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Department of physics

Seminar Ia

Scanning Tunnelling Microscopy

Observations of Superconductivity

Author: Tim Verbovšek

Mentor: dr. Rok Žitko

Co-Mentor: dr. Erik Zupanič

Ljubljana, February 2013

Abstract We introduce the phenomenon of conventional and high-temperature superconductivity and briefly

describe some basic theories that have been proposed to describe these effects. We explain quantum

tunnelling and its application in scanning tunnelling microscopy (STM) and spectroscopy (STS). We

describe some existing STM and STS measurements of conventional superconductors and analyse the

results. Finally some measurements on high temperature superconductors (HTSC) are presented.

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Contents

Abstract ......................................................................................................................................................... 1

Contents ........................................................................................................................................................ 2

1 Superconductivity ...................................................................................................................................... 3

1.1 The BCS Theory ................................................................................................................................. 4

1.1.1 The Energy Gap ........................................................................................................................... 5

1.2 Magnetic Vortices ............................................................................................................................... 6

1.3 Unconventional superconductors ........................................................................................................ 6

2 Tunnelling of Electrons .............................................................................................................................. 7

2.1 Operation of STM and STS ................................................................................................................ 8

2.2 The STM System .............................................................................................................................. 10

3 STS Observations of Superconducting States .......................................................................................... 10

3.1 NbSe2 Studies .................................................................................................................................... 10

3.2 Measurements on HTSC ................................................................................................................... 12

3.2.1 Energy Gap ................................................................................................................................ 12

3.2.2 Magnetic Vortices ...................................................................................................................... 13

Summary ..................................................................................................................................................... 15

References ................................................................................................................................................... 15

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1 Introduction While measuring the resistivity of mercury at low temperatures in 1911, Kamerlingh Onnes discovered

that at a specific temperature (the critical temperature TC) the resistivity sharply falls to zero; the material

enters the superconducting state. Defining properties of the superconducting state are zero-resistivity, the

Meissner effect (perfect diamagnetism) and the energy gap in the density of states of electrons. Since their

discovery many theories were developed in order to explain the phenomenon and they succeeded in doing

so.

In 1986, however, Bednorz and Muller discovered the superconducting state in cuprates with the

surprisingly high critical temperature at around 80 K (at that time no known superconductors had the TC

higher than 30 K). These superconductors (the high temperature superconductors) do not obey the

existing theories and so either new theories must be developed or the existing ones adapted to explain the

HTSC. Scanning tunnelling microscopy and spectroscopy are techniques which show promise in

unlocking the mysteries of the HTSC.

2 Superconductivity There are several different theories, based on various approaches, trying to explain the phenomenon of

superconductivity.

The first was the phenomenological London theory, which is able to explain the Meissner effect. It was

developed by London brothers in 1935. They assumed the free electrons in a superconductor are either

superconducting or non-superconducting. By combining this assumption with the Maxwell equations,

they found that the magnetic field inside the superconductor must diminish exponentially away from the

surface, with the characteristic length being the penetration length :

2

2

12

2

0

1;

,e

s

B B

m

n e

(2.1)

where ns is the density of superconducting electrons [1]. The superconducting material near the surface

responds to the external magnetic field with internal currents, which cancel the magnetic field inside the

superconductor. This is known as the Meissner effect.

The phenomenological Ginzburg-Landau theory (1950) is a thermodynamic theory describing the phase

transition to a superconducting state. The parameter describing the superconducting state is the order

parameter and it describes the free energy density of the material in the superconducting state as [1]

22 2 4( )

( ) ( ) ( ) ( ) ,2 2

S N

Tf f T

m

r r r (2.2)

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where ( )T is given as

0( ) ( ).CT T T (2.3)

The order parameter can now be found by minimizing the free energy density. By using the variational

approach and substituting

S S Sf f f (2.4)

the variation of free energy density of the superconducting state is written as

22

22

2 22 22 2

( ) ( )2

( ) ( )2

,2 2

Sf bm

bm

b bm m

(2.5)

where the last step uses the first Green’s law. If the variation of the free energy is to equal zero the

following equation must be true:

222 0.

