Solid State Physics Lecture Notes PDF

Solid State physics phys.371 Presented by Dr. Rania Saber Hafez [email protected] 1  Solid State physics phys.371 Total 100/100 20 Grades 20 100 Midterm exam...

Solid State physics phys.371 Presented by Dr. Rania Saber Hafez [email protected] 1  Solid State physics phys.371 Total 100/100 20 Grades 20 100 Midterm exam 60 Final exam 2  References Core book: 1- Solid state physics, An introduction, Philip Hofmann. 2-Solid state physics, R.K.Puri and V.K.Babbar. www.philiphofmann.net 3  References Other books at a similar level: 1-Solid state physics, Kittel (Wiley) 2-Solid state physics, Blakemore (Cambridge) 3-Fundamentals of solid state physics, Christman (Wiley) More advanced: Solid state physics, Ashcroft and Mermin 4  Content Chapter 1. Crystal Structures Chapter 2. Bonding in solids Chapter 3. Mechanical Properties Chapter 4. Thermal Properties of the Lattice Chapter 5. Electronic Properties of Metals: Classical Approach 5  Introduction What is solid state physics? Macroscopic view = Experiment results Physics Microscopic view= Theoretical models 6  Introduction What is solid state physics? It is the branch of physics dealing with physical properties of solids. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties (macroscopic properties) of solid materials result from their atomic-scale properties (microscopic properties). How 7  Introduction here we investigate the electrical resistivity of three states of solid matter: 1. Graphite 2. Diamond 3. Buckminster-fullerene (metal) (insulator) (superconductor) They are all just carbon! 8  Understanding the electrical properties of solids is right at the heart of modern society and technology. Thus, solid-state physics forms a theoretical basis of materials science. It also has direct applications, for example in the technology of transistors and semiconductors. Important physical properties depend on crystal structure and crystal orientation e.g conductivity, magnetic properties, strength etc. 9  Chapter 1 (Crystal structure)  Introduction  General description of crystal structures  Some important crystal structures  Crystal structure determination 10  introduction matter solid liquid gas 11  introduction Solids Solids consist of atoms or molecules executing thermal motion about an equilibrium position fixed at a point in space. Solids (at a given temperature, pressure, and volume) have stronger bonds between molecules and atoms than liquids. Solids require more energy to break the bonds. 12  introduction Liquids Similar to gases, liquids haven’t any atomic/molecular order and they assume the shape of the containers. Applying low levels of thermal energy can easily break the existing weak bonds. Gases Gases have atoms or molecules that do not bond to one another in a range of pressure, temperature and volume. These molecules haven’t any particular order and move freely within a container it takes the shape and the volume of the container. 13 introduction 14  introduction Solids materials Non-crystalline crystalline (amorphous) Single crystal polycrystalline Each type is characterized by the size of ordered region within the material. An ordered region is a spatial volume in which atoms or molecules have a regular geometric arrangement or periodicity. 15 Single crsytal polycrystalline amorphous 16  introduction Crystalline state of solids is characterized by regular or periodic arrangements of atoms or molecules that arranged in a definite, repeating pattern in three dimension. Most of solids are crystalline in nature due to the energy released during the formation of ordered structure is more than that released during the formation of a disordered structure (Crystalline state is a low energy state). The crystalline solids may be sub-divided into: single crystals polycrystalline solids 17  introduction Single crystal ideally have a high degree of order, or the periodicity of atoms extend throughout the material (entire volume of the material) ex: diamond, quartz, etc. has an atomic structure that repeats periodically across its whole volume. Even at infinite length scales, each atom is related to every other equivalent atom in the structure by translational symmetry 18  introduction Polycrystalline material is a material made up of an aggregate of many small single crystals (also called crystallites or grains) with random orientations separated by well-defined boundaries. These ordered regions, or single crystal regions, vary in size and orientation wrt one another. These regions are called as grains ( domain) and are separated from one another by grain boundaries. The atomic order can vary from one