Electrons in a Periodic Magnetic Field Induced by a Regular Array of Micromagnets P. D.

VOLUME 74,NUMBER 15

PHYSICAL REVIEW LETTERS

10ApRIL 1995

Electrons in a Periodic Magnetic Field Induced _by a Regular Array of Micromagnets P. D.

Ye,^'

D.

Weiss,^'

R. R.

Gerhardts, ^' M. Seeger,

K.

von Klitzing, ^'

K.

Eberl,^' and H. ^Nickel-'

'Max Plan-ck Insti-tut fur Festkorperforschung, D 705-69Stuttgart, Germany Max Plan-ck Insti-tut fur Metallforschung, Institut fur Physik, D 7056-9 Stuttgart, Germany

-'Forschungsinstitut der Deutschen Bundespost, D-64295 Darmstadt, Germany (Received 2 December 1994)

The deposition offerromagnetic microstructures on top ofa high-mobility two-dimensional electron gas (2DEG) allows the investigation ofelectron transport in aperiodic magnetic field which alternates on a length scale small compared to the elastic mean free _path of the electrons. The longitudinal resistance of the 2DEG displays, as a function of the externally applied field, the long-predicted magnetic commensurability oscillations which result from the interplay between the two characteristic length scales ofthe system, the classical cyclotron radius R,^. ofthe electrons and the period a ofthe magnetic field modulation.

PACS numbers: 73.50.Jt,73.20.Dx, 75.50.Rr Transport properties

of

electrons in a two-dimensional electron gas (2DEG) subjected to a periodic magnetic field have attracted considerable theoretical interest

[1—

6].

Depending on the strength

of

the local magnetic field, the electron motion in the plane

of

the 2DEG can be tuned from regular to chaotic. The motion

of

ballistic electrons in a periodic magnetic field is also believed to be closely related to the motion

of

composite fermions in a density modulated 2DEG in the fractional quantum Hall regime

[7].

Distinct theoretical predictions exist for the limit

of

a weak one-dimensional

(1D)

magnetic modulation (modulation amplitude (B ^~ && ~BO~,the external magnetic field) where the magnetoresistance

p„

^{is expected to}

oscillate with minima appearing at magnetic fields given by [2

—5]

guiding center drift

of

the cyclotron orbits which vanishes ifthe flat-band condition holds

[11].

This classical picture is easily extended to include in addition to a modulation

V„(x) = _V„cos

_Kx

_of

_the electrostatic potential energy

of

an electron a weak modulation

B„,

(x)

= _B

_cosKx

_of

_the

z component

of

the magnetic field

(K = _2tr/a).

Aver- aging the modulation induced drift

of

the guiding centers over the unperturbed cyclotron orbits at field Bo

[11, 12],

we obtain for the resulting change in the resistivity

27.² n 2

where _{po is the} zero-field resistivity

of

the unmodulated 2DEG, ^7.is the scattering time, and

S=U

cos KR,

— — ~

kFa

_h~

_{sin KR,}.

——

277

(3)

where A

=

_{0, 1,... is an integer} oscillation index, kF

=

$2~n,

the Fermi wave number, with n, the carrier den- sity

of

the 2DEG, and a the period

of

the 1D modula- tion in the x direction. This predicted effect is closely related to the commensurability oscillations observed re- cently in the resistivity p

of

a 2DEG with weak elec- tric modulation

[8

^—

10].

Similar to the electric case, the magnetic modulation leads to a modified energy spec- trum [1

— 5].

The degenerate Landau levels are transformed into bands

of

finite width. The dispersion

of

these Lan- dau bands provides an additional contribution to the re- sistivity

p,

^{, which vanishes only when the bandwidth}

becomes zero

("

flat-band condition'*)

[9, 10].

In contrast to

Eq. (1),

which isthe fiat-band condition formagnetic mod- ulation, the flat-band condition for weak electrical modu- lation reads

2R, =

_(A

^—

_1/4)a with A

=

_{1,2,. ... Hence,}

p

of

a 2DEG with a weak electric modulation displays minima at Bo fields where in a weak magnetic modulation

of

the same period a maxima are expected.