2b

m

(2.6)

The main result of the GL theory is the coherence length ξ, which represents the distance over which the

superconducting state decays due to a local perturbation. It can be defined in the example of a boundary

between a superconductor and a normal metal. The boundary conditions in this case are (0) 0 and

( ) 1 . From the boundary conditions and Eq. (2.6) the form of the order parameter and the coherence

length can be determined as

2

0( ) tanh ; ( ) .2 ( )2 ( )

xx T

m TT

(2.7)

2.1 The BCS Theory The microscopic origin of the phenomenon was described by physicists Bardeen, Cooper and Schrieffer

in 1957. Leon Cooper has shown (in 1956) that because of the phonon-electron interactions in materials a

bound state of two electrons is possible under certain conditions. This bound state, the Cooper pair,

behaves like a boson and it moves in a material without energy dissipation [1]. The BCS theory attributes

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the origin of superconductivity to these Cooper pairs and predicts an upper limit for critical temperatures

TC of superconductors.

The study of superconductivity became even more complicated with the discovery of the HTSC. While

these unconventional superconductors exhibit similar properties to conventional superconductors, the

origins of high-temperature superconductivity are still not fully understood as the BCS theory does not

apply to them.

2.1.1 The Energy Gap

The BCS theory predicts the density of electron states inside a superconductor as [2]

2 2

( ) Re ,E

EE

(2.8)

where presents the pair potential in a Cooper pair. From Eq. (2.8) it follows that the LDOS is zero for

E ; the superconducting gap (in the order of a few meV). The BCS theory as such applies to s-wave

superconductors. Because the pairing potential is considered constant, the energy gap is isotropic and the

energy gap width is given as [1]

1.764 .B Ck T (2.9)

But in the general case, this is not true, due to complex crystal structures and electron interactions. The

pair potential is usually not constant, and in the k-space it is momentum-dependent. For two dimensional

d-wave superconductors, the pair potential can be written as

0( ) cos2 , (2.10)

where is the angle between a crystallographic axis and the direction of the Cooper pair’s momentum.

The LDOS is then given as

2

0

1( ) ( , ).

2E d E

(2.11)

The energy gap in the zero temperature limit is flat for s-wave superconductors and V-shaped for d-wave

superconductors (calculated from Eq. (2.9) and Eq. (2.10)).

Because of the energy gap, interesting phenomena occur at the normal-material/superconductor (N/S)

boundaries. Because states with energies less than are forbidden inside a superconductor, an electron

with E cannot cross from the normal state material to the superconductor. It can, however, bind to

another electron to form a Cooper pair and then enter the superconductor. This process, the Andreev

reflection [2], creates a hole in the normal-material, travelling in the opposite direction of the incident

electron. The opposite process is also possible; a hole is “reflected” as an electron.

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In an S/N/S junction, it is possible for an electron to be reflected from one boundary as a hole, travel

across the normal-material and be reflected from the other boundary as an electron. This results in a

bound state, analogous to a particle in a potential well. This creates additional available states inside such

a junction, resulting to an increase in the LDOS near such boundaries.

2.2 Magnetic Vortices Superconductors can be distinguished based on their behaviour in the presence of a magnetic field. In

type I superconductors the superconducting state suddenly collapses at the critical field Hc and the

material enters the normal state. In type II superconductors, the magnetic field penetrates the material in

the form of magnetic vortices for fields above the lower critical field HC1. If the field’s intensity is further

increased to the upper critical field HC2, the superconducting state collapses, like in type I

superconductors.

The GL theory can be expanded to include magnetic vortices. The theory predicts the density of magnetic

vortices in a superconductor as a function of the external magnetic field to be

2

.vN eB

A h (2.12)

Abrikosov showed that near the critical magnetic field HC2 (where the superconducting state collapses in

type II superconductors) the vortices form a triangular (or under some circumstances square [3]) lattice.

Because the magnetic field penetrates the material in these magnetic vortices, its superconducting state

there is disrupted and the material is in a normal state. The distinction between type I and type II can also

be made in the GL theory with the Ginzburg-Landau parameter , defined as

1 ,2

1 ,2

type I

type II

(2.13)

Some superconductors are shown in Fig. 1 along with their critical temperature TC, penetration length λ,

coherence length ξ and the GL parameter κ.

2.3 Unconventional superconductors

Despite it being a successful theory of superconductivity, the BCS theory does not apply to all

superconductors; those are called unconventional superconductors. Some groups of unconventional

superconductors are the HTSC, heavy fermion metals (UPt3), organic superconductors and ferromagnetic

superconductors (ZrZn2) [1].

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Fig. 1: Table of some superconducting materials along with their characteristic quantities [1].

Cuprates are layered crystals in which the superconducting state is reached in the CuO2 planes, which are

separated by insulating layers. The two most studied cuprates are YBCO (YBa2Cu3O7-x) and BSCCO

(Bi2Sr2CaCu2O8+x). The orthorhombic unit cell of the latter is shown in Fig. 2a [5]. The origin of

superconductivity is because of their complex structures and strong electron coupling different from that

of conventional superconductors and still not fully understood. In the following years, more HTSC were

discovered, as shown in Fig. 2b.

