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Materials & Properties II: Thermal & Electrical Characteristics Sergio Calatroni - CERN Outline (we will discuss mostly metals) • Electrical properties Electrical conductivity o Temperature dependence o Limiting factors Surface resistance - • o Relevance for accelerators o Heat exchange by radiation (emissivity) Thermal properties Thermal conductivity o Temperature dependence, electron & phonons o Limiting factors Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 3 The electrical resistivity of metals changes with temperature Copper T Constant T-5 Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 4 All pure metals… 100 Electrical resistivity of Be Electrical resistivity of Al 10 10 1 1 Electrical resistivity [µ.cm] Electrical resistivity [µ.cm] Electrical resistivity [µ.cm] 10 10-1 10-1 1 10-1 10-2 Electrical resistivity of Ag 10-2 10-3 10-2 10-3 1 10 100 Temperature [K] 1000 1 10 100 1000 1 10 Temperature [K] Properties II: Thermal & Electrical 100 1000 Temperature [K] CAS Vacuum 2017 - S.C. 5 Alloys? Resistivity of Fe Alloys 90.00 80.00 Resistivity [µOhm.cm] 70.00 60.00 AISI 304 L 50.00 AISI 316 L 40.00 Invar 36 30.00 20.00 10.00 0.00 0.00 50.00 100.00 150.00 200.00 250.00 300.00 Temperature [K] Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 6 Some resistivity values (in µ.cm) (pure metals) Variation of a factor ~70 for pure metals at room temperature Even alloys have seldom more than a few 100s of µcm We will not discuss semiconductors (or in general effects not due to electron transport) Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 7 Definition of electrical resistivity R length The electrical resistance of a real object (for example, a cable) section 1 The electrical resistivity is measured in Ohm.m Its inverse is the conductivity measured in S/m mv m e 2 F 2e ne ne Changes with: temperature, impurities, crystal defects Electron relaxation time Electron mean free path Constant for a given material Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 8 Basics (simplified free electron Drude model) Electrical current = movement of conduction electrons - Properties II: Thermal & Electrical + CAS Vacuum 2017 - S.C. 9 Defects Defects in metals result in electron-defect collisions They lead to a reduction in mean free path ℓ, or equivalently in a reduced relaxation time . They are at the origin of electrical resistivity - Properties II: Thermal & Electrical + CAS Vacuum 2017 - S.C. 10 Possible defects: phonons Crystal lattice vibrations: phonons Temperature dependent - Properties II: Thermal & Electrical + CAS Vacuum 2017 - S.C. 11 Possible defects: phonons Crystal lattice vibrations: phonons Temperature dependent - Properties II: Thermal & Electrical + CAS Vacuum 2017 - S.C. 12 Possible defects: impurities Can be inclusions of foreign atoms, lattice defects, dislocations Not dependent on temperature - Properties II: Thermal & Electrical + CAS Vacuum 2017 - S.C. 13 Possible defects: grain boundaries Grain boundaries, internal or external surfaces Not dependent on temperature - Properties II: Thermal & Electrical + CAS Vacuum 2017 - S.C. 14 The two components of electrical resistivity Temperature dependent part It is characteristic of each metal, and can be calculated Proportional to: - Impurity content - Crystal defects - Grain boundaries Does not depend on temperature Total resistivity Properties II: Thermal & Electrical Varies of several orders of magnitude between room temperature and “low” temperature CAS Vacuum 2017 - S.C. 15 Temperature dependence: Bloch-Grüneisen function d ph (T ) T 13 d hvs 2 N 6 2k B V T T d T 5 T d 5 d T 0 x5 dx x x (e 1)(1 e ) Debye temperature: ~ maximum frequency of crystal lattice vibrations (phonons) Given by total number of high-energy phonons proportional ~T