-beam-

induced cu rrents, X-ray an d Auger elec-

trons,

electron channelling effects, and

cathodoluminescence are discussed to

evaluate specific contrasts an d to ob tain

quantitative information.

K.Yosida

Theory

o f

Mag net ism

1st ed.

1996.

Corr. 2nd printing 1998.

% 320pp. 47 figs.

(Springer Series in Solid-StateSciences,

Vol.

122)

Hardcover 37.50

ISBN 3-540-60651-3

An important subfield of the theory of

solids that has guided a long history of

research into the phenomenon of magne-

tism. This book provides the foundation

for further developm ent in this field.

p

Prices subject to change without notice.

In

EU countries the local

VAT

is effective

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