domain to the next. The grains are usually 100 nm - 100 microns in diameter. Polycrystals with grains that are 10 ) electrons. (1845-1923) Bertha Röntgen’s Hand 8 Nov, 1895 131  X-Ray Diffraction (XRD) X-ray production X rays can be produced in a highly evacuated glass bulb, called an X-ray tube, that contains essentially two electrodes (an anode made of platinum, tungsten, or another heavy metal of high melting point, and a cathode) When a high voltage is applied between the electrodes, streams of electrons (cathode rays) are accelerated from the cathode to the anode and produce X rays as they strike the anode. Evacuated glass bulb Anode Cathode 132 2 types of X-ray can be produced: Bremsstrahlung“breaking radiation” X-ray X-rays are produced when the electrons are suddenly decelerated upon collision with the metal target. Characteristic X-ray If the bombarding electrons have sufficient energy, they can knock an electron out of an inner shell of the target metal atoms. Then electrons from higher states drop down to fill the vacancy, emitting x-ray photons with precise energies determined by the electron energy levels. 133  X-Ray Diffraction (XRD) Absorption of X-rays A larger atom is more likely to absorb an X-ray photon because larger atoms have greater energy differences between orbitals (the energy level more closely matches the energy of the photon). Smaller atoms, where the electron orbitals are separated by relatively low jumps in energy, are less likely to absorb X-ray photons. The soft tissue in your body is composed of smaller atoms, and so does not absorb X-ray photons particularly well. The calcium atoms that make up your bones are much larger, so they are better at absorbing X-ray photons. 134  X-Ray Diffraction (XRD) Diffraction is a wave phenomenon in which the apparent bending and spreading of waves when they meet an obstruction. Unlike visible light, X-ray cannot be diffracted by ordinary optical grating because of their very short wavelengths. In 1912, a German physicist Max Von Lau suggested the use of a single crystal to produce diffraction of X-rays. X-Ray Diffraction (XRD) The atomic planes of a crystal cause an incident beam of X-ray to interfere with one another as they leave the crystal. The phenomenon is called X-ray diffraction. Max Von Lau (1879-1960) 135  Solid State Physics deals how the waves are propagated through such periodic structures. In this chapter we study the crystal structure through the diffraction of X-ray. Every crystalline substance gives a pattern: the same substance always gives the same pattern, and in a mixture of substances each produced its pattern independently of the others The X-ray diffraction pattern of a pure substance is like a fingerprint of the substance. The X-ray diffraction is non-destructive analytical technique for identification and quantitative determination of the various crystalline forms, known as phases Beam diffraction takes place only in certain specific directions, much as light is diffracted by a grating. By measuring the directions of the diffraction and the corresponding intensities, one obtains information concerning the crystal structure responsible for diffraction. 136  X-Ray Diffraction (XRD) X-ray crystallography is a technique in crystallography in which the pattern produced by the diffraction of x-rays. A crystal behaves as a 3-D diffraction grating for x-rays In a diffraction experiment, the spacing of lines on the grating can be deduced from the separation of the diffraction maxima Information about the structure of the lines on the grating can be obtained by measuring the relative intensities of different orders Similarly, measurement of the separation of the X-ray diffraction maxima from a crystal allows us to determine the size of the unit cell and from the intensities of diffracted beams one can obtain information about the arrangement of atoms within the cell. 137  X-ray diffraction working principle A stream of X-rays directed at a crystal diffract and scatter as they encounter atoms. The scattered rays interfere with each other and produce spots of different intensities that can be recorded on film 138  X-ray diffraction working principle The conditions for a crystal to diffract X-ray can be determined by using Bragg’s law. In 1913 English physicists Sir W.H. Bragg and his son put a model which generates the conditions for diffraction in a very simple way. Bragg law Sir William Henry Bragg (1862-1942), identifies the angles of the incident radiation William Lawrence Bragg (1890-1971) relative to the lattice planes for which diffraction peaks occurs. Bragg derived the condition for constructive interference of the X-rays scattered from a set of parallel lattice planes. 