Given by total number of phonons at low energy ~T3 and their scattering efficiency T2 Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 16 Low-temperature limits: Matthiessen’s rule total (T ) phonons (T ) impurities grain boundaries ... Or in other terms 1 1 1 total (T ) ... phonons impurities grain boundaries 1 Every contribution is additive. Physically, it means that the different sources of scattering for the electrons are independent Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 17 Effect of added impurities (copper) (Cu)(300K)=1.65 µ.cm Note: alloys behave as having a very large amount of impurities embedded in the material Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 18 An useful quantity: RRR total (300 K ) phonons (300 K ) impurities grain boundaries total (4.2 K ) impurities grain boundaries Fixed number total (300 K ) phonons (300 K ) 0 RRR total (4.2 K ) 0 0 phonons (300 K ) Practical formula ( RRR 1) Experimentally, we have a very neat feature remembering that R(300 K ) total (300 K ) RRR R(4.2 K ) total (4.2 K ) Depends only on “impurities” Dominant in alloys R length section Independent of the geometry of the sample. Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 19 Final example: copper RRR 100 Copper 𝑅𝑅𝑅 = 𝜌 300 𝐾 𝜌 <10 𝐾 = 100 300K = phonon. = 1.55x10-8 m 10K = 1.55x10-10 m If this is due only to oxygen: imp. = 5.3 x10-8 m / at% of O 1.55𝑥10−10 = 0.003 at % of O 5.3𝑥10−8 30 ppm atomic ! This is Cu-OFE Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 20 Estimates of mean free path • • • • me vF m ne 2 ne 2 Typical values? Example of Cu at room temperature Let’s assume one conduction electron per atom. = 1.55 x 10-8 m. density = 89400 kg/m3 m = 9.11 x 10-31 kg, e = 1.6 x 10-19 C, A = 63.5, NA = 6.022 x 1023 Exercise ! Solution: • 2.5 x 10-14 s. Knowing that vF = 1.6 x 106 m/s we have • ℓ 4 x 10-8 m at room temperature. It can be x100 x1000 larger at low temperature Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 21 Interlude: LHC 8.33 T dipoles (nominal field) @ 1.9 K Beam screen operating from 4 K to 20 K SS + Cu colaminated, RRR ≈ 60 Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 22 Magnetoresistance The LHC Electron trajectories are bent due to the magnetic field Cyclotron radius: r mvF eB B x RRR [T] B-field ℓ𝑒𝑓𝑓 ℓ eff eff ℓ⟶ℓ 4 𝜏⟶𝜏 4 r sin r sin r 2 2 r Properties II: Thermal & Electrical ℓ ℓ𝑒𝑓𝑓 ≈ ℓ eff r sin 8 2 eff 4 8 CAS Vacuum 2017 - S.C. 23 Fermi sphere • The real picture: the whole Fermi sphere is displaced from equilibrium under the electric field E, the force F acting on each electron being –eE • This displacement in steady state results in a net momentum per electron 𝛿𝒌 = 𝑭𝜏/ℏ thus a net speed increment 𝛿𝒗 = 𝑭𝜏/𝑚 = − 𝑒𝑬𝜏/𝑚 • 𝒋 = 𝑛𝑒𝛿𝒗 = 𝑛𝑒 2 𝑬𝜏/𝑚 and from the definition of Ohm’s law 𝒋 = 𝜎𝑬 we have 𝜎 = 𝑛𝑒 2 𝜏 𝑚 Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 24 The speed of conduction electrons • Fermi velocity vF = 1.6 x 106 m/s 𝜎𝑬 𝑛𝑒 • 𝛿𝒗 = 𝒋/𝑛𝑒 thus 𝛿𝒗 = As an order-of-magnitude, in a common conductor, we may have a potential drop of ~1V over ~1m 𝑉 𝑑 = 𝑒𝜏𝑬 𝑚 • 𝑬= ≈1 V/m and as a consequence 𝛿𝒗 ≈4 x 10-3 m/s • The drift velocity of the conduction electrons is orders of magnitude smaller than the Fermi velocity (Repeat the same exercise with 1 A of current, in a copper conductor of 1 cm2 cross section) Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 25 Square resistance and surface resistance Consider a square sheet of metal and calculate its resistance to a transverse current flow: a current a a R da d d This is the so-called square resistance often indicated as 𝑅∎ Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 27 Square resistance and surface resistance And now imagine that instead of DC we have RF, and the RF current is 2 confined in a skin depth: 0 a current a R da d a d Rs 0 2 