prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: Where is McPherson County located?

Read the following text and answer briefly.

Context 1: McPherson County (standard abbreviation: MP) is a county located in the U S  state of Kansas   As of the 2020 census, the county population was 30,223  The largest city and county seat is McPherson  The county is named for Civil War General James B  McPherson 

History

Early history

For many millennia, the Great Plains of North America was inhabited by nomadic Native Americans  From the 16th century to 18th century, the Kingdom of France claimed ownership of large parts of North America  In 1762, after the French and Indian War, France secretly ceded New France to Spain, per the Treaty of Fontainebleau  In 1802, Spain returned most of the land to France, but keeping title to about 7,500 square miles 

In 1803, most of the land for modern day Kansas was acquired by the United States from France as part of the 828,000 square mile Louisiana Purchase for 2 83 cents per acre  In 1848, after the Mexican–American War, the Treaty of Guadalupe Hidalgo with Spain brought into the United States all or part of land for ten future states, including southwest Kansas  In 1854, the Kansas Territory was organized, then in 1861 Kansas became the 34th U S  state 

19th century

From the 1820s to 1870s, the Santa Fe Trail passed through, what is now McPherson County  The trail entered the county, east of Canton, then south of Galva, then north of Inman, and west towards Lyons  In 1855, Charles O  Fuller established a ranch adjacent to the Running Turkey Creek Crossing about two miles south and one mile east of Galva  Fuller's Ranch provided accommodations for travelers on the Santa Fe Trail and was probably the first white settlement in McPherson County 

Peketon County was established in 1860, by the passage of a bill by S  N  Wood:  An act to establish Peketon County  Section 1  - That all that territory west of the sixth principal meridian and south of Township 16, in Kansas Territory, be and the same is hereby erected into a county, to be known by the name of Peketon County  On February 17, 1865, Peketon County was abolished, and McPherson County was made a part of Marion County, which extended from the west line of Chase County to the present western boundary of Kansas 

In 1868, Solomon Stephens and L  N  Holmberg were appointed Justices of the Peace—the first officers in what is now McPherson County  The next year (1869) occurred the first election for the township, now the county of McPherson  McPherson was regularly organized as a county in the spring of 1870, a mass meeting being held at Sweadal  Sweadal, the county seat thus selected, was located about one mile and a half southwest of the present site of Lindsborg  In September, however, the County Commissioners resolved to meet at the latter place, McPherson which had already been located some two years 

In April, 1873, a petition was filed for the county seat re-location  It was signed by 483 voters, and a special election was accordingly ordered for June 10  Upon that day, McPherson received 605 votes, New Gottland 325, King City 3 and Lindsborg 1; McPherson's majority over all, 276  In May the McPherson Town Company had offered, as an inducement for the location of the county seat at this point, the free use of rooms for ten years, and the donation of two squares of land on the town site  The offer was accepted the next month, the County Commissioners selecting blocks 56 and 65  Thus the county seat was established at McPherson and has remained since 

As early as 1875, city leaders of Marion held a meeting to consider a branch railroad from Florence  In 1878, Atchison, Topeka and Santa Fe Railway and parties from Marion County and McPherson County chartered the Marion and McPherson Railway Company   In 1879, a branch line was built from Florence to McPherson, in 1880 it was extended to Lyons, in 1881 it was extended to Ellinwood  The line was leased and operated by the Atchison, Topeka and Santa Fe Railway  The line from Florence to Marion, was abandoned in 1968  In 1992, the line from Marion to McPherson was sold to Central Kansas Railway  In 1993, after heavy flood damage, the line from Marion to McPherson was abandoned  The original branch line connected Florence, Marion, Canada, Hillsboro, Lehigh, Canton, Galva, McPherson, Conway, Windom, Little River, Mitchell, Lyons, Chase, then connected with the original AT&SF main line at Ellinwood 

In 1887, the Chicago, Kansas and Nebraska Railway extended its main line from Herington to Pratt   This main line connected Herington, Ramona, Tampa, Durham, Waldeck, Canton, Galva, McPherson, Groveland, Inman, Medora, Hutchinson, Whiteside, Partridge, Arlington,  Langdon, Turon, Preston, Natrona, Pratt   In 1888, this main line was extended to Liberal   Later, this line was extended to Tucumcari, New Mexico and Santa Rosa, New Mexico, where it made a connection with the Southern Pacific from El Paso, Texas   The Chicago, Kansas and Nebraska Railway was absorbed by the Chicago, Rock Island and Pacific Railway  This line is also called the "Golden State Route" 