139  W.L. Bragg considered crystals to be made up of parallel planes of atoms. Incident waves are reflected specularly from parallel planes of atoms in the crystal, with each plane is reflecting only a very small fraction of the radiation, like a lightly silvered mirror. In mirrorlike reflection the angle of incidence is equal to the angle of reflection. When the X-rays strike a layer of a crystal, some of them will be reflected. We are interested in X-rays that are in-phase with one another. X-rays that add together constructively in x-ray diffraction analysis in- phase before they are reflected and after they reflected. In phase waves Bragg’s law is fulfilled out phase waves Bragg’s law is not fulfilled 140  These two x-ray beams travel slightly different distances. The difference in the distances traveled is related to the distance between the adjacent layers. Connecting the two beams with perpendicular lines shows the difference between the top and the bottom beams. The length DE is the same as EF, so the total distance traveled by the bottom wave is expressed by: EF = d sinθ DE = d sinθ Constructive interference of the DE + EF = 2 d sinθ radiation from successive planes n λ = 2 d sinθ occurs when the path difference is an integral number of wavelenghts. This is the Bragg Law. n λ = 2 d sinθ Where , d: the spacing of the planes n : is the order of diffraction 141  X-Ray Diffraction (XRD) Bragg reflection can only occur for wavelength n λ = 2 d sinθ sinθ = (nλ/2d) nλ ≤ 2d since sinθ ≤ 1 Taking d ≈ 10-10 m, we obtain λ ≤ 10-10m or 1Ao. X-ray having wavelength in this range. This is why we cannot use visible light. No diffraction occurs when the above condition is not satisfied. The diffracted beams (reflections) from any set of lattice planes can only occur at particular angles predicted by the Bragg law. 142  For structural analysis, X-rays of known wavelength λ are employed and the angles for which reflections θ take place are determined experimentally. Considering only the first order reflections the d values corresponding to these reflections are then obtained from: 2 d sinθ = λ By knowing d, one can proceed to determine the size of the unit cell and the distribution of atoms within the unit cell. In XRD nth order reflection from (hkl) plane is considered as 1st order reflection from (nh nK nl) plane. Note that the smaller the spacing the higher the angle of diffraction, i.e. the spacing of peaks in the diffraction pattern is inversely proportional to the spacing of the planes in the lattice. The diffraction pattern will reflect the symmetry properties of the lattice. 143  Since Bragg's Law applies to all sets of crystal planes, the lattice can be deduced from the diffraction pattern, making use of general expressions for the spacing of the planes in terms of their Miller indices. 144 145  h2 + k2 + l2 SC FCC BCC DC In certain lattice types 1 100 (non primitive), the 2 110 110 arrangement and spacing 3 111 111 111 of lattice planes produces diffractions from certain 4 200 200 200 planes that are always 5 210 exactly 1800 out of phase 6 211 211 (extinction rule). 7 8 220 220 220 220 The ratio of (h2 + k2 + l2) 9 300, 221 derived from extinction 10 310 310 rules is proportional to 11 311 311 311 Sin2  which can be used 12 222 222 222 in the determination of 13 320 the lattice type 14 321 321 15 Allowed reflections 16 400 400 400 400 in SC*, FCC*, 17 410, 322 BCC* & DC 18 411, 330 411, 330 146 crystals 19 331 331 331 147 148 149 150 151  X-Ray Diffraction method X-ray diffraction methods Rotating Laue’s Powder crystal method method method 152  The Laue’s Method The Laue method is mainly used to determine the orientation of large single crystals while radiation is reflected from, or transmitted through a fixed crystal. In this method a single crystal specimen is held stationary in the path of white X-ray having wavelengths ranging from 0.2 to 2 A0. The Bragg angle is fixed for every set of planes in the crystal. Each set of planes picks out and diffracts the particular wavelength from the white radiation that satisfies the Bragg law for the values of d and θ involved. The diffracted beams form arrays of spots, that lie on curves on the film. 153  The Laue’s Method This method is not useful for determination of lattice parameters because the wave length of a particular reflection is unknown. It is used in the determination of crystal symmetry. There are two modes for Laue’s method: Transmission Laue Method Back-reflection Laue Method 154  Transmission Laue Method In the transmission Laue method, the film is placed behind