For the case

of

a pure electric modulation, the addi- tional contribution to

p,

^{- has} been related to the classical

with co

=

_eB /m* (m* _{is the} effective electron mass

of

GaAs). The

"+"

_{in Eq.}

₍₃₎

_{holds ifBo}

)

^{0,and the}

"—"

sign holds

if

the applied field (and thus the direction

of

the cyclotron motion) is reversed, Bp

(

^{0, while V (x)}

and

B

(x) are fixed

[13].

^{The zeros}

^of

^Eq.

(3)

^and hence the Rat-band positions now depend on the relative strengths

of

electric and magnetic modulations,

U„,

^and

h

co,

respectively.

Over the last years a variety

of

experimental at- tempts were made

(e.

g.^,

[5])

to establish a periodic magnetic field on the length scale

of

a few hundred nanometer, so far, however, without success. In this Letter we report a novel method to investigate electron transport in a periodic magnetic field. By depositing an array

of

ferromagnetic dysprosium (Dy) strips with widths

of

a few hundred nanometer on top

of

a semicon- ductor heterojunction (Fig. 1),we generate a 1D periodic magnetic field in the plane

of

the 2DEG. By ^increasing the strength

of

these micromagnets via the externally ap- plied field we show that the long-desired magnetic com- mensurability oscillations appear ⁱⁿ the resistivity p

VOLUME 74,NUMBER 15

PHYSICAL REVIEW LETTERS 10

APRIL

1995

NiCr

c.

)

~

^II

contacts

I

NiCr

era===

I!

FIG. 1. (a) ^Sketch of the one-dimensional ferromagnetic Dy grating on top of a GaAs-A1GaAs heterojunction. (b) Electron micrograph of the Dy strips evaporated across a mesa edge: a

=

1 pm, height of a Dy strip: 200 nm. (c) Device geometry containing the ferromagnetic grating and an unpatterned reference Hall bar.

Our samples were prepared from high-mobility GaAs- AlGaAs heterojunctions where the 2DEG was located ap- proximately

100

nm underneath the sample surface. The carrier density n, and electron mobility p, at

4.

2

K

were

-2.

2 X

10"

_cm ² and 1.3 X 106 cm2/Vs, respectively, corresponding to ^an elastic mean free path

of —

_10p,m.

50p, m wide Hall bars, sketched in Fig.

1(c),

^{were fab-} ricated by ^standard photolithographic techniques. Al- loyed AuGe/Ni pads contact the 2DEG. A

10

nm thin NiCr film, evaporated on top

of

the devices, defines an equipotential plane to avoid electric modulation

of

the 2DEG. However, strain due to different thermal expan- sion coefficients

of

the ferromagnetic grating and the heterojunction always results in a weak electric periodic potential as the sample is cooled down to cryogenic tem- peratures (see below). The Dy gratings ^with period

of 500

nm and 1 p, m were defined by electron beam lithog- raphy on top

of

one

of

the NiCr gates. After developing the exposed PMMA resist a 200 nm Dy film was evapo- rated. After lift-off in acetone micromagnet arrays, like the one shown in Fig. 1(b), ^{were obtained.} For magne- tization measurements, large area (5 ^{X 5 mm}

)

^{Dy wire} arrays with comparable wire width were prepared by holo- graphic lithography. Four point resistance measurements were performed in a 4He cryostat with superconducting coils using standard aclock-in techniques. Forall experi- ments, the external magnetic field Bp was applied normal (z direction) tothe plane

of

the 2DEG.

Measurements

of

the macroscopic stray fields

of

the large array

of

holographically prepared Dy wires

(a =

760nm) were measured ^with a Quantum Design SQUID magnetometer. The resulting hysteresis loop shown in Fig. 2 indicates hard magnetic behavior

of

the Dy wires.

The coercive field

B, ^of 0.