This is a (simplified) definition of surface resistance Rs (We will discuss this in more details at the tutorials) Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 28 Surface impedance in normal metals • The Surface Impedance 𝑍𝑠 is a complex number defined at the interface between two media. • The real part 𝑅𝑠 contains all information about power losses (per unit surface) 1 Rs I 2 1 P 2 2 Rs H 02 d 2 • The imaginary part 𝑋𝑠 contains all information about the field penetration in the material 2 0 Xs • For copper ( = 1.75x10-8 µ.cm) at 350 MHz: • 𝑅𝑠 = 𝑋𝑠 = 5 m and = 3.5 µm Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 29 Why the surface resistance (impedance)? • It is used for all interactions between E.M. fields and materials Q0 Rs • In RF cavities: quality factor • In beam dynamics (more at the tutorials): - Longitudinal impedance and power dissipation from wakes is Ploss MI b2 Re Z seff where Z seff is a summation of 2R 2b Z s over the bunch frequency spectrum - Transverse impedance: ZT Properties II: Thermal & Electrical 2R c Zs 3 b CAS Vacuum 2017 - S.C. 30 From RF to infrared: the blackbody Thermal exchanges by radiation are mediated by EM waves in the infrared regime. Peak ≈ 3000 µm x K Schematization of a blackbody Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 31 Blackbody radiation • A blackbody is an idealized perfectly emitting and absorbing body (a cavity with a tiny hole) • Stefan-Boltzmann law of radiated power density: P T4 A σ ≈ 5.67 × 10−8 W/(m2K4) • At thermal equilibrium: • is the emissivity (blackbody=1) • • A “grey” body will obey: 1 r (t ) Thus for a grey body: P T 4 A Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 32 From RF to infrared in metals • Thermal exchanges by radiation are mediated by EM waves in the infrared regime. • At 300 K, peak ≈ 10 µm of wavelength -> ≈ 1013 Hz or RF ≈ 10-13 s • The theory of normal skin effect is usually applied for: RF 1 • But it can be applied also for: RF 1 • In the latter case it means: RF • For metals at moderate T we can then use the standard skin effect theory to calculate emissivity Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 33 Emissivity of metals 1 1 r • From: • Thus we can calculate emissivity from reflectivity: 1 r 4 Rs Rvacuum Rs 0 Rvacuum 2 0 376.7 0 • The emissivity of metals is small • The emissivity of metals depends on resistivity • Thus, the emissivity of metals depends on temperature and on frequency Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 34 Practical case: 316 LN Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 35 Thermal conductivity of metals Copper peak T-1 constant Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 36 Thermal conductivity: insulators Determined by phonons (lattice vibrations). Phonons behave like a “gas” peak T-3 constant Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 37 Thermal conductivity: insulators Thermal conductivity K ph from heat capacity C ph (as in thermodynamics of gases) 1 1 K ph C ph vs C ph vs2 3 3 C ph 3 Nk BT 3RT C ph T d T 12 4 Nk B 5 d 1 T const. 3 T d K ph const T K ph d T d 3 Properties II: Thermal & Electrical T d T d T d for ultra-pure crystals 1 K peak C ph vs 3 = max dimension of specimen CAS Vacuum 2017 - S.C. 38 Thermal conductivity: metals Determined by both electrons and phonons. Thermal conductivity K el from heat capacity Cel 1 2 nk B2T 2 nk B2T K el Cel vF vF 3 3mvF2 3m 1 T const. K el K ph T d T d impurities T d Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 39 Thermal conductivity of metals: total Copper Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 40 Wiedemann-Franz Proportionality between thermal conductivity and electrical conductivity K el 2 kB 2 T LT 3 e T d L = 2.45x10-8 WK-2 (Lorentz number) Useful for simple estimations, if one or the other quantity are known