Materials & Properties II: Thermal & Electrical Characteristics. Sergio Calatroni - CERN

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2 Materials & Properties II: Thermal & Electrical Characteristics Sergio Calatroni - CERN

3 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 o Temperature dependence, electron & phonons Limiting factors Properties II: Thermal & Electrical CAS Vacuum S.C. 3

4 The electrical resistivity of metals changes with temperature Copper T Constant T -5 Properties II: Thermal & Electrical CAS Vacuum S.C. 4

5 Electrical resistivity [µ.cm] Electrical resistivity [µ.cm] Electrical resistivity [µ.cm] All pure metals Electrical resistivity of Be 10 Electrical resistivity of Al Electrical resistivity of Ag Temperature [K] Temperature [K] Temperature [K] Properties II: Thermal & Electrical CAS Vacuum S.C. 5

6 Resistivity [µohm.cm] Alloys? Resistivity of Fe Alloys AISI 304 L AISI 316 L Invar Temperature [K] Properties II: Thermal & Electrical CAS Vacuum S.C. 6

7 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 S.C. 7

8 Definition of electrical resistivity R length section The electrical resistance of a real object (for example, a cable) 1 The electrical resistivity is measured in Ohm.m Its inverse is the conductivity measured in S/m m v ne m ne e F e Changes with: temperature, impurities, crystal defects Electron relaxation time Electron mean free path Constant for a given material Properties II: Thermal & Electrical CAS Vacuum S.C. 8

9 Basics (simplified free electron Drude model) Electrical current = movement of conduction electrons - + Properties II: Thermal & Electrical CAS Vacuum S.C. 9

10 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 S.C. 10

11 Possible defects: phonons Crystal lattice vibrations: phonons Temperature dependent - + Properties II: Thermal & Electrical CAS Vacuum S.C. 11

12 Possible defects: phonons Crystal lattice vibrations: phonons Temperature dependent - + Properties II: Thermal & Electrical CAS Vacuum S.C. 1

13 Possible defects: impurities Can be inclusions of foreign atoms, lattice defects, dislocations Not dependent on temperature - + Properties II: Thermal & Electrical CAS Vacuum S.C. 13

14 Possible defects: grain boundaries Grain boundaries, internal or external surfaces Not dependent on temperature - + Properties II: Thermal & Electrical CAS Vacuum S.C. 14

15 The two components of electrical resistivity Proportional to: - Impurity content - Crystal defects - Grain boundaries Does not depend on temperature Total resistivity Temperature dependent part It is characteristic of each metal, and can be calculated Varies of several orders of magnitude between room temperature and low temperature Properties II: Thermal & Electrical CAS Vacuum S.C. 15

16 Temperature dependence: Bloch-Grüneisen function ph ( T ) T d 5 d T 0 ( e x 5 x 1)(1 e x ) dx d hvs k B 6 N V 1 3 Debye temperature: ~ maximum frequency of crystal lattice vibrations (phonons) T T d Given by total number of high-energy phonons proportional ~T T 5 T d Given by total number of phonons at low energy ~T 3 and their scattering efficiency T Properties II: Thermal & Electrical CAS Vacuum S.C. 16

17 Low-temperature limits: Matthiessen s rule total ( T ) ( T )... phonons impurities grain boundaries Or in other terms total ( T ) 1 phonons 1 impurities 1 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 S.C. 17

18 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 S.C. 18

19 An useful quantity: RRR total total (300K) (4.K) phonons impurities (300K) grain boundaries impurities grain boundaries Fixed number RRR total total (300K) (4.K) phonons (300K) 0 0 Depends only on impurities Dominant in alloys 0 phonons (300K) ( RRR 1) Practical formula Experimentally, we have a very neat feature remembering that R(300K) RRR R(4.K) total total (300K) (4.K) length R section Independent of the geometry of the sample. Properties II: Thermal & Electrical CAS Vacuum S.C. 19

20 Final example: copper RRR 100 Copper RRR = ρ 300 K ρ <10 K = K = 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 1.55x x10 8 = at % of O 30 ppm atomic! This is Cu-OFE Properties II: Thermal & Electrical CAS Vacuum S.C. 0