Fig. 2: (a) The orthorhombic unit cell of BSCCO (Bi2Sr2CaCu2O8+x)

with its dimensions [5] and (b) known

superconductors plotted with the year of their discovery (horizontal axis) and critical temperature (vertical axis) [6].

3 Tunnelling of Electrons Tunnelling is a quantum-mechanical phenomenon. If a particle is situated on one side of a potential

barrier of finite width and height, there is a non-zero probability for the particle to be found on the other

side of the barrier even in the case where its energy is lower than the height of the potential barrier; the

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particle “tunnels” through the barrier [7]. Fig. 3.a shows the potential well and the wave function of the

particle. The blue line represents the wave function of an electron tunnelling from the tip to the sample

and the red line vice-versa.

Fig. 3: (a) The wave function of a tunnelling electron and (b) a schematic of the STM experiment with its tip and the

sample surface [5].

3.1 Operation of STM and STS There are different microscopy and spectroscopy measurements possible with the STM. The most

important advantages of using the STM are in the very local nature of measurements performed and in the

very high energy resolution, limited only by the temperature of the tunnelling junction.

The STM, Fig. 3b, uses the tunnelling phenomenon to measure the local density of electron states

(LDOS) on the surface of metals and thus their structure. An atomically sharp tip is driven across the

sample surface by precise piezos. The tip probes the sample surface’s occupied or unoccupied states,

depending on the tunnelling voltage polarity.

The tunnelling current is a function of the voltage difference between the sample and the tip VT, the tip

distance from the sample surface d and the densities of electron states in the sample s and the tip t

[8]:

2

2

2

( ) ( )( ( ) ( )) ( , ) ;

( , ) exp 2 ( 2 ) ,

T t T s t T s T

eT t s T

I eV f eV f M eV d

mM eV d eV

(3.1)

where ( )f is the Fermi distribution and t and s are the work functions of the tip and the sample,

respectively. The tunnelling current roughly decreases exponentially with increasing sample-tip distance

d and it is usually in the order of a few nA.

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There are two basic modes of imaging sample surfaces with an STM: the constant-height scanning and

the constant-current scanning. In the constant-height mode the STM tip is first lowered just above the

sample surface (0.4 – 0.7 nm). The tunnelling current is then observed as a function of the x and y

position of the tip, while keeping the height of the tip constant. In the constant-current mode, the

electronic feedback keeps the tunnelling current constant by adjusting the height of the tip. Its height is

then observed. Both modes of operation can be seen in Fig. 4 [5]. The constant-height mode is faster,

because adjusting the tip height is relatively slow, but working with this mode requires that the sample

surface is very flat so the tip does not crash into the sample surface.

Fig. 4: The two modes of scanning: (a) the constant-current and (b) the constant-height mode [5].

The STS consists of observing the LDOS of surface electrons as a function of bias (voltage VT) between

the STM tip and the sample surface, while keeping the position (x, y and z) of the tip constant. By

calculating a derivative of Eq. (3.1) with respect to the tunnelling voltage and assuming a constant LDOS

of the tip, the differential current in the low-temperature limit can be written as [8]

( ) ( ) ( ).

T

Tt s T t s T

T V

dIeV d eV

dV

(3.2)

We see that the differential tunnelling current dI/dVT is proportional to the LDOS of surface electrons. It

can therefore be observed either by first measuring the tunnelling current as a function of the voltage

difference between the tip and the sample and finding the function’s derivative or with the lock-in

technique, where a small sinusoidal voltage is added to the static voltage difference between the sample

and the tip, so the tunnelling current can be written as

22 2

2

( ) ( )( ) ( sin( )) ( ) sin( ) sin ( ) ...T T

T m T m m

dI V dI VI V I V V t I V V t V t

dV dV (3.3)

The differential current can then be extracted with a lock-in amplifier and a good signal-to-noise ratio can

be reached.

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3.2 The STM System In order to maximize the STM’s precision several conditions must be met. The first is the ultra-high

vacuum. Air molecules trapped inside the system could adsorb to the sample or the STM tip and ruin the

clean surface of the sample. The second important condition for the precise measurements with the STM

is its low working temperature. Scanning metallic surfaces takes time (several minutes for quality

images). To ensure that the sample surface remains static (to minimize thermal fluctuations of atoms),

low temperatures are desired. Cooling can be done using liquid nitrogen to 77 K (the boiling temperature

of nitrogen). Lower temperatures are reached using liquid helium (its boiling temperature being 4.2 K)

and even lower temperatures using Joule-Thomson (1 K) or dilution refrigerators (few 10 mK).

