# Presentation: 127680

## Experimental Methods in Condensed Matter Physics 

***Overview and Introduction***
**Paul C. Canfield**
**Professor,  Department of Physics **
**Senior Physicist, Ames Laboratory**
**Iowa State University**
**August 25 and 27 2008**

## and how to use them for the identification of basic phase transitions and ground states such as:

- Antiferromagnetic
- Ferromagentic
- Superconductor
- Metal-to-Insulator
- Charge Density Wave
- Meta-magnetic

- We will use contemporary compounds and review published data.

- 	In this lecture we will review basic types measurements:
- Resistivity, 
- Specific heat, Cp
- Magnetization, M, and magnetic susceptibility, .

- To some extent this lecture sets the agenda for the semester.  We will review each of these measurements, as well as the thermodynamic variables associated with them, in detail.

## V

- V

- I

- I

- R = V/I
- R = l/A
- Care in measuring V, I, l and A is, of course, needed

- l

- We can look at this in two simple ways:
-  = ne2/m	 = 1/
- In this case we can make inferences about n and / or 
- Another, slightly more sophisticated formula is the Kubo formula:
-  = (1/43)(e2/3ħ)∫dSF   	(which is basically an integral of the electronic				    mean free path, , over the Fermi surface)
- NOTE:  the measured electrical resistivity will contain an impurity scattering term, 0 which appears additively.  This is often associated with chemical impurities as well as a variety of structural defects.

- Resistivity / Resistance

## For a simple metal the resistivity can often be modeled by:

-  = f(T) + 0 and
- Where f(T) ~ AT2 at lowest T
- and f(T) ~ BT at intermediate T
- 0 is the manifestation of the finite nature of .
- The residual resistivity ratio: RRR, is a measure of how defect free (or defected) a metallic sample is.
- RRR = (high T) / (low T)
- Often RRR = (300 K)/(2 K)
- NOTE:  RRR is also R(HT)/R(LT) since factors of l/A cancel out.
- High RRR means low 0.

- Resistivity / Resistance

## 			If the defect density of a compound can be controlled then 0 (and therefore ) can be systematically changed, leading to a constant off-set in resistivity curves.  This can be done by judicious annealing of the sample (in some cases).

- 			If the defect density of a compound can be controlled then 0 (and therefore ) can be systematically changed, leading to a constant off-set in resistivity curves.  This can be done by judicious annealing of the sample (in some cases).

- Superconductivity is easy to spot in (T)  data since it manifests as a sudden drop in  to zero.

- Resistivity / Resistance

## Resistivity measurements can be used to learn about the superconducting state, for example in MgB2.

- Resistivity measurements can be used to learn about the superconducting state, for example in MgB2.
- (T,H) data can be used to delineate the superconducting / normal phase boundary:  Hc2(T).
- It is important to clearly state the criterion or criteria used to determine a transition point.  In this case we illustrated the effect of using an onset, mid-point, and off-set criterion.

- Resistivity / Resistance

## 0 decreases with increased order.  Structurally this means less defects.  The conduction electrons can also couple to localized magnetic moments, such as on rare earths.  

- 0 decreases with increased order.  Structurally this means less defects.  The conduction electrons can also couple to localized magnetic moments, such as on rare earths.

- Effects of local moments, ordered as well as disordered.

- When the moments change from disordered (paramagnetic) to ordered (ferromagnetic, antiferromagnetic, or more complex order) there is a decrease in scattering.  This is called a loss of spin-disorder scattering.

- Resistivity / Resistance

## Loss of spin disorder scattering can be seen in high RRR samples as well as in low RRR samples, but it does get harder and harder to see as 0 increases (RRR approaches unity).

- Loss of spin disorder scattering can be seen in high RRR samples as well as in low RRR samples, but it does get harder and harder to see as 0 increases (RRR approaches unity).

- Resistivity / Resistance

## Multiple transitions can be easily detected and identified:  In this case we have a rare example of TN ~ 10 K with the loss of spin disorder scattering, followed at lower temperatures by Tc ~ 6K and a total loss of resistivity.

- Multiple transitions can be easily detected and identified:  In this case we have a rare example of TN ~ 10 K with the loss of spin disorder scattering, followed at lower temperatures by Tc ~ 6K and a total loss of resistivity.

