Chapter 1 Introduction
1.1 Superconductors
1.1.2 Type-I and Type-II superconductors
From the last section we learned that a magnetic flux will be excluded from inside of the superconductor due to the Meissner effect. But the Meissner effect can be destroyed when a strong enough magnetic field is applied to the superconductor. That critical magnetic field, Hc, can be used to explain different behaviors of type Ⅰand typeⅡ superconductors shown in figure 1.1.3.
Usually for type Ⅰ superconductors, the superconductivity will totally disappear when the applied magnetic field is above a critical value Hc; namely type Ⅰ superconductor is back to the normal state with zero magnetization M. Pure metals, such as lead, aluminum, and mercury [7], typically exhibit type I superconductivity.
Figure 1.1.2: Description of the perfect diamagnetism. [7]
In case of Type Ⅱ superconductors, they are normally alloys with three different states, superconducting, normal, and mixed states. The mixed state, as the name implies, means that the superconducting and normal states coexist at the same time. So there are two different critical fields, Bc1 and Bc2, in the type-II superconductor. All of the high-temperature superconductors belong to type-II; EBCO is no exception.
Figure 1.1.3: Magnetization M and applied field H for type-I and type-II Superconductors [9].
1.2 Graphene
Graphene is a single atomic layer of graphite. The structure of graphene contains a honeycomb lattice of carbon atoms in a perfect two-dimensional (2D) layer. It was not known to exist in an isolated form until 2004. The electron transport properties were reported from some research groups [10-12].
Due to the unusual properties of graphene, such as quasiparticle, massless Dirac fermion and the relativistic-like behaviors [11, 12], ambipolar field effect [10], “half-integer” quantum Hall effect [11], high crystal quality, and ballistic transport at submicron distances [10, 11]. More explorations of graphene have been conducted since its discovery for the years to come.
(a) (b)
Figure 1.2.1: (a) Honeycomb lattice of a carbon monolayer [15]. (b) Ambipolar electric field effect in single-layer graphene. [16]
Recently, several methods of producing graphene films were proposed. Graphene can be fabricated by exfoliation, chemical vapor deposition (CVD) on copper (Cu) [13], or epitaxially grown on Silicon carbide (SiC) [14]. Exfoliated graphene is the simplest one; formed by mechanically using tape cleavage. Nevertheless, there is an inherent problem in this technique: the size of graphene can be only in microns. It is not a scalable technique and really hard to locate the graphene film. CVD graphene and epitaxial graphene can be used to produce large area pieces. Moreover, epitaxy produces higher quality and uniform films than CVD, which is not detailed in this study. The epitaxial graphene was formed on the Si face of a high-purity SiC wafer by thermal annealing and that is the main source of graphene in this thesis. We will show more measurements results regarding to epitaxial graphene [17, 18] in the later chapters.
References
[1] B. D. Josephson, Proceedings of the IEEE 62, 6, 838 - 841, (1974).
[2] B. D.J osephson, Phys. Lett 1,7, 251-253 (1962).
[3] H. K. Onnes, Commun. Phys. Lab. Univ. Leiden 12, 120, (1911).
[4] Charles P. Poole, Jr., Horacio A. Farach, Richard J. Creswick, “Superconductivity”, (Academic Press, 1995).
[5] Nao Takeshita, Ayako Yamamoto, Akira Iyo, and Hiroshi Eisaki, J. Phys. Soc.
Jpn. 82, 023711 (2013).
[6] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Physical Review 108, 1175 (1957).
[7] Webpage” Lecture 8: Perfect Diamagnetism”, Terry P. Orlando, Massachusetts Institute of Technology, 2003, Retrieved April 11, (2012).
http://web.mit.edu/6.763/www/FT03/Lectures/Lecture8.pdf [8] Allister M Forrest, Eur J. Phys. 4, 17 (1983).
[9] Webpage” Fichier: Magnetisation_and_superconductors.png”, Retrieved April 11, (2012).
http://fr.wikipedia.org/wiki/Fichier:Magnetisation_and_superconductors.png [10] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I.