20th century
The National Old Trails Road, also known as the Ocean-to-Ocean Highway, was established in 1912, and was routed through Windom, Conway, McPherson 

Geography

According to the U S  Census Bureau, the county has a total area of , of which  is land and  (0 3%) is water 

Adjacent counties
 Saline County (north)
 Dickinson County (northeast)
 Marion County (east)
 Harvey County (southeast)
 Reno County (southwest)
 Rice County (west)
 Ellsworth County (northwest)

Major highways
  Interstate 135
  U S  Route 56
  U S  Route 81
  K-4
  K-61
  K-153

Demographics

The McPherson Micropolitan Statistical Area includes all of McPherson County 

2000 census
As of the census of 2000, there were 29,554 people, 11,205 households, and 7,966 families residing in the county   The population density was 33 people per square mile (13/km2)   There were 11,830 housing units at an average density of 13 per square mile (5/km2)   The racial makeup of the county was 96 53% White, 0 81% Black or African American, 0 34% Native American, 0 32% Asian, 0 06% Pacific Islander, 0 79% from other races, and 1 16% from two or more races   1 94% of the population were Hispanic or Latino of any race  37 1% were of German, 12 9% Swedish, 12 1% American, 6 7% English and 6 3% Irish ancestry according to Census 2000 

There were 11,205 households, out of which 33 00% had children under the age of 18 living with them, 62 50% were married couples living together, 6 00% had a female householder with no husband present, and 28 90% were non-families  25 50% of all households were made up of individuals, and 11 80% had someone living alone who was 65 years of age or older   The average household size was 2 49 and the average family size was 2 99 

In the county, the population was spread out, with 25 40% under the age of 18, 10 30% from 18 to 24, 25 20% from 25 to 44, 21 80% from 45 to 64, and 17 30% who were 65 years of age or older   The median age was 38 years  For every 100 females there were 95 90 males   For every 100 females age 18 and over, there were 92 90 males 

The median income for a household in the county was $41,138, and the median income for a family was $48,243  Males had a median income of $33,530 versus $21,175 for females  The per capita income for the county was $18,921   About 4 20% of families and 6 60% of the population were below the poverty line, including 5 20% of those under age 18 and 8 10% of those age 65 or over 

Government

Presidential elections
McPherson county is often carried by Republican candidates  The last time a Democratic candidate has carried this county was in 1964 by Lyndon B  Johnson 

Laws
Following amendment to the Kansas Constitution in 1986, the county remained a prohibition, or "dry", county until 1996, when voters approved the sale of alcoholic liquor by the individual drink with a 30 percent food sales requirement 

Education

Colleges
 McPherson College in McPherson
 Bethany College in Lindsborg
 Central Christian College in McPherson

Unified school districts
 Smoky Valley USD 400
 McPherson USD 418
 Canton-Galva USD 419
 Moundridge USD 423
 Inman USD 448

School district office in neighboring county
 Goessel USD 411
 Little River-Windom USD 444

Museums
 Birger Sandzén Memorial Gallery in Lindsborg
 McCormick-Deering Days Museum in Inman
 McPherson Museum in McPherson
 Lindsborg Old Mill & Swedish Heritage Museum in Lindsborg
 Kansas Motorcycle Museum in Marquette

Communities

Cities

 Canton
 Galva
 Inman
 Lindsborg
 Marquette
 McPherson (county seat) 
 Moundridge
 Windom

Unincorporated communities
† means a Census-Designated Place (CDP) by the United States Census Bureau 
 Conway
 Elyria†
 Groveland
 Johnstown
 New Gottland
 Roxbury†

Ghost towns
 Alta Mills
 Battle Hill
 Christian
 Doles Park
 Elivon
 King City
 Sweadal

Townships
McPherson County is divided into twenty-five townships   The cities of Lindsborg and McPherson are considered governmentally independent and are excluded from the census figures for the townships   In the following table, the population center is the largest city (or cities) included in that township's population total, if it is of a significant size 

See also
 List of people from McPherson County, Kansas
 National Register of Historic Places listings in McPherson County, Kansas
 McPherson Valley Wetlands
 Maxwell Wildlife Refuge

References

Notes

Further reading

 Wheeler, Wayne Leland  "An Analysis of Social Change in a Swedish-Immigrant Community: The Case of Lindsborg, Kansas " (PhD dissertation, University of Missouri-Columbia; ProQuest Dissertations Publishing, 1959  5905657) 