21 Estimates of mean free path m ne mevf ne Typical values? Example of Cu at room temperature Let s assume one conduction electron per atom. = 1.55 x 10-8 m. density = kg/m 3 m = 9.11 x kg, e = 1.6 x C, A = 63.5, N A = 6.0 x 10 3 Exercise! Solution:.5 x s. Knowing that v F = 1.6 x 10 6 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 S.C. 1

22 Interlude: LHC 8.33 T dipoles (nominal 1.9 K Beam screen operating from 4 K to 0 K SS + Cu colaminated, RRR 60 Properties II: Thermal & Electrical CAS Vacuum S.C.

23 Magnetoresistance Electron trajectories are bent due to the magnetic field The LHC Cyclotron radius: r mv eb F B x RRR [T] B-field l l l lτ4 eff l l eff l τ ττ4 eff rsin r eff r rsin eff eff rsin Properties II: Thermal & Electrical CAS Vacuum S.C. 3

24 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 Eτ/m and from the definition of Ohm s law j = σe we have σ = ne τ m Properties II: Thermal & Electrical CAS Vacuum S.C. 4

25 The speed of conduction electrons Fermi velocity v F = 1.6 x 10 6 m/s δv = j/ne thus δv = σe = eτe ne m As an order-of-magnitude, in a common conductor, we may have a potential drop of ~1V over ~1m E = V d 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 cm cross section) Properties II: Thermal & Electrical CAS Vacuum S.C. 5

26 Square resistance and surface resistance Consider a square sheet of metal and calculate its resistance to a transverse current flow: current a a R a d a d d This is the so-called square resistance often indicated as R Properties II: Thermal & Electrical CAS Vacuum S.C. 7

27 Square resistance and surface resistance And now imagine that instead of DC we have RF, and the RF current is confined in a skin depth: 0 current a a R a d a d d R s 0 This is a (simplified) definition of surface resistance R s (We will discuss this in more details at the tutorials) Properties II: Thermal & Electrical CAS Vacuum S.C. 8

28 Surface impedance in normal metals The Surface Impedance Z s is a complex number defined at the interface between two media. The real part R s contains all information about power losses (per unit surface) 1 RsI 1 P R H s 0 d The imaginary part X s contains all information about the field penetration in the material 0 X s For copper ( = 1.75x10-8 µ.cm) at 350 MHz: R s = X s = 5 m and = 3.5 µm Properties II: Thermal & Electrical CAS Vacuum S.C. 9

29 Why the surface resistance (impedance)? It is used for all interactions between E.M. fields and materials In RF cavities: quality factor Q 0 R s In beam dynamics (more at the tutorials): - Longitudinal impedance and power dissipation from wakes is P loss eff MIb Re Z s Z s s the bunch frequency spectrum eff where is a summation of R b Z over - Transverse impedance: Z R c T Z 3 s b Properties II: Thermal & Electrical CAS Vacuum S.C. 30

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 S.C. 31

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 A 4 T σ W/(m K 4 ) At thermal equilibrium: is the emissivity (blackbody=1) A grey body will obey: 1 r ( t) Thus for a grey body: P 4 T A Properties II: Thermal & Electrical CAS Vacuum S.C. 3

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 -> Hz or RF 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 S.C. 33

33 Emissivity of metals From: 1 1 r Thus we can calculate emissivity from reflectivity: 1 r 4 R R s vacuum R R s vacuum 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 S.C. 34

34 Practical case: 316 LN Properties II: Thermal & Electrical CAS Vacuum S.C. 35

35 Thermal conductivity of metals peak Copper T -1 constant Properties II: Thermal & Electrical CAS Vacuum S.C. 36

36 Thermal conductivity: insulators Determined by phonons (lattice vibrations). Phonons behave like a gas peak T -3 constant Properties II: Thermal & Electrical CAS Vacuum S.C. 37