V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[11] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V.
Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
[12] Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature 438, 201 (2005).
[13] X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni,I. Jung, E.
Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, Science 324, 1312 (2009).
[14] Johannes Jobst, Daniel Waldmann, Florian Speck, Roland Hirner, Duncan K.
Maude, Thomas Seyller, and Heiko B. Weber1, Phys. Rev. B 81, 195434 (2010).
[15] C. W. J. Beenakker, Rev. Mod. Phys. 80, 1337-1354 (2008).
[16] A. K. Geim, K. S. Novoselov, Nature Materials 6, 183 (2007).
[17] Gregory M. Rutter, Nathan P. Guisinger, Jason N. Crain, Phillip N. First, and Joseph A. Stroscio, Phys. Rev. B 81, 245408 (2010).
[18] Y.-M. Lin, C. Dimitrakopoulos, K. A. Jenkins, D. B. Farmer, H.-Y. Chiu, A. Grill, Ph. Avouris, Science 327, 5966 (2010).
Chapter 2 Josephson effect
The Josephson effect is the phenomenon of a supercurrent; a current that flows through a junction without any applied voltage. Devices known as the Josephson junction consist of two superconductors sandwiching a weak link, which can be an insulating layer (superconductor / insulator / superconductor junction, or S-I-S), a non-superconducting metal (S-N-S), or a thin layer of graphene (S-G-S). From another perspective, the Josephson effect can be viewed as an example of a macroscopic quantum phenomenon, which will be explained in the following section.
2.1 Macroscopic quantum model
When a superconductor is at the temperature below its critical temperature, two electrons with opposite momentums and spin directions near the Fermi surface can be combined into Cooper pairs, which are composite bosons [5]. It is not affected by the limit of the Pauli exclusion principle which allows different bosons in the same quantum state. Large number of Cooper pairs will condense into the same quantum state, the ground state. A macroscopic phenomenon, Josephson effect, can be observed when most of the bosons with the same behavior stay in the same state. Therefore, the behavior of Josephson junction related to the Cooper pairs can be described by using a single wave function. A macroscopic quantum model followed can be utilized to explain this quantum phenomenon. Josephson tunneling junction formed by combining two superconductors exhibits a barrier to the electrons. A certain phase difference, attributed to the tunneling effect and phase interference, between the macroscopic wave functions of the device, cause the Josephson effect.
2.2 Josephson equation
A schematic description of a Josephson junction obtained by inserting an insulating layer between two superconductors is shown in figure 2.2.1. When the insulating layer is very thin, some degree of weak coupling between two superconductors exists. A phase correlation, due to the weak coupling, between superconductors makes Cooper pairs penetrating this barrier from superconductor 1 into superconductor 2. In order to realize the tunneling of the Cooper pairs in a Superconductor/Insulator/Superconductor (SIS) device, the thickness of the insulating layer needs to be within approximately 10 Å due to the slim probability of tunneling in the SIS structure. Figure 2.2.2 (a) and (b) illustrate two possible schemes to observe the tunneling phenomenon. Figure 2.2.2 (b) shows another structure called a Superconductor/Non-superconducting metal/Superconductor (SNS) device. As long as the thickness of the non-superconducting metal is kept below about 100 nm, there is the opportunity to observe the tunneling effect of the Cooper pairs.
Figure 2.2.1: A schematic view of Josephson junction.
For a mathematical explanation, let us consider two superconductors, denoted as 1 and 2, respectively, separated by an insulating barrier (SIS structure). From a macroscopic quantum model, the time-dependent Schrödinger equation for the superconductor can be given by:
iℏ∂ψ∂t = 𝐻ψ, (2.2.1)
where 𝜓 and 𝐻 are the wave functions and Hamiltonians for superconductors, respectively.
So we can write down the wave functions of the electrons in the superconductors with the weak coupling as follows:
iℏ∂ψ∂t1 = 𝐸ψ1+ Kψ2, (2.2.2) iℏ∂ψ∂t2 = 𝐸ψ2+ Kψ1, (2.2.3) where K is the coupling constant, deciding the tunneling probability of Cooper pairs. Due to the weak coupling existing between the superconductors, the constant K is very small.