- Resistivity / Resistance

## 	Resistivity is also very sensitive to changes in the Fermi Surface:

- 	Resistivity is also very sensitive to changes in the Fermi Surface:
- This can be seen in CDW (and SDW) transitions, where nested parts of the Fermi surface become gapped below TCDW.  This leads to an decrease in  due to a decrease SF.  For a partial gapping of the F.S. the sample remains metallic and ultimately returns to (T) with positive slope.

- Resistivity / Resistance

- Kubo formula:    = (1/43)(e2/3ħ)∫dSF

## 	If the Fermi surface is fully gapped, SF goes to zero, then the resistivity becomes infinite. 

- 	If the Fermi surface is fully gapped, SF goes to zero, then the resistivity becomes infinite.
- Another way of thinking of this is that the number of carriers, n, goes to zero:			= ne2/m	 = 1/
- This is a metal-to-insulator transition and is seen in some CDW / SDW systems as well as several 3-d transition metal oxide systems.

- Resistivity / Resistance

- Kubo formula:    = (1/43)(e2/3ħ)∫dSF

## In a non-magnetic sample, specific heat can provide information about the electronic and vibrational excitations modulo kBT. 

- In a non-magnetic sample, specific heat can provide information about the electronic and vibrational excitations modulo kBT.
- At low temperatures the specific heat of a metal can be modeled as
- Cp = T + T3
- A plot of C/T versus T2 allows for the clean separation of  and .
- In simple models  is proportional to N(EF) or, alternately, the electron mass, and  is proportional to the Debye temperature and the characteristic frequency of lattice vibrations.

- Specific heat

## Specific heat can also be used to locate and characterize phase transitions.  As seen earlier, we can suppress Tc with an applied field, so the Cp feature can be more clearly seen via comparison to the non-superconducting (in high applied magnetic field) compound.

- Local moment ordering can be seen even more clearly (larger entropy).  Shown here are a series of transitions in antiferromagnetic DyAgGe.

- Specific heat

## Specific heat is intimately related to entropy

- ∫(Cp/T)dT = S
- The means that we can determine how much entropy (change) is associated with a given state.  For magnetic systems we need to use the magnetic Cp.  We often do this by subtracting off the Cp(T) data from a non-magnetic analogue (e.g. LuAgGe from TmAgGe).

- For these local moment systems S ~ R ln D

- Specific heat

## Given the Hund’s rule, groundstate multiplet J we would expect   R*ln(2J + 1) entropy associated with the magnetic state.  This is what we find for Gd:  (S = 7/2, L = 0 and J = 7/2) S ~ R ln(8).  For other rare earths the spin-orbit coupling give rise to crystalline electric field (CEF) splitting.  

- Given the Hund’s rule, groundstate multiplet J we would expect   R*ln(2J + 1) entropy associated with the magnetic state.  This is what we find for Gd:  (S = 7/2, L = 0 and J = 7/2) S ~ R ln(8).  For other rare earths the spin-orbit coupling give rise to crystalline electric field (CEF) splitting.

****** ****= 18 K**

- e.g.:  Ce (J =5/2) in cubic point symmetry

- 

- This can be seen in the Cp as a Schottky anomaly.  This is clearly shown at the right in PrAgSb2.  This is modeled as a two level system.

- Specific heat

##  = (2/3)*kB2T *N(EF) = kB2T *(mkF/3ħ2)

-  = (2/3)*kB2T *N(EF) = kB2T *(mkF/3ħ2)

- Specific heat

- The electronic specific heat can be used as a measure of the electron effective mass, based on the free electron result of

- Heavy Fermions are compounds with exceptionally high values of  for T < TK, the Kondo temperature.

- Strictly speaking, they are defined as compounds with     > 400 mJ/mol-K2, about 400 times the value of  for Cu.  Large  means large electron mass, ergo, heavy fermion.