At low enough temperatures and with precise building of the STM system, atomic resolution images can

be taken.

4 STS Observations of Superconducting States

4.1 NbSe2 Studies

In 1981 Hess et al. studied NbSe2, a type II superconductor with a critical temperature TC = 7.2 K [9].

Results of the STS, shown in Fig. 5a, show the predicted energy gap with the width of 1.11 meV at 1.45

K. The width of the energy gap varies with temperature and it completely vanishes above TC (inset).

Fig. 5: (a) The STS spectrum as a function of the bias voltage for the type II superconductor NbSe2 (at T=1.45 K) with the

width of the energy gap as a function of temperature (inset) and (b) an image of the triangular Abrikosov lattice of

magnetic vortices [9].

The sample was then exposed to an external magnetic field. The bias voltage was set to 1.3 mV, the

magnetic field’s intensity to 1 T and the sample temperature to 1.8 K. The STM image, shown in Fig. 5b,

clearly shows the magnetic vortices forming a triangular Abrikosov lattice, with vortex spacing 479 Å,

consistent with Eq. (2.12). The lattice was found to expand when lowering the magnetic field and persist

to fields down to 0.02 T.

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At 0.02 T the vortices were separated to about 3460 Å and could therefore be observed individually.

Three STS spectra were measured; one far from the vortex (2000 Å), one near (75 Å) and one in the

centre of the vortex. The three spectra are shown in Fig. 6a. The results clearly show an increase in the

LDOS in the centre of the vortex at zero-bias. To inspect the electronic structure over the vortex centre

more precisely, another STS scan was measured with the magnetic field’s intensity at 0.03 T and

temperature 1.8 K. In a 1000 Å line crossing the vortex centre, 128 points were chosen and an STS

spectrum was measured in each of them. The vortex half-width was found to be 77 Å, consistent with the

predicted coherence length. The results are shown in Fig. 6b.

In a normal vortex core the material is in a non-superconducting state and one would expect a featureless

LDOS spectrum there. This zero-bias conductance peak (ZBCP) was later explained with Andreev

reflections. Given the limited STS resolution, the discrete states inside the vortex are not seen and the

ZBCP appears continuous.

Further experiments on NbSe2 [10] showed the magnetic vortices are star-shaped with sixfold symmetry,

shown in Fig. 7. Size and orientation of these vortices depend strongly on the bias voltage of the STM. At

0 mV the points of the vortices are aligned to the crystallographic axis (Fig. 7a) and at 0.5 mV the points

are rotated by 30° to the crystallographic axis (Fig. 7b). N. Hayashi et al. [11] showed that an anisotropic

pair potential with six-fold symmetry 0( ) (1 cos6 )Ac with a varying anisotropy parameter cA

can explain this phenomenon.

Fig. 6: (a) Three STS spectra; taken in the centre of a magnetic vortex (top curve), 75 Å from the centre (middle) and far

away from the vortex (lower). (b) A more precise measurement of spectra in 128 points, taken along a magnetic vortex.

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Fig. 7: Star-shaped magnetic vortex in NbSe2 at (a) VT=0 mV and (b) VT=0.5 mV, H=50 mT [10].

4.2 Measurements on HTSC Because of the HTSC’s long penetration depths (relative to their coherence lengths), the STS and STM

techniques are superior to techniques sensitive to magnetic fields for observing the HTSC, as the changes

in the LDOS are a direct result of changes in the order parameter . Despite having high critical

temperatures and easily cleavable structures, the HTSC are difficult to image for several reasons.

Coherence lengths are usually much shorter than in ordinary superconductors, so the vortices are difficult

to observe. The crystalline structures are complicated (Fig. 2a) and growing big enough homogeneous and

defect free samples is not easy.

4.2.1 Energy Gap

Doping plays an important role in the HTSC. Undoped cuprates are Mott insulators (they behave as

insulators). When doping is increased, they become superconductors and the critical temperature TC first

increases to a maximum (at optimal doping). If doping is increased even more, the critical temperature

vanishes. Varying doping levels also affect the superconducting gap, as shown in Fig. 8 (BSCCO). The

energy gap width and the critical temperature decrease with increasing doping level. The reduced energy

gap, defined as 2B Ck T

, ranges from 4.3 up to 28. This contradicts the BCS theory which predicts a

constant ratio 1.764, presented in Eq. (2.9) [5].