Then we express the wave functions in terms of the pair density as the following:
Figure 2.2.2: Different types of weak links: (a) SIS tunneling junction; (b) SNS tunneling junction. [1]
𝜓1 = √𝜌1𝑒𝑖𝜃1 and (2.2.4)
𝜓2 = √𝜌2𝑒𝑖𝜃2, (2.2.5)
where 𝜌1and 𝜌2 are the densities of super electrons in superconductors.
If the wave functions in equations (2.2.4) and (2.2.5) are substituted into equations (2.2.2) and (2.2.3), we can separate the real and the imaginary parts and obtain equations as the following: can have the relation between equations (2.2.6) and (2.2.8) as the following:
∂ρ1
∂t = −∂ρ∂t2. (2.2.10)
The equation shows the increasing rate of the Cooper pairs density in superconductor 1 is equal to the decreasing rate of the Cooper pairs density in superconductor 2. The current density of the Cooper pairs can be shown as the following:
js = 2e∂ρ∂t1 =2𝑒𝐾ℏ 𝜌 sin 𝜙
js = jcsin 𝜙, (2.2.11)
where 𝑗𝑐 =2𝑒𝐾𝜌ℏ is the critical current density, ϕ is the phase difference, and K is the
If the potential difference between the superconductors is V, the energy difference for an electron traveling through them is 2 eV; therefore,
∂ϕ
∂t =2𝑒𝑉ℏ . (2.2.13)
Eqs. (2.2.11) and (2.2.13) are called “Josephson relations”, and that describe the basic tunneling behavior of Cooper pairs across the junction.
2.3 RCSJ model
The resistively and capacitively shunted junction (RCSJ) model is an easy and useful equivalent-circuit model to quantify the behavior of a Josephson junction. It is composed of a resistor and a capacitor, parallelly connected to an ideal Josephson junction with a critical current. A supercurrent exists when we apply a constant current I < Ic because there is zero resistance in the junction. On the contrary, if we apply a constant current I >
Ic flowing through the Josephson junction, a normal current In must start flowing because the supercurrent cannot exceed Ic. That can be seen more clearly in figure 2.3.1.
Figure 2.3.1: Shows the I-V characteristic for a Josephson junction at T = 0 K. Ic is the maximum supercurrent at zero voltage, and In is the normal-state current [1].
Then, we can simplify RCSJ model illustrated in figure 2.3.2. By the mathematical derivation followed, more details of this model will be revealed. When the current flowing through the Josephson junction is I > Ic, the voltage V across the junction is not zero while the superconductors are still in the superconducting state. Not only the Josephson current but also the normal current flows through the junction. The time-dependent current passing through the junction in terms of capacitance can be represented as follows:
I = I𝑑+ I𝑛+ I𝑠 = 𝐶𝑑𝑉(𝑡)𝑑𝑡 +𝑉(𝑡)𝑅 + I𝑐sin 𝜙, (2.3.1) where Is , In, and Id are the supper, normal, and displacement currents, respectively. (I𝑑 = 𝐶𝑑𝑉(𝑡)𝑑𝑡 )
From the AC Josephson equation:
d𝜙
dt =2𝑒𝑉(𝑡)ℏ (2.3.2)
, eqn. 2.3.1 can be written as:
I =ℏ𝐶2𝑒𝑑2𝑑𝜙(𝑡)2𝑡 +2𝑒𝑅ℏ d𝜙(𝑡)dt + I𝑐sin 𝜙. (2.3.3) In fact, Josephson junction in an electric circuit can be substituted by the following equivalent circuit shown in figure 2.3.2. The equivalent circuit model is called resistively and capacitively shunted junction; abbreviated as RCSJ model.
2.4 Andreev reflection
In this thesis, we study the Josephson effect in a high-temperature/graphene hybrid junction. Andreev reflection in our devices happened when the temperature of device is below Tc, at which electrodes were in the superconducting state leaving the graphene in the normal state; therefore, a superconducting state/normal state (S/N) interface formed.