County
 Through the Years: A Pictorial History of McPherson County; McPherson Sentinel' Heritage House Publishing Co; 1992 
 McPherson County First Courthouse Built About 1869 or 1870; Lindsborg News-Record; March 30, 1959 
 Pioneer Life and Lore of McPherson County, Kansas; Edna Nyquist; Democratic-Opinion Press; 1932 
 A History of the Church of the Brethren in Kansas (includes McPherson College history); Elmer LeRoy Craik; McPherson Daily; Republican Press; 397 pages; 1922 
 Portrait and Biographical Record of Dickinson, Saline, McPherson, and Marion Counties, Kansas; Chapman Bros; 614 pages; 1893 
 Standard Atlas of McPherson County, Kansas; Geo  A  Ogle & Co; 82 pages; 1921 
 Plat Book of McPherson County, Kansas; North West Publishing Co; 50 pages; 1903 
 Edwards' Atlas of McPherson County, Kansas; John P  Edwards; 51 pages; 1884 

Trails
 The Story of the Marking of the Santa Fe Trail by the Daughters of the American Revolution in Kansas and the State of Kansas; Almira Cordry; Crane Co; 164 pages; 1915  (Download 4MB PDF eBook)
 The National Old Trails Road To Southern California, Part 1 (LA to KC); Automobile Club Of Southern California; 64 pages; 1916  (Download 6 8MB PDF eBook)

Mennonite Settlements
 Impact of Mennonite settlement on the cultural landscape of Kansas; Brenda Martin; Kansas State University; 1985/1988  
 Mennonite settlement : the relationship between the physical and cultural environment; Susan Movle; University of Utah; 1975/1886 
 Status of Mennonite women in Kansas in their church and home relationships; Eva Harshbarger; Bluffton College; 1925/1945 

External links

County
 
 McPherson County - Directory of Public Officials
Historical
 , from Hatteberg's People'' on KAKE TV news
Maps
 McPherson County Maps: Current, Historic, KDOT
 Kansas Highway Maps: Current, Historic, KDOT
 Kansas Railroad Maps: Current, 1996, 1915, KDOT and Kansas Historical Society

 
Kansas counties
1867 establishments in Kansas

Context 2: Hugh Hilton Goodwin (December 21, 1900 – February 25, 1980) was a decorated officer in the United States Navy with the rank of Vice Admiral  A veteran of both World Wars, he commanded escort carrier  during the Mariana Islands campaign  Goodwin then served consecutively as Chief of Staff, Carrier Strike Group 6 and as Air Officer, Philippine Sea Frontier and participated in the Philippines campaign in the later part of the War 

Following the War, he remained in the Navy and rose to the flag rank and held several important commands including Vice Commander, Military Air Transport Service, Commander, Carrier Division Two and Commander, Naval Air Forces, Continental Air Defense Command 

Early life and career

Hugh H  Goodwin was born on December 21, 1900, in Monroe, Louisiana and attended Monroe High School there (now Neville High School)  Following the United States' entry into World War I in April 1917, Goodwin left the school without receiving the diploma in order to see some combat and enlisted the United States Navy on May 7, 1917  He completed basic training and was assigned to the battleship   Goodwin participated in the training of armed guard crews and engine room personnel as the Atlantic Fleet prepared to go to war and in November 1917, he sailed with the rest of Battleship Division 9, bound for Britain to reinforce the Grand Fleet in the North Sea 

Although he did not complete the last year of high school, Goodwin was able to earn an appointment to the United States Naval Academy at Annapolis, Maryland in June 1918  While at the academy, he earned a nickname "Huge" and among his classmates were several future admirals and generals including: Hyman G  Rickover, Milton E  Miles, Robert E  Blick Jr , Herbert S  Duckworth, Clayton C  Jerome, James P  Riseley, James A  Stuart, Frank Peak Akers, Sherman Clark, Raymond P  Coffman, Delbert S  Cornwell, Frederick J  Eckhoff, Ralph B  DeWitt, John Higgins, Vernon Huber, Albert K  Morehouse, Harold F  Pullen, Michael J  Malanaphy, William S  Parsons, Harold R  Stevens, John P  Whitney, Lyman G  Miller and George J  O'Shea 

Goodwin graduated with Bachelor of Science degree on June 3, 1922, and was commissioned Ensign in the United States Navy  He was subsequently assigned to the battleship  and took part in the voyage to Rio de Janeiro, Brazil, before he was ordered to the Naval Torpedo Station at Newport, Rhode Island for submarine instruction in June 1923  Goodwin completed the training several weeks later and was attached to the submarine   He then continued his further training aboard submarine  and following his promotion to Lieutenant (junior grade) on June 3, 1925, he qualified as submariner 