the crystal to record beams which are transmitted through the crystal. One side of the cone of Laue reflections is defined by the transmitted beam. The film intersects the cone, with the diffraction spots generally lying on an ellipse. Most suitable for determination the crystal orientation. Also used in determination of symmetry of single crystals 155  Back-reflection Laue Method In the back-reflection method, the film is placed between the x-ray source and the crystal. The beams which are diffracted in a backward direction are recorded. One side of the cone of Laue reflections is defined by the transmitted beam. The film intersects the cone, with the diffraction spots generally lying on an hyperbola. This method is the only method for the study of large and thick crystals. Single Crystal X-Ray Film 156  Laue Pattern The symmetry of the spot pattern reflects the symmetry of the crystal when viewed along the direction of the incident beam. Crystal orientation is determined from the position of the spots. Each spot can be indexed, i.e. attributed to a particular plane, using special charts. The Laue technique can also be used to assess crystal perfection from the size and shape. This method is not useful for determination of lattice parameters because the wave length of a particular reflection is unknown. 157  Rotating crystal method In the rotating crystal method, a single crystal is mounted with an axis normal to a monochromatic x-ray beam. A cylindrical film is placed around it and the crystal is rotated about the chosen axis. As the crystal rotates, sets of lattice planes will at some point make the correct Bragg angle for the monochromatic incident beam, and at that point a diffracted beam will be formed. The reflected beams are located on the surface of imaginary cones. By recording the diffraction patterns (both angles and intensities) for various crystal orientations, one can determine the shape and size of unit cell as well as arrangement of atoms inside the cell. 158  Rotating crystal method Lattice constant of the crystal a can be determined by means of this method. How for a given wavelength λ if the angle at which a reflection occurs is known θ, dhkl can be determined. a d  h2  k 2  l 2 159  Powder method The sample used in this method is in the form of a fine powder containing a large number of tiny crystallites with random orientations. Here a monochromatic X-ray beam is incident on a powdered or polycrystalline sample. This method is useful for samples that are difficult to obtain in single crystal form. The powder method is used to determine the value of the lattice parameters accurately. Lattice parameters are the magnitudes of the unit vectors a, b and c which define the unit cell for the crystal. For every set of crystal planes one or more crystals will be in the correct orientation to give the correct Bragg angle to satisfy Bragg's equation. Every crystal plane is thus capable of diffraction. 160  Powder method  A asample If monochromatic the sample of some x-ray consists ofbeam hundreds someis of directed crystals at aa single tens of randomly (i.e. powdered crystal, orientated then sample) single only one crystals, show thator the the two diffracted diffracted diffracted beamsbeams are may seen result. form continuous to lie on cones. the surface A circle of severalis cones. film used The to record cones may the emerge in pattern diffraction all directions, as shown. forwards Each and backwards. cone intersects the film giving diffraction lines. The lines are seen as arcs on the film. 161  Powder method The experimental arrangement used to produce diffraction is shown in the following figure: Debye-Scherrer camera. 162  Powder method It consists of a cylindrical camera, called the Debye-Scherrer camera. A very small amount of powdered material is sealed into a fine capillary tube made from glass that does not diffract x-rays. The specimen is placed in the Debye Scherrer camera and is accurately aligned to be in the centre of the camera. X-rays enter the camera through a collimator. The powder diffracts the x-rays in accordance with Braggs law to produce cones of diffracted beams. These cones intersect a strip of photographic film located in the cylindrical camera to produce a characteristic set of arcs on the film. 163  Powder method When the film is removed from the camera, flattened and processed, it shows the diffraction lines and the holes for the incident and transmitted beams. 