The Andreev reflection occurs at the S/N interface. A hole in the N region will be reflected when an incoming electron from the N region hits the S/N interface as shown in figure 2.4.1. Therefore an electron pair (Cooper pair) will transport in S region. In other words, if the electron in the conduction band is in the excited state below the superconducting gap, two possible ways occur. The electron can be either normally reflected at S/N interface or paired with another electron possessing opposite momentum and spin to form a Cooper pair tunneling through the S/N interface. Each Andreev reflection transfers a charge of 2e across the interface and causes a retroreflection (AR) [3] of a hole of opposite spin and momentum to the incident electron.
Figure 2.4.1: Shows a schematic of Andreev reflection in S/N interface.
But difference between using the normal metal or graphene as the sandwiched layer exists. The Andreev reflection of a normal metal behaves the same as mentioned above in figure 2.4.1. If graphene is used to replace the metal between the superconductors, retroreflection (AR) and specular Andreev reflection (SAR), which will be present in the superconductor/graphene/superconductor devices [2], can be observed.
Figure 2.4.2: Shows different scenarios of Andreev reflection in graphene: (a) EF≫ eV , retro-reflection. (b) EF= 0 , spercular reflection. (c) the incoming electron,
The basic idea of Andreev reflection in graphene is shown in Fig 2.4.1. It is known that any Cooper pair will exist only at a finite bias; therefore, the applied bias (V) should not exceed the superconducting gap in order to keep the superconducting state. Two conditions can be considered. Firstly, when Fermi energy (E𝐹) is much larger than the superconducting gap and the bias of junction is smaller than the superconducting gap (V < △≪ E𝐹) as shown in figure 2.4.2 (a) and (c), referring to the retro-reflection (AR).
Secondly, if the Fermi energy is smaller than the superconducting gap, normally at EF = 0 possibly achieved by tuning the gate, fabricated by graphene, to adjust the junction bias below the superconducting gap and the electron energy above Fermi energy (0 = EF <
eV < △), corresponding to the specular reflection (SAR) as shown in Fig 2.4.1 (b) and (d).
In order to understand how the hybrid HTS/graphene device, structured by EBCO/graphene/EBCO works, the aforementioned fundamental properties of superconductors and graphene are necessary to investigate and analyze the results of the devices in this study.
References
[1] Andrei Marouchkine, “Room-Temperature Superconductivity,” (Cambridge International Science Publishing, (2004).
[2] Katsuyoshi Komatsu, Chuan Li, S. Autier-Laurent, H. Bouchiat, and S. Gueron, Phys. Rev. B 86, 115412 (2012)
[3] Pablo Burset Atienza, Superconducting proximity effect and nonlocal transport in graphene and carbon nanotubes (April 2012).
[4] Xing Lan Liu, Quantum Dots and Andreev Reflections in Graphene (August 2010).
[5] M. de Llano, F. J. Sevilla and S. Tapia, Int. J. Mod. Phys. B 20, 2931 (2006).
Chapter 3 Experimental technique and Sample fabrication
In this thesis, two novel materials were fabricated to form the EBCO/grphene/EBCO structure. The configuration of the RF sputtering system, used to grow EBCO thin film, will be introduced first; followed by the fabrication of graphene on the EBCO bridge and then the measurement.
3.1 Fabrication of EBCO thin films
In order to produce EBCO/graphene/EBCO devices, we need to grow EBCO thin film on SrTiO3 (STO) substrate. The RF sputtering system used is shown in figure 3.1.1.
Figure 3.1.1: A schematic of the RF sputtering system.