Useful also (very very approximately) to estimate contact resistances Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 41 The LHC collimator Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 42 Contact resistance (both electrical and thermal) • Complicated… and no time left n depends on: Plastic deformation Elastic deformation Contact area: A ~ P • n n O(1) Roughness “height” and “shape” Contacts depend also on oxidation, material(s) properties, temperature… Example for electric contacts: • • Theoretically: - RP-1/3 in elastic regime - RP-1/2 in plastic regime Experimentally: - RP-1-1/2 (same as for thermal contacts) Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 43 References • Charles Kittel, “Introduction to solid state physics” • Ashcroft & Mermin, “Solid State Physics” • S. W. Van Sciver, “Helium Cryogenics” • M. Hein, “HTS thin films at µ-wave frequencies” • J.A. Stratton, “Electromagnetic Theory” • Touloukian & DeWitt, “Thermophysical Properties of Matter” Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 44 The end. Questions? 45 Plane waves in vacuum Plane wave solution of Maxwell’s equations in vacuum: E E0 e i ( kz t ) H H0 e Where (in vacuum): So that: k 2 E E0 e i ( kz t ) The ratio Z E H i ( kz t ) c H E0 0 0 H E0 k 0 e i ( kz t ) 0 0 ε0 i ( kz t ) e μ0 0 376.7 is often called impedance of the free 0 space and the above equations are valid in a continuous medium Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 46 Plane waves in normal metals More generally, in metals: Z k i 2 2 E H k This results from taking the full Maxwell’s equations, plus a supplementary equation which relates locally current density and field: J ( x , t ) E( x , t ) ne 2 ne 2 0 me vF me 2 2 In metals k i and the wave equations become: k i i E E0 ei ( kz t ) E0 ei (z t ) e z With 1 is the damping coefficient of the wave inside a 2 metal, and is also called the field penetration depth. Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 47 V~ z x y S=d2 E z (0) Surface impedance Zs H z (0) V I V dEz (0) x;; I d J z y dy J z ( y) Z s Rs iX s E z (0) J y dy 0 1 J z y dy J z (0) 0 E z ( 0) E z ( 0) J z (0) H x (0) z 0 1 1 2 Ptot (t ) Rs I Rs d 2 H x2 2 2 1 1 Brf 2 P / S Prf Rs H rf Rs 2 2 o Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 2 48 Normal metals in the local limit J z y J z (0)e y i t J t J ( 0 ) e 2 o n E (0) 1 Z n Rn iX n z (1 i) J z (0) n Rn X n 1 n n o o n 2 n 2 Properties II: Thermal & Electrical o (1 i) 2 n ( Rn ) CAS Vacuum 2017 - S.C. 49 Limits for conductivity and skin effect 2 0 ~ 1. Normal skin effect if: e.g.: high temperature, low frequency 2. Anomalous skin effect if: e.g.: low temperature, high frequency Note: 1 & 2 valid under the implicit assumption 1 1 1 & 2 can also be rewritten (in advanced theory) as: It derives that 1 can be true for 1 2 2 34 1 and also for 1 Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 50 Mean free path and skin depth Mean free path ~ 1 neff n0 Skin depth 2 0 ne 2 ne 2 0 eff me vF me vF Properties II: Thermal & Electrical 2 0 const. CAS Vacuum 2017 - S.C. 51 Anomalous skin effect Anomalous skin effect Asymptotic value Normal skin effect Understood by Pippard, Proc. Roy. Soc. A191 (1947) 370 Exact calculations Reuter, Sondheimer, Proc. Roy. Soc. A195 (1948) 336 Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 52 Debye temperatures Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 53 Heat capacity of solids: Dulong-Petit law Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 54 Low-temperature heat capacity of phonon gas (simplified plot in 2D) Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 55 Phonon spectrum and Debye temperature Density of states D : How many elemental oscillators of frequency Assuming constant speed of sound Properties II: Thermal & Electrical CAS Vacuum 2017 - S.C. 56