37 Thermal conductivity: insulators Thermal conductivity K from heat capacity C (as in thermodynamics of gases) ph ph K ph 1 3 C ph v s 1 3 C ph v s C C ph ph 3Nk B T 3RT Nk B T d 3 T T d d 1 T const. T T d d K K ph ph const T d T T d 3 K C v d for ultra-pure crystals peak 1 3 ph = max dimension of specimen s Properties II: Thermal & Electrical CAS Vacuum S.C. 38

38 Thermal conductivity: metals Determined by both electrons and phonons. Thermal conductivity Kel from heat capacity Cel K el 1 3 C el v F nkbt 3mv F v F nkbt 3m K el K ph 1 T const. T T d d impurities T d Properties II: Thermal & Electrical CAS Vacuum S.C. 39

39 Thermal conductivity of metals: total Copper Properties II: Thermal & Electrical CAS Vacuum S.C. 40

40 Wiedemann-Franz Proportionality between thermal conductivity and electrical conductivity Kel 3 k e B T L T T d L =.45x10-8 WK - (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 S.C. 41

41 The LHC collimator Properties II: Thermal & Electrical CAS Vacuum S.C. 4

42 Contact resistance (both electrical and thermal) Complicated and no time left n Contact area: A ~ P n O(1) n depends on: Plastic deformation Elastic deformation 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/ in plastic regime Experimentally: - RP -1-1/ (same as for thermal contacts) Properties II: Thermal & Electrical CAS Vacuum S.C. 43

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 S.C. 44

44 The end. Questions? 45

45 Plane waves in vacuum Plane wave solution of Maxwell s equations in vacuum: E E i( kzt ) i( kzt ) 0 e H H0 e H E k i( kzt ) 0 e 0 Where (in vacuum): k c So that: E E i( kzt ) 0 i( kzt ) 0 e H E0 e μ0 ε The ratio Z E H 0 is often called impedance of the free space and the above equations are valid in a continuous medium Properties II: Thermal & Electrical CAS Vacuum S.C. 46

46 Plane waves in normal metals More generally, in metals: k i Z H E k This results from taking the full Maxwell s equations, plus a supplementary equation which relates locally current density and field: In metals J( x, t) E( x, t) k and the wave equations become: 1 E E i ne m v 0 i( kzt ) i( zt ) z 0 e E0 e e With is the damping coefficient of the wave inside a metal, and is also called the field penetration depth. e k F ne m e i i Properties II: Thermal & Electrical CAS Vacuum S.C. 47

47 V~ Surface impedance x y z S=d (y) J z (0) E z (0) H z Z s V V I de J z 0 y ( 0) x; I d J dy z ; 1 (0) 0 J z y dy z Z s R s ix s 0 E J z z (0) y dy E J z z (0) (0) E H z x (0) (0) P 1 1 tot ( t) RsI Rsd H x P / S P rf 1 R H s rf 1 R s B rf o Properties II: Thermal & Electrical CAS Vacuum S.C. 48

48 Normal metals in the local limit J z y J (0) e z y o n J t J (0) e it Z n R n ix n E J z z (0) (0) 1 n (1 i) o (1 i) n R n X n 1 n o o n n n ( R n ) Properties II: Thermal & Electrical CAS Vacuum S.C. 49

49 Limits for conductivity and skin effect 0 ~ 1 1. Normal skin effect if: e.g.: high temperature, low frequency. Anomalous skin effect if: e.g.: low temperature, high frequency Note: 1 & valid under the implicit assumption 1 1 & can also be rewritten (in advanced theory) as: It derives that 1 can be true for 1 and also for 1 Properties II: Thermal & Electrical CAS Vacuum S.C. 50

50 Mean free path and skin depth Mean free path ~ 1 n eff n 0 Skin depth 0 ne 0 eff mevf ne m v e F 0 const. Properties II: Thermal & Electrical CAS Vacuum S.C. 51

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 S.C. 5

52 Debye temperatures Properties II: Thermal & Electrical CAS Vacuum S.C. 53

53 Heat capacity of solids: Dulong-Petit law Properties II: Thermal & Electrical CAS Vacuum S.C. 54

54 Low-temperature heat capacity of phonon gas (simplified plot in D) Properties II: Thermal & Electrical CAS Vacuum S.C. 55

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 S.C. 56