Two 2-inch in diameter targets made of stoichiometric superconducting Eu1Ba2Cu3O compound were installed in the RF sputtering chamber. A mixture of Ar and O2 at a ratio of 9 (Ar/O2) was used as the sputtering gas at the pressure of about 350 mTorr. The STO substrates were mounted on the quartz holder. The growth temperature of the substrates was controlled by three 1kW quartz lamps underneath the holder. The growth time of the EBCO films was two hours by activating two 50W RF sputter guns. The post-growth annealing time is approximately 1 hour at the same deposition output power of three 1kW quartz lamps. The profile of the cooling process consists of two stages: (1) the temperature was cooled down to 250℃ from the deposition temperature at the rate of 5℃/min, (2) followed by a rate of 15℃/min down to the room temperature.
3.2 Fabrication of SNS devices
Once the growth of EBCO thin film on STO was completed, the fabrication of the EBCO/Au/EBCO devices shown in figure 3.2.2 was followed. Extra care should be exerted at each step to ensure the successful fabrication of the SNS devices since no one had ever tried it at the time. We patterned our EBCO thin film as shown in figure 3.2.1 (b) by photolithography. Figure 3.2.1 (a) is a schematic of our mask, and (b) is the SEM image for the region that circled by dotted line in (a) and (b). The edge accumulation was observed during the fabrication since the size of the STO substrate was about 3.3×3.3 mm2; too small for a homogeneous spin coating. Nevertheless, it would not affect the
(a) (b)
(c)
Figure 3.2.1: Optical lithography technique for the SNS device. (a) Mask. (b) Patterned EBCO thin film on STO. (c) The SEM image for the superconductor bridge is the region that circled by dotted line in (a) (b). W is the width of bridge, W~10μm.
W
The focus ion beam (FIB) was employed to cut the superconducting bridge as shown in figures 3.2.2 (b) and (c) after patterning the EBCO thin film. The depth of each cut is 1μm, which will assure the separation of the EBCO on both sides of the bridge from each other. Before the deposition of Au to fill up the region cut by FIB, a thermal
(a)
(b) (c)
Figure 3.2.2: (a) Shows the schematic of the EBCO/Au/EBCO structure, (b) is the SEM diagram of the EBCO bridge cut by focus ion beam (FIB), and (c) is the SEM image of the squared region in (b), where W, ~10μm, is the width of bridge, and L,
~107 nm, is the length of the gap.
L W
L W
3.3 Fabrication of SGS devices
In the bridge structure, the Au contact can be replaced by graphene to form SGS devices. Exfoliated graphene was used for the purpose of feasible fabrication of EBCO/graphene/EBCO structure. Most of the SGS devices were done in an opposite fashion as shown in figure 3.3.1 (a); depositing superconductors on the graphene thin film transferred onto the Si/SiO2 substrate. The reason why we reversed the process was due to the high-temperature superconductors used for investigation. Unlike metallic superconductors, the growth of high-temperature superconductors needs a lattice-matched substrate. This is why EBCO was positioned under graphene.
Figure 3.3.1: Shows a schematic representation of different structures of SGS devices; (a) structure previously used for metallic superconductors, and (b) a novel device structure for high-temperature superconductors, where SC is the abbreviation of superconductor.
3.4 Four-terminal resistance measurement
The four-terminal resistance measurement is a common technique for the high accuracy, especially in cryogenic experiments. This technique can minimize the measurement errors caused from the contact resistance. If we perform a two-terminal measurement, the contact resistance would usually severely compromise the measuring results; therefore, we chose a four-terminal resistance measurement. However, the resistance measured by the four-terminal resistance measurement method are obtained from the van der Pauw method.
Figure 3.4.1: The configuration for van der Pauw method with an arbitrary shape and four contacts located at the peripheral. The resistance RMN,OP is measured across
The van der Pauw method is usually used for measuring the resistivity, and this method can be applied to any sample with an arbitrary shape, but the thickness of the sample need to be uniform and the area of contacts should be as small as possible.
As shown in figure 3.4.1, current will pass through contacts O and P (IOP). The potential difference between contacts M and N (VMN) is measured. By Ohm’s law, the resistance between contacts M and N can be found as in eq. (3.4.1).
𝑅𝑀𝑁,𝑂𝑃 =𝑉𝐼𝑀𝑁
𝑂𝑃.
