1

Molecular electronics: beyond Coulomb blockade

• Resonant tunneling diodes / transistors

• Switching / memory based on wavefunction engineering

• Switching / memory based on steric properties

What kind of physics could lead to useful / interesting molecular devices, beyond the Coulomb blockade effects we’ve been examining?

Progress in experiments

A few basic techniques have been used to make most of the experimental progress in this area:

• Mechanical break junctions

• Nanopores

• Scanned probe microscopy

• Crossed wire structures

We’ll examine each of these in turn.

2

Mechanical break junctions

Basic idea:Mechanical advantage: lateral motion from bending can be ~ 1000x smaller than vertical motion.

Breaking can be done in UHV to avoid contamination.

Bare metal tunneling current allows calibration of position.

Images from van Wees group, Netherlands

Zhou et al., APL 67, 1160 (1995).

MBJ: Reed

Reed et al., Science 278, 252 (1997)Start with unbroken wire, precoated with self-assembled monolayer (SAM) of interesting molecule.

Break in UHV and allow molecules to rearrange.

Bring junction back together for measurements, knowing what piezo voltage corresponds to the correct interelectrode spacing.

3

MBJ: Reed

Results:

Fairly reproducible IV curves that look like those at right.

Differential conductance (blue trace) has features that can be identified with resonant tunneling through molecular levels.

Theory must account for specific bonding of S to Au to get shape close.

Theory still overestimates conduction by ~ 20x.

Reed et al., Science 278, 252 (1997)

di Ventraet al., PRL 84, 979 (2000)

MBJ: KarlsruheReichert et al., PRL 88, 176804 (2002)

Same basic plan, but this time do self-assembly on already-broken junction.

Idea: compare symmetric molecule and asymmetric molecule cases, to see if the molecular structure is really what’s determining conduction properties.

4

MBJ: KarlsruheReichert et al., PRL 88, 176804 (2002)

Result:

• Asymmetric molecule always gives asymmetric IV traces.

• Symmetric molecule usually gives symmetric IV traces.

MBJ: KarlsruheReichert et al., PRL 88, 176804 (2002)

One other major point:

Even with symmetric molecule, asymmetric curves can result.

Moving electrodes while molecule is bonded either distorts bond angles at S-Au point, or rearranges last few Au atoms on surface.

Confirms that molecular conduction characteristic depend critically on the atomic-scale details at the metal-molecule junctions.

5

MBJ: Crossed wiresKushmericket al., PRL 89, 086802 (2002)

Same point can be made in an even simpler crossed-wire geometry at room temperature.

MBJ: Crossed wires and vibrations

Kushmericket al., Nano Lett. 4, 639 (2004)

• Junctions made this way are stable enough to allow measurements of d2I/dV2.

• Inelastic electron tunneling features make sense.

6

MBJ: Gating

• Possible (though hard!) to make mechanical break junctions in gate-able configurations!

• Potentially very important technique.

Champagne et al., cond-mat/0409134

MBJ summary

• Highly productive research technique for examining single molecules.

• Shows that single molecule conduction is possible, though generally poor.

• Single molecule conduction depends crucially on atomic-scale details of bonding and metal surfaces.

• Gating is possible under certain circumstances.

Endemic problems with technique:

• Mechanical stability essential.

• Very difficult to do temperature sweeps - everything moves due to differential contraction. This is a problem b/c standard way ofdeducing conduction mechanisms uses G(V,T).

7

Nanopores Images from Reed group, Yale

Not a single molecule technique - more like ~ 1000 in parallel.

Originally developed to study ~ 10-30 nm diameter metal junctions.

NanoporesImages from Reed group, Yale

Requires self-assembly followed by evaporation of top electrode.

Noone knows what interface looks like.

Yield very very low (~ 1-2 %).

8

Nanopore devices: NDR

Chen et al., Science 286, 1550 (1999)Donhauser et al., Science 293, 2303 (2001)Seminario et al., JACS 122, 3015 (2000)

Nanopore devices made with above molecule exhibit negative differential resistance, as shown.

Mechanism not clear: temperature dependence is significant and steep.

Proposed mechanism: detailed coupling of HOMO/LUMO to leads.

Nanopore devices: NDR

With modification of molecule, can see NDR that persists up to room temperature.

Other possible mechanism besides simple electronic structure: orientation of phenyl rings.

9

Nanopore devices: memory Reed et al., APL 78, 3735 (2001).

Hysteretic IV curves.

Initial ramp up: “high” conductance state.

Ramp back down: “low” conductance state, until reset.