He then served aboard submarine  off the coast of California, before he was ordered for the recruiting duty to San Francisco in September 1927  While in this capacity, Goodwin applied for naval aviation training which was ultimately approved and he was ordered to the Naval Air Station Pensacola, Florida in August 1928  Toward the end of the training, he was promoted to lieutenant on December 11, 1928, and upon the completion of the training in January 1929, he was designated Naval aviator 

Goodwin was subsequently attached to the Observation Squadron aboard the aircraft carrier  and participated in the Fleet exercises in the Caribbean  He was transferred to the Bureau of Aeronautics in Washington, D C  in August 1931 and served consecutively under the architect of naval aviation William A  Moffett and future Chief of Naval Operations Ernest J  King 

In June 1933, Goodwin was ordered to the Naval War College at Newport, Rhode Island, where he completed junior course in May of the following year  He subsequently joined the crew of aircraft carrier  and served under Captain Arthur B  Cook and took part in the Fleet exercises in the Caribbean and off the East Coast of the United States 

He was ordered back to the Naval Air Station Pensacola, Florida in June 1936 and was attached to the staff of the Base Commandant, then-Captain Charles A  Blakely  When Blakely was succeeded by William F  Halsey in June 1937, Goodwin remained in Halsey's staff and was promoted to Lieutenant Commander on December 1, 1937  He also completed correspondence course in International law at the Naval War College 

Goodwin was appointed Commanding officer of the Observation Squadron 1 in June 1938 and attached to the battleship  he took part in the patrolling of the Pacific and 
West Coast of the United States until September 1938, when he assumed command of the Observation Squadron 2 attached to the battleship  

When his old superior from Lexington, now Rear Admiral Arthur B  Cook, was appointed Commander Aircraft, Scouting Force in June 1939, he requested Goodwin as his Aide and Flag Secretary  He became Admiral Cook's protégé and after year and half of service in the Pacific, he continued as his Aide and Flag Secretary, when Cook was appointed Commander Aircraft, Atlantic Fleet in November 1940 

World War II

Following the United States' entry into World War II, Goodwin was promoted to the temporary rank of Commander on January 1, 1942, and assumed duty as advisor to the Argentine Navy  His promotion was made permanent two months later and he returned to the United States in early 1943 for duty as assistant director of Planning in the Bureau of Aeronautics under Rear admiral John S  McCain  While still in Argentina, Goodwin was promoted to the temporary rank of Captain on June 21, 1942 

By the end of December 1943, Goodwin was ordered to Astoria, Oregon, where he assumed command of newly commissioned escort carrier USS Gambier Bay  He was responsible for the initial training of the crew and was known as a strict disciplinarian, but the crew appreciated the skills he taught them that prepared them for combat  Goodwin insisted that everyone aboard has to do every job right every time and made us fight our ship at her best 

During the first half of 1944, Gambier Bay was tasked with ferrying aircraft for repairs and qualified carrier pilots from San Diego to Pearl Harbor, Hawaii, before departed on May 1, 1944, to join Rear admiral Harold B



ground_truth: McPherson County is located in the U.S. state of Kansas.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: When did Goodwin become a Naval aviator?

Read the following text and answer briefly.

Context 2:  Sallada's Carrier Support Group 2, staging in the Marshalls for the invasion of the Marianas 

The air unit, VC-10 Squadron, under Goodwin's command gave close air support to the initial landings of Marines on Saipan on June 15, 1944, destroying enemy gun emplacements, troops, tanks, and trucks  On the 17th, her combat air patrol (CAP) shot down or turned back all but a handful of 47 enemy planes headed for her task group and her gunners shot down two of the three planes that did break through to attack her 

Goodwin's carrier continued in providing of close ground support operations at Tinian during the end of July 1944, then turned her attention to Guam, where she gave identical aid to invading troops until mid-August that year  For his service during the Mariana Islands campaign, Goodwin was decorated with Bronze Star Medal with Combat "V" 

He was succeeded by Captain Walter V  R  Vieweg on August 18, 1944, and appointed Chief of Staff, Carrier Division Six under Rear admiral Arthur W  Radford  The Gambier Bay was sunk in the Battle off Samar on October 25, 1944, during the Battle of Leyte Gulf after helping turn back a much larger attacking Japanese surface force 

Goodwin served with Carrier Division Six during the Bonin Islands raids, the naval operations at Palau and took part in the Battle of Leyte Gulf and operations supporting Leyte landings in late 1944  He was later appointed Air Officer of the Philippine Sea Frontier under Rear admiral James L  Kauffman and remained with that command until the end of hostilities  For his service in the later part of World War II, Goodwin was decorated with Legion of Merit with Combat "V"  He was also entitled to wear two Navy Presidential Unit Citations and Navy Unit Commendation 