## Magnetic susceptibility

- For BH / kBT << 1 the magnetization of a paramagnetic, local moment system is linear in H.  For such low H,  = M(T) / H.
- 	Note:  gross energy scales:  (1T)*(1B) ~ (kB)*(1 K)

- Data taken on paramagnetic TmNi2B2C (for T > 10 K)

## Curie Weiss Law:  The temperature dependence of the magnetic susceptibility in the paramagnetic state is given by:

- = [C/(T - )] + 0
- C = Npeff2B2/3kB

- Magnetic susceptibility

*Note:  Gd is isotropic since L = 0 means no CEF splitting.*

- 0 = roughly temperature independent term that is the sum of core diamagnetism, Pauli paramagnetism and Landau diamagnetism

## Magnetic susceptibility

- Ce

- Pr

- Lu

- Sm

- Pm

- Nd

- Er

- Ho

- Dy

- Tb

- Gd

- Eu

- Yb

- Tm

- The rare earth series offers a 14 position knob for tuning magnetism.  By controlling S, L, J as well as point symmetry (CEF) we can get a wide variety of magnetic properties: eff, sat and anisotropy.

- Ashcroft / Mermin Solid State Physics

## Magnetic susceptibility

- Kittel, Introduction to Solid State physics 4th edition

## Magnetic susceptibility

- For R with finite L (i.e not Gd3+ or Eu2+) there will be CEF splitting that can give rise to significant anisotropy.  The degree of anisotropy will depend on the R as well as the point symmetry it is in.

- This anisotropy can be clearly seen in -1 as well as  plots.

## Magnetic susceptibility

- In addition allowing for evaluation of peff and  the magnetic susceptibility can be used to determine the antiferromagnetic ordering temperature, TN.  Although it is tempting to simply take the temperature of the maximum  value, this only gives a rough estimate.  Near TN, d(T)/dT has the same temperature dependence as Cp, so the temperature of the maximum in d(T)/dT is considered a more reliable criterion for determination of TN.

## d/dT can also show similar temperature dependencies near TN, but only for a limited number of transitions.  In general though, d/dT can be used as a criterion for TN or TC, but this will depend on the noise level of the data.

- d/dT can also show similar temperature dependencies near TN, but only for a limited number of transitions.  In general though, d/dT can be used as a criterion for TN or TC, but this will depend on the noise level of the data.

- A detour back to resistivity for a moment

- M. E. Fisher, J. S. Langer, Philos. Mag., 7 (1962) 1731.

## The transition to the ferromagnetic state is conspicuous, but the actual TC value is hard to extract from the M(T) data.  (It can actually be much clearer in the Cp(T) or (T) data.)

- The transition to the ferromagnetic state is conspicuous, but the actual TC value is hard to extract from the M(T) data.  (It can actually be much clearer in the Cp(T) or (T) data.)
- One way of estimating TC from magnetization data is to plot M(H) isotherms as an “Arrot Plot”:  M2 vs. H/M.  The data linearly go through the origin when T ~ TC.
- Grossly this is similar to seeing when a spontaneous magnetization appears in low field.

- Magnetic susceptibility

***Ferromagnetic transitions***

## Magnetic susceptibility

- Using susceptibility, detailed information about R-series can be determined.  These information, combined with resistivity and specific heat data can often provide enough insight for choosing samples for specific purposes (Ising spin-glass, metamagnetism, etc.).

## We have now seen how (T), Cp(T), and (T) can be used to determine magnetic ordering temperatures (as well as many other useful parameters).

- We have now seen how (T), Cp(T), and (T) can be used to determine magnetic ordering temperatures (as well as many other useful parameters).

- GdCo2Zn20

- What about multiple transitions?

## HoNi2B2C has a cascade of magnetic transitions (AF) between 6 and 5 K.  They are clear in both Cp and  data.

- HoNi2B2C has a cascade of magnetic transitions (AF) between 6 and 5 K.  They are clear in both Cp and  data.

- P. C. Canfield et al. / Physica C  (1994),  230(3-4),  397-406  AND

- Multiple transition temperatures

- HoNi2B2C

- HoNi2B2C

- HoNi2B2C

## Multiple transitions in temperature and field

- TbNi2Ge2 has two clear transitions in zero (low) field

## Multiple transitions in temperature and field

- When a magnetic field is applied along the c-axis for T<TN there are a cascade of meta-magnetic (field stabilized) phase transitions.

## Multiple transitions in temperature and field

- M(H) and M(T) data can be used to construct a H-T phase diagram that delineates regions of differing magnetic order.