Like NDR, effect is strongly temperature dependent.

Requires very large electric fields to actuate….

Nanopore devices: memory

Can clearly use this kind of hysteretic IV curve as a memory!

Again, potential mechanisms are either wavefunction oriented (shape of electron distribution in HOMO and LUMO) or steric (reorientation of side groups).

10

Nanopore devices: simple tunneling + vibrationsWang et al., PRB 68, 035416 (2003)

Wang et al., Nano Lett. 4, 643 (2004)

• Recent nanopore papers on very “boring” molecules show nice results.

• Conduction really is via tunneling.

• Inelastic features make sense.

Scanned probe microscopy: STM

Paul Weiss, PSU

Another means of looking for interesting molecules and testing their (2-terminal) conductive properties is to use scanned probe microscopes.

• Allows highly controlled positioning of electrodes previously decorated with molecules.

• Can obtain I-V curves for single molecules.

• Can quickly examine many molecules

11

Scanned probe microscopy: STM

Downsides:

• Mechanical stability

• Difficult to do T-dep. measurements.

• Interpretation of contacts difficult -tunneling conductance is supposed to be proportional to product of local single-particle density of states of tip and surface.

• As result, tricky to deconvolve topography from electronic properties.

One time it’s not too hard: time-varying behavior

Donhauseret al., Science 292, 2303 (2001).

Scanned probe microscopy: AFM

Cui et al., Science 294, 571 (2001)

Conducting AFM is also a useful tool.

Downside: contact area is typically significantly larger than single molecule.

Upside: can vary contact force and see what happens.

Surprisingly nice result at right: looks like as Au nanoparticle is pressed harder, more and more molecules are in contact in the circuit.

Tricky to reconcile this with the data shown previously demonstrating that contact geometry is huge influence.

12

Nanopore and SPM summary

• Nanopores have produced some impressive results - NDR, memory w/ T-dependent measurements of ~ 1000 molecules in ||.

• Problem is, yields are terrible, and diagnostics on final molecule condition is essentially impossible.

• SPM techniques are much faster, and confirm some of the nanopore work (NDR, switching of conductance states).

• Unfortunately, SPM has its own set of experimental issues.

• Mechanisms behind these properties still argued.

Crossed wire structuresCollier et al., Science 285, 391 (1999)Collier et al., Science 289, 1172 (2000)

On industrial side, HP and UCLA have been working on “crossed wire” structures.

Suffer from same diagnostic concerns as nanopore devices, though much easier to fabricate (larger overlap).

•Molecules have hysteretic IVcurves - may be used for nonvolatile memory.

• Limited to 2-terminal devices without gain, but that’s not so crucial for nonvolatile memory schemes.

13

Crossed wire structuresHP press release, 9/9/02

• Combine nanoimprint lithography with this sort of device design.

• Result: 64-bit nonvolatile memory in ~(3 µm)2, 10x higher density than commercially available NVRAM.

• Also built multiplexer for RAM readout.

• Predict possible market-ready products w/in 5-10 years.

Crossed wires: Big Problems!

• Problems have surfaced.

• Switching in a number of devices has nothing to do with molecules!

• “People have suddenly gotten religion” --a nano researcher who shall remain nameless.

Science, Oct. 24, 2003

Lau et al., Nano Lett. 4, 569 (2004)

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Nanoparticle dimers

Dadoshet al., Nature 436, 677 (2005)

Nanoparticle dimers

Dadoshet al., Nature 436, 677 (2005)

Blinking and fluctuations!

Role of protective ligands on nanocrystals?

15

Nanocell concept

The researchers: Tour (Rice), Reed (Yale), Allara (Penn State), Mallouk (Penn State)

Nanocell concept:

• Self-assemble large number of functionalized nanoparticles + molecules (e.g. NDR and memory molecules) across an array of leads.

• Training + field programming: apply voltage pulses; measure currents; train system to have desired functionality (!).

Nanocell conceptinput terminals

output terminals

• Every device is unique (analogous to detailed wiring of brain).

• Architecture that permits on-the-fly adaptation and training would be inherently robust + defect tolerant.

• A huge departure from standard electronics manufacturing practice!

Molecular Electronics Corp. / Motorola paradigm: self-assembly + software architecture to allow training of circuits.

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Summary and prospects

• Several research tools exist for trying to examine molecular conduction beyond Coulomb blockade.

• Basic qualitative features involving resonant tunneling through levels are understood.

• Details (contacts, quantitative results with predictive power) are still lacking.

• Basic physics responsible for certain interesting properties (NDR, switching) still under debate.