5

T

is high compared to pure polycrystalline Dy

(B, (

^{50 mT). Hence, the demagne-}

FIG. 2. Hysteresis of the magnetic polarization

J

measured from a large array (5X 5 mm2) of Dy wires prepared by holographic lithography on a GaAs substrate. The inset sketches the normal component B (x)ofthe stray field induced by the micromagnets in the plane ofthe 2DEG. Reversing the external magnetic field results in a phase shift ofthe modulation with respect tothe grating (see text).

tization process is controlled either by the pinning or the nucleation mechanism

[14].

The stray field

B

_(x)in the z direction, sketched in the inset

of

Fig. 2, is expected to alternate with an average value close to zero

[4].

We ^re- turn tothis point below.

Figure

3(a)

^{shows the} resistivity

p„of

^{the 2DEG}

underneath the Dy superlattice ^with period a

=

1 p,m.

Since the devices with a

=

₅₀₀nm display compara- ble behavior, we concentrate on the larger period sam- ple below. The traces labeled 1

—10 T

are obtained as follows: After the initial cooldown we first sweep to 1

T

and measure the p trace labeled 1

T

from 1 to

— _0.

₂₅

_T.

Then we sweep to

B " =

₂ T and take the next p trace in the same Bp interval. By successively sweeping to higher

B '"

_{(up to}

_{10 T)}

_the magnetic polari- zation

1

ⁱⁿ the Dy strips and hence the strength

of

the re- sulting periodic field

B

_(x)are increased. This enhanced strength affects the magnetoresistance traces: _p displays oscillations with a dramatically growing maximum be- tween Bp

= _0.

₁₅ and

0.

75

T

(the superimposed oscilla- tions above

0.

5

T

are Shubnikov

—

_{de Haas} oscillations).

Forpositive Bp, the minima in p appear at Bpvalues ex- pected from Eq.

(1)

^{for magnetic modulation}

[15].

^From

the amplitude Ap,

of

the maxima at Bp

— _0.

_{3T we esti-}

mate the amplitude

of

the imposed magnetic field

B

from Eq. (2) ^with

2R, =

_0.

_75a.

_{This gives}

_B

_{values, plot-}

ted in the inset

of

Fig.

3(a),

which range between 13mT

fortheB "= 1Ttraceand40mTfortheB "= 10T

trace. At low Bp the traces are not symmetric with respect to Bp

=

0as is expected for a broken time reversal sym- metry

[16].

This can be seen clearly in Fig. 3(b) ^where the low Bp regime

of

Fig.

3(a)

is magnified. With in- creasing

B "

a richer oscillatory structure unfolds in p indicating a growing magnetic field modulation. The p minima for Bp

)

⁰

^of

trace e almost perfectly coincide

3014

VOLUME 74,NUMBER 15

PH YS ICAL REVIEW LETTERS 10

APRiL

1995

300 40 _10T

a.)

I I

T=4.2K

200

100

0-0.25 60

I I I I

0.00 0.25 B,(T}

0.50 0.75 1.00

50

( 40

30

20-0.25 20

2. 3.

I I

-0.05 B,(T)

3. 2.

I

0.05 0.15 0.25

15

43 10

CO

0.03

I

-0.25 -0.15 0.25

2 34

I I I

43 2

I I I I I

-0.05 0.05 0.15 B,(T)

FIG.3. ^{~a~~ ~}p ^vs external ma_gnetic ₀

v ~

q()

i e triangles with ositions con ition in a periodic ma neti

d l th t t}1of^{t}1 ('} )

arge resistance maximumimum atat

-0 —

_0.3_{3T and (ii) the} os'

„minima

in (b) around

—

_{0 16}

lfi io

of()

^{a showingn}

^—

^{the.shiftando0.12}

increasing B~'" (from a

=

1 T to e

=

_10T

kth o itio fthe magnetic Hat-band c ( ^'v

CC

n riang esmark the electric o ). ^Th o i I h' hl' _h

used to evaluate the B~'s from the zeros of E . s ig ig t the osition

conditions.

again mark magnetic and electric flat-band

with whatt isis expected from Eq.