1:powder 2: incident beam 3: reflected beam The angle θ corresponding to a particular pair of arcs is related to the distance S between the arcs as 4θ(radians) = S/R Where R is the radius of the camera. If θ measured in degree 4θ(degrees)= 57.3 S/R If taking the radius of the camera in multiples of 57.296, R=57.3 mm θ(degrees)= S(mm)/4 For first order reflections from all possible planes 2d sinθ=λ then calculate d to determine the space lattice of the crystal 164 165  Application of XRD XRD is a nondestructive technique. Some of the uses of x-ray diffraction are; 1. Differentiation between crystalline and amorphous materials; 2. Determination of the structure of crystalline materials; 3. Determination of electron distribution within the atoms, and throughout the unit cell; 4. Determination of the orientation of single crystals; 5. Determination of the texture of polygrained materials; 6. Measurement of strain and small grain size…..etc 166 Advantages and disadvantages of X-rays Advantages; X-ray is the cheapest, the most convenient and widely used method. X-rays are not absorbed very much by air, so the specimen need not be in an evacuated chamber. Disadvantage; They do not interact very strongly with lighter elements. 167  Reciprocal lattice In crystal lattice there are many sets of parallel planes with different orientations and different interplaner spacing which can cause diffraction. Some times it is difficult to deal with all such plane. To solve this problem P.P.Ewald developed a new type of lattice known as reciprocal lattice. The idea of reciprocal lattice was that each set of parallel planes could be represented by a normal to these planes having length equal to the reciprocal of the interplaner spacing. Thus the direction of each normal represents the orientation of corresponding parallel planes. The normals are drawn with reference to an arbitrary origin and point are marked at their ends. These points form regular arrangement which is called reciprocal lattice 168 Reciprocal lattice 169 Animated example showing how to obtain the reciprocal points from a direct lattice 170  Reciprocal lattice Reciprocal lattice vectors A reciprocal lattice vector, σhkl, is defined as vector having magnitude equal to the reciprocal of the interplaner spacing dhkl and direction coinciding with normal to the (hkl) planes. σhkl = (1/dhkl) ^n Unit vector normal to the (hkl) plan It should now be clear that the direct lattice, and its planes are directly linked with the reciprocal lattice. Like a direct lattice, a reciprocal lattice also has a unit cell. If the unit cell axes and angles of the direct cell are known by the letters a, b, c, α, β, γ, the corresponding parameters for the reciprocal cell are written with the same symbols, adding an asterisk: a*, b*, c*, α*, β*, γ*. 171  Reciprocal lattice The basis vectors of the reciprocal lattice are defined as: 2π Is the volume of the direct cell 2π 2π The position lattice vector in direct crystal T = n1a + n2b+ n3c The position lattice vector in reciprocal lattice G = ha* +kb* + lc* 172  Properties of reciprocal lattice (1) The direct lattice is the reciprocal of its own reciprocal lattice. (2) Each point in a reciprocal lattice corresponds to particular set of parallel planes of the direct lattice. (3) The reciprocal lattice of sc lattice is sc lattice. (4) The reciprocal lattice of fcc lattice is bcc lattice and vice versa. (5) The volume of a unit cell of the reciprocal lattice is inversely proportional to the volume of the direct lattice. (6) The distance of a reciprocal lattice point from any arbitrary fixed point is inversely proportion to the interplaner spacing in direct lattice. 173  Reciprocal lattice of SC lattice a (2π/a) 174 175  Structure factor (Fhkl) it is a mathematical function describing the amplitude and phase of wave diffracted from crystal lattice planes. Or it is defined as the ratio of the amplitude of radiation scattered by the entire unit cell to the amplitude of radiation scattered by a single point electron placed at origin. f α Zn (Z is the atomic number) Fhkl = ∑ fj e 2πi(huj+kvj+lwj) = fi S Atomic scattering factor (f) Geometrical structure factor Amplitude of scattered ray from an atom (it indicates the presence or f = Amplitude of scattered ray from an electron absence of a particular reflection in the diffraction pattern) It describes how atomic arrangement (uvw) influences the intensity of the scattered beam. (the intensity of the scattered beam depends on the atomic number) i.e. It tell us which reflections (peaks, hkl) to expect in a diffraction pattern from a given crystal structure with atoms located at positions u,v,w. 176  Diffraction Methods Diffraction methods X-ray Neutron Electron diffraction diffraction diffraction Different radiation source of neutron or electron can also be used in diffraction experiments. The physical basis for the diffraction of electron and neutron beams is the same as that for the diffraction of X rays, the only difference being in the mechanism of scattering. 