- M(H)

- M(T)

## CeSb

- CeSb can be grown in exquisite purity from Sn flux.  At first glance  and  data seem to be a bit noisy at low T....

- Multiple transitions in temperature and field

- Actually CeSb has a formidable number of phase transition in zero and applied field and excellent (Sn flux grown) crystals allow us to examine them in detail.

## CeSb

- Both  and  data reveal a multitude of transitions.

## CeSb

- These multiple transition appear in finite field as well.

- Multiple transitions in temperature and field

## CeSb

- Multiple transitions in temperature and field

- M(H), M(T), (H) and (T) data can be used to assemble an H-T phase diagram of fantastic detail.  This system was studied extensively in the 70’s and 80’s by several neutron scattering groups as well as serving as the inspiration for the ANNNI model.  The precise origin of this complexity is still an open question.

## Solids

- Liquids

***but instead,***

## mapping out the electronic / magnetic phases.

- Phase diagrams can involve more than applied field and temperature (or composition and temperature).

- SC

- normal

- PM

- FM

- AFM

## Angular dependent effects

- Recall that we found TbNi2Ge2 to be quite anisotropic, but the magnetic ordering helps to obscure just how anisotropic the local moments really are.

- By diluting Tb into the non-magnetic LuNi2Ge2 we can see just how anisotropic the low T magnetization is in the paramagnetic state.  The local Tb moments are extremely axial and their longitudinal magnetization is simply their projection along the field direction:  M() = M0cos().  This makes TbNi2Ge2 a model, Ising system, i.e. the moment is either spin up or spin down along the Ising axis.

- T = 2 K

## Angular dependent effects

- Hc ~ 1/cos() *(There is a critical field along the c-axis that must be exceeded.)*
- Msat ~ cos() *(The longitudinal moment is just the projection of easy axis moment along field direction.)*

## Angular dependent effects

- HoNi2B2C manifests a more interesting anisotropy.  It is tetragonal and has Ho in a tetragonal point symmetry.  At low temperatures the local moment is confined to the basal plane.

- Until recently there was a prejudice that if the local moment was not Ising in nature, then it would be x-y, meaning that there would be no anisotropy in the easy plane....We found a very different situation:  a 4-state clock system:  the moments only point N, S, E, W; or up, down, right, left; or along the four (110) directions.

## Angular dependent effects

- The T < TN M(H) data show clear, _*in-plane*_ anisotropies with three metamagnetic transitions that vary with .

- The highest field transition appears to diverge as H approaches the (111) direction

## Angular dependent effects

- All three critical fields show compellingly simple angular dependencies.

## Angular dependent effects

## Angular dependent effects:  H –  phase diagrams

- Angular dependent effects:  H –  phase diagrams

- These angular dependencies can be easily captured by assuming that each state consists of a net distribution of highly anisotropic moments along the four, in-plane easy axes.

## Summary

- (T,H,),
- Cp(T,H,),
- M(T,H,).
*These techniques combined with the ability to design, discovery, and grow novel compounds opens the way to the discovery and description of novel physics.*

## What else can we do?  (part 1)

- 	Magneto-transport
- (H) and Kohler’s rule
- 		Hall efffect
- 	Quantum oscillations
- 		dHvA measurements in M(H)
- 		SdH measurements in (H)
*Other measurements that are basic and useful:*
- 	Thermoelectric power
- 	Thermal Conductivity

- What else can we do?  (part 1)

## What else can we do?  (part 2)

- Low T:  down to 1.5 K, down to 0.3 K , down to 0.03 K.  Each gets harder and more expensive.
- High d.c. magnetic field:  5T, 10 T, 15 T, 20 T.  Each step doubles cost (or more)
- Pulsed magnetic fields:  30 T, 60 T, 90 T
*Extremes help us learn more by controllably perturbing the sample:*
- Pressure:  use of hydrostatic pressure to tune compounds is clean and useful.  10 – 30 kbar is not too hard, 30 – 100 kbar is difficult, and greater than 100 kbar is very difficult.  Resistivity under pressure is the easiest measurement.  Low pressure (~ 10 kbar) magnetic susceptibility,  is not too hard either.

- What else can we do?  (part 2)

- Some of these measurements can be done in your own lab, some with friends, some at specialized facilities.

## That’s All Folks

*That’s All Folks*