(1),

^indicatin

dic

' "'t"

^{fi ld d}

potential. Th

e ominates overer the weak electric

h o iive

fild

e

B,

^around

—

_0.5

_{T [17}

_{. H}

f

11o i t i

F

3(b

serves oscillations in

ig.

)

to ne ative

B

p one still ob- a ions ^m

p„.

^{But now the minima in}

=

_{1,2)appear at Bopositio h}

ima for pure electric m d 1 a

ns w ere one ex ects mo u ation with eriod a by open triangles The or

tion is strain caused b the d

s. e origin

of

this electric modula- coefficients

of

Dy and GaAs

y t e different thermal exexpansiona y an a s. This effect, re orted

r—

viously for other materialia corn mations

[5,

18

19

1 A

e an edges in GaAs v ssuming a sinusoi e rom t e amplitude

of

the res

at Bp

— ^— _0.

_{12T t(trace}

_a)

_{an amplitude}

^e

^resistance

_of

_th ^maximum

1 ri o ⁿ

il

^{ia V}

^{of —}

^0.3meV

[20]. ^T

e res

of

the weak electric modulation en o

Bm~x al

mo u ation which is independent

of

a'ows one to "calibrate" the stre

a1

"

_{e e strength}

_of

_{the stray}

"

and hence

B

are incre

g es

.

oth a periodic potential an magnetic field

p ial and an oscillating ic e act on the electrons, the

fIa-

d"" ^in'd

^by t^he zeros

of E .

3 .

a^—an condition depends on the ratio hen hen and hence the field

B

d

f

h

e

of

the microma e p position

of

the lt hown in tht

einsetof

Fi .

3a.

e ana yze the position

of

the mmima between 1

and m s d

ere, the subscripts

of

the oscillation index A, e

e

an m, stand for the electric and the ma netic conditions, respectivel

c

ive _y

f

or reversed Bp, ^and betwe and 1 for positive

B

. Th

eween

2,

e p. e inset also includes the tween 1 and 0 These

d and reverse mag- ic e s in Fig. 3(b) is also observed fo

with higher ^A's. Th

for the minima e origin

of

this asym y

q. p is reversed. This e in

Eq.

3

if B

ge o t e sign at Bp

=

0 can be viewed as a e eectric modulation V x and

t}1 d t'

fB

^inset

^'"

^{o ig.}

^2.

^{Here, we ut}

hile the phas wit respect to the ferroma netic independent

of

the

B

d

gne ic grating) is

p o _p

e p irection, the hase

o:

^{or Bp}

)

^{0the local maxima}

are underneath theeD_y sstrips while for

— _B ( ^B

^p

( ^(Othe

^{0 h}

VOLUME 74, NUMBER 15

PH YS ICAL REVIEW LETTERS 10

ApRI~ 1995 maxima are between the strips. This allows an abso-

lute measurement

of

the phase

of

the electric modulation:

Since the sign in Eq.

(3)

^is

"+"

_{for Bp}

)

^0while

^B )

^0,

we obtain the correct shift

of

the minima for V

~

0,

i.e.

,

ifn, is higher underneath the strips

[21].

In Fig.

3(c)

we display model calculations

of p,

^based

on Eq.

(2).

^The

Ap„oscillations

^calculated for four dif- ferent hew /V ratios closely resemble the experimental data in Fig.

3(b).

^{As in} the experiment the oscillation minima shift with increasing

B

from the electric ^Hat- band condition to the magnetic Oat-band condition. For Jib /V

= _0.

2, corresponding to

B —

_40mT in our ex- periment, the p minima are determined _by the magnetic fiat-band conditions (filled triangles) indicating the domi- nance

of

the periodic

B

field. The asymmetry ^with re- spect to the shift

of

the minima (see, e._g., the shift from

1,

1 for Bo

~0

and from

2,

^{1 for} Bo

)0

^with

increasing

B )

^{is adjusted to the experiment} by choosing V

(

⁰ (see above).