Postwar service

Following the surrender of Japan, Goodwin assumed command of Light aircraft carrier  on August 24, 1945  The ship was tasked with air missions over Japan became mercy flights over Allied prisoner-of-war camps, dropping food and medicine until the men could be rescued  She was also present at Tokyo Bay for the Japanese surrender on September 2, 1945 

Goodwin returned with San Jacinto to the United States in mid-September 1945 and he was detached in January 1946  He subsequently served in the office of the Chief of Naval Operations until May that year, when he entered the instruction at National War College  Goodwin graduated in June 1947 and served on Secretary's committee for Research on Reorganization  Upon promotion to Rear admiral on April 1, 1949, Goodwin was appointed Chief of Staff and Aide to Commander-in-Chief, Atlantic Fleet under Admiral William H  P  Blandy 

Revolt of the Admirals

In April 1949, the budget's cuts and proposed reorganization of the United States Armed Forces by the Secretary of Defense Louis A  Johnson launched the wave of discontent between senior commanders in the United States Navy  Johnson proposed the merging of the Marine Corps into the Army, and reduce the Navy to a convoy-escort force 

Goodwin's superior officer, Admiral Blandy was call to testify before the House Committee on Armed Services and his harsh statements for the defense of the Navy, costed him his career  Goodwin shared his views and openly criticized Secretary Johnson for having power concentrated in a single civilian executive, who is an appointee of the Government and not an elected representative of the people  He also criticized aspects of defense unification which permitted the Joint Chiefs of Staff to vote on arms policies of individual services, and thus "rob" the branches of autonomy 

The outbreak of the Korean War in summer 1950 proved the proposal of Secretary Johnson as incorrect and he resigned in September that year  Also Secretary of the Navy, Francis P  Matthews resigned one month earlier 

Later service

Due to the Revolts of the admirals, Blandy was forced to retire in February 1950 and Goodwin was ordered to Newport, Rhode Island for temporary duty as Chief of Staff and Aide to the President of the Naval War College under Vice admiral Donald B  Beary in April 1950  Goodwin was detached from that assignment two months and appointed member of the General Board of the Navy  He was shortly thereafter appointed acting Navy Chief of Public Information, as the substitute for Rear Admiral Russell S  Berkey, who was relieved of illness, but returned to the General Board of the Navy in July that year  Goodwin served in that capacity until February 1951, when he relieved his Academy class, Rear admiral John P  Whitney as Vice Commander, Military Air Transport Service (MATS) 

While in this capacity, Goodwin served under Lieutenant general Laurence S  Kuter and was co-responsible for the logistical support of United Nations troops fighting in Korea  The MATS operated from the United States to Japan and Goodwin served in this capacity until August 1953, when he was appointed Commander Carrier Division Two  While in this assignment, he took part in the Operation Mariner, Joint Anglo-American exercise which encountered very heavy seas over a two-week period in fall 1953 

Goodwin was ordered to the Philippines in May 1954 and assumed duty as Commander, U S  Naval Forces in the Philippines with headquarters at Naval Station Sangley Point near Cavite  He held that command in the period of tensions between Taiwan and China and publicly declared shortly after his arrival, that any attack on Taiwan by the Chinese Communists on the mainland would result in US participation in the conflict  The naval fighter planes under his command also provided escort for passing commercial planes  Goodwin worked together with retired Admiral Raymond A  Spruance, then-Ambassador to the Philippines, and accompanied him during the visits to Singapore, Bangkok and Saigon in January 1955 

On December 18, 1955, Goodwin's classmate Rear admiral Albert K  Morehouse, then serving as Commander, Naval Air Forces, Continental Air Defense Command (CONAD), died of heart attack and Goodwin was ordered to CONAD headquarters in Colorado Springs, Colorado to assume Morehouse's position  While in this capacity, he was subordinated to Army General Earle E  Partridge and was responsible for the Naval and Marine Forces allocated to the command designated for the defense of the Continental United States 

Retirement

Goodwin retired on June 1, 1957, after 40 years of active service and was advanced to the rank of Vice admiral on the retired list for having been specially commended in combat  A week later, he was invited back to his Monroe High School (now Neville High School) and handed a diploma showing that he had been graduated with the class of 1918  He then settled in Monterey, California where he taught American history at Stevenson school and was a member of the Naval Order of the United States 