177  Neutron Diffraction Neutron, possesses wave properties λ ~1A°; Energy E~0.08 eV. Neutron does not interact with electrons in the crystal. Thus, unlike the x-ray, which is scattered entirely by electrons, the neutron is scattered entirely by nuclei Although uncharged, neutron has an magnetic moment, so it will interact strongly with atoms and ions in the crystal which also have magnetic moments (information about magnetic structure of the crystal). Neutrons are more useful than X-rays for determining the crystal structures of solids containing light elements. Neutron sources in the world are limited so neutron diffraction is a very special tool. 178  Neutron Diffraction Neutron diffraction has several advantages over its x-ray counterpart; 1. Neutron diffraction is an important tool in the investigation of magnetic ordering that occur in some materials. 2. Light atoms such as H are better resolved in a neutron pattern because, having only a few electrons to scatter the X ray beam, they do not contribute significantly to the X ray diffracted pattern. 3. Penetrates deeply into most materials (allows for samples with large volumes ≈ cm3) 179  Electron Diffraction Electron diffraction has also been used in the analysis of crystal structure. The electron, like the neutron, possesses wave properties; λ ~2A°; Energy E~40 eV. Electrons are charged particles and interact strongly with all atoms. So electrons with an energy of a few eV would be completely absorbed by the specimen. In order that an electron beam can penetrate into a specimen , we use a beam of very high energy (50 keV to 1MeV) as well as the specimen must be thin (100-1000 nm) 180  Electron Diffraction If low electron energies are used, the penetration depth will be very small (only about 50 A°), and the beam will be reflected from the surface. Consequently, electron diffraction is a useful technique for surface structure studies such as oxide layers on metals. Electrons are scattered strongly in air, so diffraction experiment must be carried out in a high vacuum. This brings complication and it is expensive as well. Compare between x-ray, neutron and electron diffraction methods. 181  Solved problems (1) An x-ray beam of wavelength 0.71 A0 is diffracted by a cubic KCl crystal of density 1.99x10-3 kgm-3. calculate the interplanar spacing for (200) planes and the glancing angle for the second order reflection from these planes. The molecular weight of KCl is 74.6 amu. solution Effective number of molecules per unit cell Effective number of molecules x molecular weightl Density = a3 183  Solved problems 2- a powder camera of radius 57.3 mm is used to obtain diffraction pattern of gold (fcc) having a lattice parameter of 4.08 Ao. The monochromatic Mo-Kα radiation of wavelength 0.71 Ao is used. Determine the first four S-values. solution 184 Solved problems 185  Solved problems 3- in powder diffraction experiment using Cu-Kα radiation of wavelength 1.54 Ao , the first five lines are observed from a monochromatic cubic crystal when the angle 2ϴ is 38.0, 44.2, 64.4, 77.2 and 81.4 degrees. Determine the crystal structure and the lattice parameter. solution Ratio 3x[sin2ϴ/sin2ϴmin] Line 2ϴ0 ϴo sin2ϴ [sin2ϴ/sin2ϴmin] [h2+k2+l2] 1 38 19 0.106 1 3 3 2 44.2 22.1 0.1415 1.33 3.99 4 3 64.4 32.2 0.284 2.68 8.04 8 4 77.2 38.6 0.3892 3.67 11.01 11 186 5 81.4 40.7 0.4252 4.01 12.03 12 Solved problems 187  Solved problems 4-The Bragg’s angle for (220) reflection from nickel (fcc) is 38.2o when x-ray of wavelength 1.54 Ao are employed in a diffraction experiment. Determine the lattice parameter of nickel. solution 188  Solved problems 5- crystal reflects monochromatic x-ray strongly when the Bragg’s glancing angle for first order reflection is 15o. What are the Bragg’s glancing angle for the second and third order reflections of the same plane? solution 189  Solved problems 6- in an x-ray diffraction experiment using Cu-kα radiation of wavelength 1.54 Ao, the first reflection from fcc crystal is observed when 2θ is 84o. Determine the indices of this reflection and the corresponding interplanar spacing. Show that only one more reflection is possible. Determine the indices of that reflection and the corresponding interplanner spacing. solution 190 Solved problems 191  Solved problems 7- Copper (fcc) has a lattice parameter o 3.61 Ao. The first order Bragg reflection from (111) planes appears at an angle of 21.7o. Determine the wavelength of x-rays used. solution 192