For

the sake

of

simplicity we used, in contrast to experiment, a Bo-independent magnetic field modulation

B

to demonstrate the shift

of

the p minima.

If

the reversed field is increased beyond the coercive field, the polarization

of

the micromagnets switches and finally follows the externally applied field. On sweeping back where we start from the corresponding

— _B "

values we obtain, as isexpected from the hysteresis loop (Fig. 2), the mirror image (with respect to the Bo

=

_{0 axis)}

_of

_the

experimental traces shown above.

Now we address the zero-field resistance

p„(Bo=

0) and the positive magnetoresistance at ~Bp~

(

30m

T.

First we note that the minimum

of

_p at Bo

=

₀ in Fig. 3(b) does not shift with increasing

B "

(shift less than 3 mT) while, on the other hand, the ampli- tude

B

becomes ashigh as 20mT at Bo

=

_{0 [seeinset}

_of

Fig.

3(a)].

Such behavior is expected for an alternating stray field

B

_(x) with an average value

of

zero (inset

of

Fig.

2).

This is consistent with the experimental finding that, independent

of B ",

the Hall resistance measured in the magnetic superlattice is the same as in the reference Hall bar. The positive low-field magnetoresistance might also be related to this alternating

B

(x) ^{field: electron} trajectories "wiggling" along a

B

_(x)

=

_{0 borderline}

(open orbits; for the electric case see,

e.

g.,

[22])

cause an increasing magnetoresistance which saturates once

~Bp~

)

^{~B ~. For trace e in Fig. 3(b)the} positive magne- toresistance saturates around 28 mT comparable to the

B

value

of

about 20mT in the inset

of

Fig.

3(a).

Finally, we note that

p„(0)

^{increases linearly with}

^B

(0), ^{as can} be extracted from Fig. 3(b) and the inset

of

Fig.

3(a).

Similar behavior is expected for electrons scattering from microinhomogeneities in a magnetic field

[23].

In summary, we have observed the long-predicted com- mensurability oscillations in the presence

of

a periodic magnetic field. The experimental technique described above opens up the way to experiments in alternating magnetic fields with periods in the nanometer regime.

During the preparation

of

this manuscript we became aware

of

similar experiments using patterned supercon- ductors on top

of

a heterojunction

[24].

We thank M. Rick and A. Gollhardt for technical support, ^and W. Dietsche, H. Kronmuller,

J.

Smet, and M. Tornow for valuable discussions.

P. D.

Ye acknowl- edges the Volkswagen Foundation for afellowship.

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[13]For

Bo)

0 and intermediate temperatures (T,

(~

T

&(T,

^.

in the notation ofRef. [3])Eq. (23) of Ref. [3]^{is equi-}

valent with our Eq.

(2).

Because ofan incorrect formula for the diffusion tensor, the value given by Eq. (16) of Ref. [4](for V

=

₀₎is too small by a factor of 2.

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Magn. Mater. 74, 291(1988).

[15] ^{The flat-band} positions were derived using first order perturbation theory involving the cosine approximation of ^the Bessel functions

[11,

12].Deviations are therefore expected for large values ofBp. This especially affects the

A

=

_{0 position} in Fig. 3(a).

[16]To account for inhomogeneities in our devices we aver- aged over

p,

^{traces taken} from potential probes ofboth sides ofthe Hall bar.

[17]^This is strictly valid only for B

'" =

_{5T.We expect B,}

values around

—

_{0.5 Tfor the other traces.}

[18]

J.

H. Davies and I.A. Larkin, Phys. Rev. B 49, 4800 (1994).

[19]P. D. Yeet al. (unpublished).

[20] D.Weiss, Phys. Scr.T35,226

(1991).

[21]GaAs is expected to shrink with respect to Dy if ^cooled down, resulting in minima ofU,

„(x)

^{under the strips.}

[22] P.H. Beton et al., Phys. Rev. B42, 9229 (1990).

[23] A. V.^Khaetskii,

J.

Phys. Condens. Matter 3, 5115

(1991).

[24] A. K.Geim (private communication).

3016