• We’re a long way from practical molecular electronic devices, though applications in niche markets may be on the horizon.

• A fascinating field in which to be working!

Organic electronics

A thriving research (and commercial !) field already exists using molecular materials.

Organic LEDs Organic FETs

IBM

17

Organic electronics

Pentacene

S S S S

R

R = C6H13

R R R

Regio-regular Poly (3-hexylthiophene)

Key point: certain molecular materials act like semiconductors.

Ordered OSCs

Small molecular crystals

Polaron bands?

Disordered OSCs

Glassy polymers

Activated hopping?

conduction band

valence band

localized states!

Historical progress

Mobilities have improved dramatically over the last 20 years.

Some OLED devices are commercially available.

However, many unanswered questions remain….

IBM

18

Unanswered questions

• What are the basic physical parameters of carriers in organic semiconductors? (Effective masses, etc.)

• What is the physics of charge injection at OSC-metal interfaces?

• How does the conduction mechanism evolve from bands to hopping as disorder is increased?

• What are the fundamental limits of OSC performance?

• How important are electronic correlations in these materials?

Next time….

Basics of magnetism and magnetic materials.

1

Basics of magnetic materials

We will briefly review magnetism and magnetic materials as necessary background for understanding magnetoelectronics, particularly the data storage industry.

• Definitions in SI

• Types of magnetic materials

• Origins of magnetic properties

• Ferromagnetism

• Domains

• Hysteresis loops

Definitions in SI

Units are the worst part of dealing with magnetic systems!

B:

• magnetic induction (usually what physicists mean by magnetic field).

• most physically significant quantity - what shows up in Lorentz force law, in determining NMR frequencies, etc.

• Unit is the Tesla. Earth’s magnetic field = 6 x 10-5 T.

• More convenient cgs unit is the Gauss = 10-4 T.

• Important boundary condition: 0=⋅∇ B

2

Definitions in SIH:

• the magnetic field.

• Caused by currents of free charge.

• Unit is the Amp/m.

• Important relation:

• With no magnetic materials around,

JH =×∇

M:

• the magnetization.

• magnetic moment per unit volume of a material.

• Unit is the Amp/m.

• For a material with a permanent magnetization M0,

0MHM += χ

HB 0µ=

“permeability of free space” = 4π x 10-7 Tm/A

magnetic susceptibility

Susceptibility and permeability

)(0 MHB += µCombining effects of external currents and material response,

We define thepermeability by: HB µ=

So, for a material with a magnetic susceptibility χ,

)1(

)1(

0

0

χµµχµµ

+=→+== HHB

Note that for real materials χ and µ are tensorial.

Relative permeability is defined as µr = µ/µ0.

Susceptibility is most useful when discussing diamagnetic (χ < 0) and paramagnetic (χ > 0) materials, rather than systems with nonzero M0.

HB rµµ0=→

3

Magnetic dipoles and magnetization

A magnetic dipole can be modeled as a loop of current: nm IA=

m

A group of identical dipoles in a plane can be replaced by one big dipole with the same current circulating around the perimeter.

Units: Am2 = J/T

Convenient number:

A 3d stack of dipoles can be replaced by thinking about a sheet current running around the perimeter.

J/T1027.92

24−×=≡m

eB

hµ

Forces, torques, and fields

m

BTorque on a magnetic dipole:

Bmτ ×=

Force on a magnetic dipole:

)( BmF ∇⋅=

What is B at the surface of a long uniformly magnetized body with magnetization M in the absence of other fields?

Answer: just µ0M, since normal component of B must be continuous. M

Energy of magnetic dipole:Bm ⋅−=U

4

Field from a dipole

5

2

0 4

)(3)(

r

r

πµ mrrm

rB−⋅=

Note that B(r) ~ r -3

Current loop only approximates a dipole field in far field limit (r>> loop radius).

Image from Purcell.

Diamagnetism

Some materials develop a magnetization that is antialignedwith the applied external field H.

Such materials are diamagnetic, and have χ < 0.

Simple classical picture for diamagnetism: Lenz’s Law

mB

Try ramping up B = µ0H.

Result is a circumferential electric field that opposes the direction of the current in the loop.

This would act to reduce the dipole moment along H, and would be diamagnetic.

Correct quantum treatment involves 2nd order perturbation theory - can end up with either sign, depending on particulars of atoms. Larmor diamagnetism or Van Vleck paramagnetism.

5

Paramagnetism

Also common is paramagnetism, when χ > 0.

Two common origins of paramagnetism:

• Curie paramagnetism - localized moments free to flip.