Another interesting phenomenon in the HTSC is the pseudogap. The energy gap does not vanish at

temperatures above TC, as shown in Fig. 9b, but a reduced density of states persists up to a higher

temperature T*. At first this pseudogap seemed similar to the superconducting gap, but it now appears

that the pseudogap does not originate from superconductivity alone [13]. For example, a schematic phase

diagram of cuprates is shown in Fig. 9a.

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Fig. 8: Behaviour of the energy gap with varying doping levels in BSCCO. The energy gap width and the critical

temperature decrease with increasing doping levels [12].

Fig. 9: The pseudogap: (a) a schematic phase diagram showing the superconducting (SC), antiferromagnetic (AF) and the

pseudogap phase. (b) Behaviour of the LDOS at various temperatures [5]. An energy gap persists for temperatures above

TC [14].

4.2.2 Magnetic Vortices

The STS spectra of magnetic vortices of the HTSC show the LDOS much different to those in

conventional superconductors. In YBCO, the results [15], shown in Fig. 10a, reveal two smaller peaks at

energies of about ±5.5 meV, instead of the ZBCP found in conventional superconductors.

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Fig. 10: (a) The STS spectra of YBCO inside the vortex (red line) and at about 20 nm away from its center (blue dashed

line). The vortex LDOS shows two smaller peaks at voltage biases ±5.5 mV [5]. (b) STS image of BSCOO with Zn

impurities (dark spots), taken at 0 mV bias. Circles represent magnetic vortices of radius 60 Å, imaged at 7 mV [16].

Due to crystal impurities and inhomogeneities the vortex lattices in YBCO [15] or BSCOO [16] do not

show long-range order. The vortices of radius 60 Å along with Zn impurities are shown in Fig. 10b,

where they are represented by black circles. It is clear that most vortices are situated at Zn impurities and

the lattice is disordered.

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Summary After the discovery of superconductivity, many attempts were made to understand it. Conventional

superconductivity is now well understood, but some aspects of high temperature superconductivity still

remain a mystery. Its high energy resolution and scanning precision and the fact that it is a direct

approach to observing electronic properties of surfaces of superconductors make tunnelling spectroscopy

an excellent technique for studying superconductors.

References [1] James F. Annett, Superconductivity, Superfluids and Condensates (Oxford University Press, New

York, 2004).

[2] S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63, 1641(2000).

[3] C. E. Sosolik, J. A. Stroscio, M. D. Stiles, E. W. Hudson, S. R. Blankenship, A. P. Fein and R. J.

Celotta, Phys. Rev. B 68, 140503(2003).

[4] J. G. Bednorz and K. A. Müller, Z. Phys. B: Condens. Matter 64, 189 (1986).

[5] Ø. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod and C. Renner, Reviews of Modern Physics 79

(2007).

[6] http://en.wikipedia.org/wiki/File:Sc_history.gif, (taken on November 21st, 2012).

[7] http://fiz.fmf.uni-lj.si/~tine/kvantna3.pdf, (taken on November 21st, 2012).

[8] Zupanič, E., 2010. Low-Temperature STM Study and Manipulation of Single Atoms and

Nanostructures, Ph.D. Thesis, Jožef Stefan International Postgraduate School

[9] H. F. Hess, R. B. Robinson, R.C. Dynes, J.M. Valles, Jr. and J.V. Waszczak, Phys. Rev. Lett. 62,

214(1989).

[10] H. F. Hess, R. B. Robinson and J. V. Waszczak, Phys. Rev. Lett. 64, 2711(1990)

[11] N. Hayashi, M. Ichioka and K. Machida, Phys. Rev. Lett. 77, 4074(1996).

[12] C. Renner, B. Revaz, J.-Y. Genoud and Ø. Fischer, J. of Low Temp. Phys. 105, 1083(1996).

[13] Rui-Hua He, M. Hashimoto, H. Karapetyan, J.D. Doralek, J.P. Hinton et al., Science 331,

1579(2011).

[14] C. Renner, B. Revaz, J.-Y. Genoud, K. Kadowaki and Ø. Fischer, Phys. Rev. Lett. 80, 149(1998).

[15] I. Maggio-Aprile, C. Renner, A. Erb, E. Walker and Ø. Fischer, Phys. Rev. Lett. 75, 2754(1995).

[16] S. H. Pan, E. W. Hudson, A. K. Gupta, K.-W. Ng, H. Eisaki, S. Uchida and J. C. Davis, Phys. Rev.

Lett. 85, 1536(2000).