Vice admiral Hugh H  Goodwin died at his home on February 25, 1980, aged 79  He was survived by his wife, Eleanor with whom he had two children, a daughter Sidney and a son Hugh Jr , who graduated from the Naval Academy in June 1948, but died one year later, when the Hellcat fighter he was piloting collided with another over the Gulf of Mexico during training 

Decorations

Here is the ribbon bar of Vice admiral Hugh H  Goodwin:

References

1900 births
1980 deaths
People from Monroe, Louisiana
Military personnel from Louisiana
United States Naval Academy alumni
Naval War College alumni
United States Naval Aviators
United States Navy personnel of World War I
United States Navy World War II admirals
United States Navy vice admirals
United States submarine commanders
Recipients of the Legion of Merit
Context 3: \section{introduction}
In the past few years, the synthesis of ferromagnetic semiconductors has become a major challenge for spintronics  Actually, growing a magnetic and semiconducting material could lead to promising advances like spin injection into non magnetic semiconductors, or electrical manipulation of carrier induced magnetism in magnetic semiconductors \cite{ohno00,Bouk02}  Up to now, major efforts have focused on diluted magnetic semiconductors (DMS) in which the host semiconducting matrix is randomly substituted by transition metal (TM) ions such as Mn, Cr, Ni, Fe or Co \cite{Diet02}  However Curie temperatures ($T_{C}$) in DMS remain rather low and TM concentrations must be drastically raised in order to increase $T_{C}$ up to room temperature  That usually leads to phase separation and the formation of secondary phases  It was recently shown that phase separation induced by spinodal decomposition could lead to a significant increase of $T_{C}$ \cite{Diet06,Fuku06}  For semiconductors showing $T_{C}$ higher than room temperature one can foresee the fabrication of nanodevices such as memory nanodots, or nanochannels for spin injection  Therefore, the precise control of inhomogeneities appears as a new challenge which may open a way to industrial applications of ferromagnetism in semiconductors 

The increasing interest in group-IV magnetic semiconductors can also be explained by their potential compatibility with the existing silicon technology  In 2002, carrier mediated ferromagnetism was reported in  MBE grown Ge$_{0 94}$Mn$_{0 06}$ films by Park \textit{et al } \cite{Park02}  The maximum critical temperature was 116 K  Recently many publications indicate a significant increase of $T_{C}$ in Ge$_{1-x}$Mn$_{x}$ material depending on growth conditions \cite{Pint05,Li05,tsui03}  Cho \textit{et al } reported a Curie temperature as high as 285 K \cite{Cho02}  
Taking into account the strong tendency of Mn ions to form intermetallic compounds in germanium, a detailed investigation of the nanoscale structure is required  Up to now, only a few studies have focused on the nanoscale composition in Ge$_{1-x}$Mn$_{x}$ films  Local chemical inhomogeneities have been recently reported by Kang \textit{et al } \cite{Kang05} who evidenced a micrometer scale segregation of manganese in large Mn rich stripes  Ge$_3$Mn$_5$ as well as Ge$_8$Mn$_{11}$ clusters embedded in a germanium matrix have been reported by many authors  However, Curie temperatures never exceed 300 K \cite{Bihl06,Morr06,Pass06,Ahle06}  Ge$_3$Mn$_5$ clusters exhibit a Curie temperature of 296 K \cite{Mass90}  This phase frequently observed in Ge$_{1-x}$Mn$_{x}$ films is the most stable (Ge,Mn) alloy  The other stable compound Ge$_8$Mn$_{11}$ has also been observed in nanocrystallites surrounded with pure germanium \cite{Park01}  Ge$_8$Mn$_{11}$ and Ge$_3$Mn$_5$ phases are ferromagnetic but their metallic character considerably complicates their potential use as spin injectors 
Recently, some new Mn-rich nanostructures have been evidenced in Ge$_{1-x}$Mn$_{x}$ layers  Sugahara \textit{et al } \cite{Sugh05} reported the formation of high Mn content (between 10 \% and 20 \% of Mn) amorphous Ge$_{1-x}$Mn$_x$ precipitates  in a Mn-free germanium matrix  Mn-rich coherent cubic clusters were observed by Ahlers \textit{et al } \cite{Ahle06} which exhibit a Curie temperatures below 200 K  Finally, high-$T_{C}$ ($>$ 400 K) Mn-rich nanocolumns have been evidenced \cite{Jame06} which could lead to silicon compatible room temperature operational devices \\
In the present paper, we investigate the structural and magnetic properties of Ge$_{1-x}$Mn$_x$ thin films for low growth temperatures ($<$ 200$^{\circ}$C) and low Mn concentrations (between 1 \% and 11 \%)  By combining TEM, x-Ray diffraction and SQUID magnetometry, we could identify different magnetic phases  We show that depending on growth conditions, we obtain either Mn-rich nanocolumns or Ge$_{3}$Mn$_{5}$ clusters embedded in a germanium matrix  We discuss the structural and magnetic properties of these nanostructures as a function of manganese concentration and growth temperature  We also discuss the magnetic anisotropy of nanocolumns and  
Ge$_3$Mn$_5$ clusters  