• Pauli paramagnetism - requires “free” electrons in a metal.

Start w/ Curie case, spin J and gyromagnetic ratio g, in magnetic flux density B.

Bg µ/)/( Jm≡TkB

1≡β

Do statistical physics here. Alignment of spin with external field lowers spin energy.

Can write down partition function and solve to find equilibrium magnetization.

Curie paramagnetism − ++≡

=

J

x

Jx

J

J

J

Jx

gJBgJnM

J

BJB

2coth

2

1

2

12coth

2

12)(

)(

B

B βµµResult:

Bj is called the Brillouin function.

As long as µBB << kBT, can expand the above to get:

3

)1()( 022 +≈ JJ

Tk

HngM

BB

µµ

where we’ve also assumed that M’s contribution to B is small, valid for dilute systems of spins.

We see that Curie paramagnets have susceptibilities like:

T

JJ

Tkng

BB

1~

3

)1()( 022 +≈ µµχ

6

Pauli paramagnetism

E

Starting from unpolarized electrons, applying a field B shifts the Fermi level for spin-up and spin-down electrons oppositely!

)(2

)(BfdVN Bup µεενε += ∫ )(

2

)(BfdVN Bdown µεενε −= ∫

Taylor expanding to find the difference,

HENNV

M FBdownupB

02 )()( µυµµ ⋅≈−=

Only good for metals - can’t have gap at Fermi surface.

Paramagnetic and diamagnetic plots

H

MH

Pauli or Van Vleck paramagnet

Larmor diamagnet

Curie paramagnet

1~0

Tk

Hg

B

Bµµ

7

Some comments

The literature on all this stuff can be terribly confusing!

People use H even when there are no “free currents” around, and often use a mishmash of SI and cgs units.

• Most insulators are weakly diamagnetic.

• Metals can be either.

• Susceptibilities are usually quoted per molar volume rather than in their dimensionless form, for experimental reasons.

• Strictly speaking, susceptibilities and permeabilities are defined as derivatives.

Some numbers

Image from Marder.

8

Ferromagnetism - toy model

Start from Curie paramagnetism picture - local moments (spins), but now allow them to respond to the local magnetic field at their position.

Assume M = χ0 (H + Hm) “molecular field”, = ηM

HM 00)1( χηχ =−

)1( 0

0

ηχχχ

−=effAt high temperatures,

Recall that χ0 ~ 1/T, so we find at high temperatures

ceff TT −

1~χ Curie-Weiss law

At Tc, the Curie temperature, the susceptibility diverges! Spontaneous magnetization = ferromagnetism.

Ferromagnetism

• Preferred direction of M can depend on crystallographic properties of material and material shape, stresses.

FM occurs because of the same sort of exchange interactionthat leads to Hund’s rules.

Effective spin-spin interaction because the spin-ordered state tends to reduce total Coulomb energy of system.

Result: In metals, different spin directions have different densities of states at the Fermi energy.

Image from Coey, Trinity College, Dublin.

9

Other types of magnetic order

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑

↑

↑

↑

↑

↑

↑

↑

↑

↑

↑

↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

↑ ↑ ↑ ↑

canted ferromagnet

antiferromagnet

canted antiferromagnet

ferrimagnet

Ferromagnetism and equilibrium configurations

One contribution (the total dipole moment of the system interacting with the applied external field) to a system’s energy will be decreased if M lies along H.

However, the field produced by the ferromagnet itself contributes to the total energy.

Competition between “external field” contribution and self-field contributions leads to complicated behavior.

As a result, ferromagnetic properties are often characterized bysweeping H along a particular direction, and measuring M.

What is seen is hysteresis - the dynamics of M at a given applied field H depend on the direction (history) of M.

10

Ferromagnetism: M vs H hysteresis

H

MH

Hc coercive field

Mr remanent magnetization

Ms saturation magnetization

slope = initial χ

slope = max. χ

“Hard” ferromagnetic materials: large values of Hc, large Mr.

“Soft” ferromagnetic materials: small values of Hc, small Mr.

Crystalline anisotropy

Because of band structure origins of FM, there can be certain crystallographic directions along which it’s energetically favorable for M to lie.

Define m as the unit vector pointing along M; u as energy density due to anisotropy.

Most common cases:

• cubic anisotropy

)()( 23

22

212

23

22

23

21

22

211 mmmKmmmmmmKu cc +++=

• uniaxial anisotropy

θθ 42

21 sinsin uu KKu += where θ is angle between m and

anisotropy axis.

11

“Stray field” energies

Energy bookkeeping for FM calculations requires keeping track ofenergy cost of the field produced by the FM itself.