(3.4.1)
RMN,OP is a resistance between contact M and N. By van der Pauw formula [4], the relation between resistances can be expressed as the following
e− πdR𝑀𝑁,𝑂𝑃𝜌 + e− πdR𝑁𝑂,𝑃𝑀𝜌 = 1 ,
(3.4.2)
where d is the sample thickness and ρ is the resistivity of the sample.
For our measurement, a high resistance resistor (~ 100 MΩ) was connected to the sample in series., as shown in figure 3.4.2. The longitudinal resistance Rxx and Hall resistance Rxy are defined as:
𝑅𝑥𝑥 = 𝑉𝑥𝑥 𝐼⁄𝑆𝐷
and 𝑅𝑥𝑦 = 𝑉𝑥𝑦 𝐼⁄𝑆𝐷,
where Vxx and Vxy are the electric potentials parallel and perpendicular to the direction of source-drain current, respectively.
3.5 Cryogenic system: Sorption pumping
3He cryostat
In order to measure the sample at the low temperature, a top loading 3He cryostat from Oxford Instrument, as show in figure 3.5.1, was utilized. The maximum magnetic field is about 15 Tesla at the temperature of approximately 0.3 K. The condensation of
3He gas, as shown in figure 3.5.2, can cool the temperature down to 1.2 K. By lowering the vapor pressure of liquid 4He in the 1K pot, which can be kept at around 2 K below the
Figure 3.4.2: The configuration of four-terminal measurement [2].
Figure 3.5.1: Shows a schematic diagram of Oxford instrument [3].
Figure 3.5.2: The process of 3He condensation. It shows releasing of 3He gas [4].
References
[1] S. H. N. Lim, D. R. McKenzie, and M. M. M. Bilek, Rev. Sci. Instrum. 80, 075109 (2009).
[2] Jheng-Cyuan Lin, Master thesis, National Taiwan University (2012).
[3] Jyun-Ying Lin, Master thesis, National Taiwan University (2006).
[4] Tzu-Lun Lin, Master thesis, National Taiwan University (2009).
Chapter 4 Results of epitaxial graphene
This part of thesis will discuss the experimental results of epitaxial graphene, i.e., the electric properties of graphene grown on the SiC substrate. The preliminary results showed weak localization, current heating, and insulator-quantum Hall transition in the epitaxial graphene. The fundamentals of these findings will be introduced first; followed by the experimental results.
4.1 Characteristics of epitaxial grapheme
At first, we should know how graphene can grow on SiC. Usually, epitaxial graphene can grow on two different faces of SiC, SiC (0001̅) carbon-terminated face (C-face) or SiC (0001) silicon-terminated face (Si-face) [1, 2]. Our epitaxial graphene was grown on (0001) surface of 6H SiC, of which the structure is shown in figure 4.1.1.
(a) (b)
Figure 4.1.1: (a) the unit cell structure of SiC, (b) a kind of SiC polytypes, 6H. “6”
refers to a stack of periodically arranged number (ABCACB), the letter “H” denotes the hexagonal symmetry of the SiC crystal. A and B only difference between shift of the lattice, their orientation do not change. But C means twisting the lattice by 60° [3].
It was easier to control the number of graphene layers grown on the Si-face than on the C-face because the growth of graphene on the Si-face is slower. For example, 2 monolayers of graphene were formed on Si-face after annealing at 1320 °C, but 16 monolayers on C-face at the same condition [4].
The epitaxial graphene was grown in a vacuum (10-6 torr) furnace by heating the SiC sample up to the temperature between 1100 °C and 1600 °C to form the graphene layer due to the sublimation of the Si atoms. This method provided a graphene layer with higher quality and more uniform film, as well as large area graphene layer, than the CVD
The epitaxial graphene was grown in a vacuum (10-6 torr) furnace by heating the SiC sample up to the temperature between 1100 °C and 1600 °C to form the graphene layer due to the sublimation of the Si atoms. This method provided a graphene layer with higher quality and more uniform film, as well as large area graphene layer, than the CVD