\section{Sample growth}

Growth was performed using solid sources molecular beam epitaxy (MBE) by co-depositing Ge and Mn evaporated from standard Knudsen effusion cells   Deposition rate was low ($\approx$ 0 2 \AA s$^{-1}$)  Germanium substrates were epi-ready Ge(001) wafers with a residual n-type doping and resistivity of 10$^{15}$ cm$^{-3}$ and 5 $\Omega cm$ respectively  After thermal desorption of the surface oxide, a 40 nm thick Ge buffer layer was grown at 250$^{\circ}$C, resulting in a 2 $\times$ 1 surface reconstruction as observed by reflection high energy electron diffraction (RHEED) (see Fig  1a)  Next, 80 nm thick Ge$_{1-x}$Mn$_{x}$ films were subsequently grown  at low substrate temperature (from 80$^{\circ}$C to 200$^{\circ}$C)  Mn content has been determined by x-ray fluorescence measurements performed on thick samples ($\approx$ 1 $\mu m$ thick) and complementary Rutherford Back Scattering (RBS) on thin Ge$_{1-x}$Mn$_{x}$ films grown on silicon  Mn concentrations range from 1 \% to 11\% Mn 

For Ge$_{1-x}$Mn$_{x}$ films grown at substrate temperatures below 180$^{\circ}$C, after the first monolayer (ML) deposition, the  2 $\times$ 1 surface reconstruction almost totally disappears  After depositing few MLs, a slightly diffuse 1 $\times$ 1 streaky RHEED pattern and a very weak 2 $\times$ 1 reconstruction (Fig  1b) indicate a predominantly two-dimensional growth  For growth temperatures above 180$^{\circ}$C additional spots appear in the RHEED pattern during the Ge$_{1-x}$Mn$_{x}$ growth (Fig  1c)  These spots may correspond to the formation of very small secondary phase crystallites  The nature of these crystallites will be discussed below 

Transmission electron microscopy (TEM) observations were performed using a JEOL 4000EX microscope with an acceleration voltage of 400 kV  Energy filtered transmission electron microscopy (EFTEM) was done using a JEOL 3010 microscope equipped with a Gatan Image Filter   Sample preparation was carried out by standard mechanical polishing and argon ion milling for cross-section investigations and plane views were prepared by wet etching with H$_3$PO$_4$-H$_2$O$_2$ solution \cite{Kaga82} 

\begin{figure}[htb]
    \center
    \includegraphics[width= 29\linewidth]{ /fig1a eps}
    \includegraphics[width= 29\linewidth]{ /fig1b eps}
    \includegraphics[width= 29\linewidth]{ /fig1c eps}
    \caption{RHEED patterns recorded during the growth of Ge$_{1-x}$Mn$_{x}$ films: (a) 2 $\times$ 1 surface reconstruction of the germanium buffer layer  (b) 1 $\times$ 1 streaky RHEED pattern obtained at low growth temperatures ($T_g<$180$^{\circ}$C)  (c) RHEED pattern of a sample grown at $T_g=$180$^{\circ}$C  The additional spots reveal the presence of Ge$_3$Mn$_5$ clusters at the surface of the film }
\label{fig1}
\end{figure}

\section{Structural properties \label{structural}}

\begin{figure}[htb]
    \center
	\includegraphics[width= 49\linewidth]{ /fig2a eps}
	\includegraphics[width= 49\linewidth]{ /fig2b eps}
	 \includegraphics[width= 49\linewidth]{ /fig2c eps}
	 \includegraphics[width= 49\linewidth]{ /fig2d eps}
    \caption{Transmission electron micrographs of a Ge$_{1-x}$Mn$_{x}$ film grown at 130$^{\circ}$C and containing 6 \% of manganese  (a) cross-section along the [110] axis : we clearly see the presence of nanocolumns elongated along the growth axis  (b) High resolution image of the interface between the Ge$_{1-x}$Mn$_{x}$ film and the Ge buffer layer  The Ge$_{1-x}$Mn$_{x}$ film exhibits the same diamond structure as pure germanium  No defect can be seen which could be caused by the presence of nanocolumns  (c) Plane view micrograph performed on the same sample confirms the columnar structure and gives the density and size distribution of nanocolumns  (d) Mn chemical map obtained by energy filtered transmission electron microcopy (EFTEM)  The background was carefully substracted from pre-edge images  Bright areas correspond to Mn-rich regions }
\label{fig2}
\end{figure}