Remember 0)(0 =+⋅∇=⋅∇ MHB µ

Therefore, can write MH ⋅−∇=⋅∇ d Looks like magnetic charge density.

Turns out that the energy associated with these stray fields can be written as:

dVdVUsample

d

allspace

dd ∫∫ ⋅−== MHH 02

0 2

1

2

1 µµ

For a particular sample geometry, one can compute to find a fictitious magnetic charge density. Then one can find Hd everywhere, and compute the energy contribution from these stray fields.

M⋅∇−

Demagnetizing factors and fields

The resulting Hd is often called a demagnetizing field.

Generally one can relate Hd to the magnetization M by the demagnetizing factor tensor:

MNH ⋅−= ˆd

One can calculate N for given sample geometries.

Special case: for ellipsoids of revolution, Hd || M.

Image from Bertram, Th. of Mag. Recording

12

Demagnetizing factors and fields

Why the name?

Inside a magnetized object, Hd opposes Hext.

Total magnetic field inside sample is Htot = Hext - Hd.

NN

N

NNS

S

S

S

S

Hext

Geometric or shape anisotropy

The demagnetizing field energy contribution leads to “geometric” or “shape” anisotropy: it costs less, energetically, to have the magnetization pointed along certain directions because that minimizes the stray fields.

Example: uniformly magnetized infinite (x-y) plate.

NNNNNNNNNNNNNNNNNNNNNNNNNN

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS

Effective surface magnetic charge density ~ Msat cos θ

angle btwM and z-axis

Resulting Hd = - Msatcosθ; θµ 220 cos2 sd Mu =

So, we find it energetically unfavorable for a thin plate to have its magnetization lying perpendicular to its surface.

13

Other energetic considerations

• FM depends strongly on the crystal structure of the materials in question.

• Particularly important in insulators, where exchange interaction between local moments is not mediated by conduction electrons.

• Result: strong coupling between deformation and magnetization: magnetostriction.

• Can be used intentionally: built-in stresses can be used to “force” easy directions to lie where desired, or to “pin” magnetization.

DomainsSo, if Tc for iron is 1073 K, and its µ0Ms = 0.17 T, why don’t we see massive magnetic effects from iron bars all the time?

• FMs can lower their total energy by spontaneously breaking up their magnetization into domains.

• Total energy must include contribution from stray fields extending all over all space.

• Domains can greatly reduce these stray fields:

NNNNN

SSSSS

closure domains

14

Domain walls

Boundary between two domains is called a domain wall.

• Because of the FM exchange interaction, it’s very energetically costly for the direction of M to change very sharply.

• Result is that local magnetization spreads out the change oversome distance = domain wall thickness.

• Two types of domain walls:

Bloch wall Neel wall

Domain wall thickness

What sets domain wall thickness(often called “width”)?

Competition between exchange interaction (wants to keep nearest neighbor spins aligned) and crystalline (or shape) anisotropy(wants to keep M oriented only along “easy” directions).

Exchange energy for pair of spins:21

2 2)2( SS ⋅−= JJSU

For small change in angle φ between neighboring spins,

Energy is minimized ( = 0) when neighboring spins are aligned.

φ2JSU ≈

For a total rotation of π spread out over a chain of N+1 spins, the total energy cost is 22 )/( NNJSU π≈

For lattice spacing a, energy density per domain wall area for Natom thick wall ~ )/( 222 NaJSπ

15

Domain wall thickness

If anisotropy energy per unit volume is given by θ2sinK

where θ is angle between M and easy axis, then averaging over the thickness of an N atom thick domain wall gives an average anisotropy energy density ofKNa

Total energy density is then KNaNaJS +)/( 222π

Can find Nm that minimizes this, and wall thickness = Nma.

)/( 322 KaJSNm π≈ )/(22 KaJSW π≈

Plugging in numbers for iron, for example, gives W ~ 100 nm.

Note that it’s easy to make structures on size scales comparableto that of domain wall thicknesses….

Domains and hysteresis

What happens to domains when an external fieldis swept?

Domains continuously rearrange themselves to minimize the total energy of the whole system.

Relative sizes of different domains can change (at right).

Individual domains can also reorient their magnetizations.

Domain walls cannot always move freely!

16

Domain wall motion

Often domain walls can be “pinned” by disorder, grain boundaries, etc.

At large enough fields, pinning energy can be overcome.

Also, kBT can push domain walls “over barriers”.

Effect of pinning: Barkhausen noise.

H

MH

Next time:

Nanoscale issues in magnetic materials