In samples grown at 130$^{\circ}$C and containing 6 \% Mn, we can observe vertical elongated nanostructures \textit{i e } nanocolumns as shown in Fig  2a  Nanocolumns extend through the whole Ge$_{1-x}$Mn$_{x}$ film thickness  From the high resolution TEM image shown in Fig  2b, we deduce their average diameter around 3 nm  Moreover in Fig  2b, the interface between the Ge buffer layer and the Ge$_{1-x}$Mn$_{x}$ film is flat and no defect propagates from the interface into the film  The Ge$_{1-x}$Mn$_{x}$ film is a perfect single crystal in epitaxial relationship with the substrate  In Fig  2c is shown a plane view micrograph of the same sample confirming the presence of nanocolumns in the film  From this image, we can deduce the size and density of nanocolumns  The nanocolumns density is 13000 $\rm{\mu m}^{-2}$ with a mean diameter of 3 nm which is coherent with cross-section measurements  In order to estimate the chemical composition of these nanocolumns, we further performed chemical mapping using EFTEM  In Fig  2d we show a cross sectional Mn chemical map of the Ge$_{1-x}$Mn$_{x}$ film  This map shows that the formation of nanocolumns is a consequence of Mn segregation  Nanocolumns are Mn rich and the surrounding matrix is Mn poor  However, it is impossible to deduce the Mn concentration in Ge$_{1-x}$Mn$_{x}$ nanocolumns from this cross section  Indeed, in cross section observations, the columns diameter is much smaller than the probed film thickness and the signal comes from the superposititon of the Ge matrix and Mn-rich nanocolumns  In order to quantify Mn concentration inside the nanocolumns and inside the Ge matrix, EELS measurements (not shown here) have been performed in a plane view geometry \cite{Jame06}  These observations revealed that the matrix Mn content is below 1 \% (detection limit of our instrument)  Measuring the surface occupied by the matrix and the nanocolumns in plane view TEM images, and considering the average Mn concentration in the sample (6 \%), we can estimate the Mn concentration in the nanocolumns  The Mn concentration measured by EELS being between 0\% and 1\%, we can conclude that the Mn content in the nanocolumns is between 30 \% and 38 \% \\
For samples grown between 80$^\circ$C and 150$^\circ$C cross section and plane view TEM observations reveal the presence of Mn rich nanocolumns surrounded with a Mn poor Ge matrix  In order to investigate the influence of Mn concentration on the structural properties of Ge$_{1-x}$Mn$_{x}$ films, ten samples have been grown at 100$^\circ$C and at 150$^\circ$C with Mn concentrations of 1 3 \%, 2 3 \%, 4 \%, 7 \% and 11 3 \%  Their structural properties have been investigated by plane view TEM observations  

\begin{figure}[htb]
    \center
    \includegraphics[width= 98\linewidth]{ /fig3a eps}
	\includegraphics[width= 45\linewidth]{ /fig3b eps}
		\includegraphics[width= 45\linewidth]{ /fig3c eps}
    \caption{Nanocolumns size and density as a function of growth conditions  Samples considered have been grown at 100$^{\circ}$C and 150$^{\circ}$C respectively  (a) Mn concentration dependence of the size distribution  (b) columns density as a function of Mn concentration  (c) Volume fraction of the nanocolumns as a function of Mn concentration }
 \label{fig3}
\end{figure}

For samples grown at 100$^\circ$C with Mn concentrations below 5 \% the nanocolumns mean diameter is 1 8$\pm$0 2 nm  The evolution of columns density as a fonction of Mn concentration is reported in figure 3b  By increasing the Mn concentration from 1 3 \% to 4 \% we observe a significant increase of the columns density from 13000 to 30000 $\rm{\mu m}^{-2}$  For Mn concentrations higher than 5 \% the density seems to reach a plateau corresponding to 35000 $\rm{\mu m}^{-2}$ and their diameter slightly increases from 1 8 nm at 4 \% to 2 8 nm at 11 3 \%  By plotting the volume fraction occupied by the columns in the film as a function of Mn concentration, we observe a linear dependence for Mn contents below 5 \%  The non-linear behavior above 5 \% may indicate that the mechanism of Mn incorporation is different in this concentration range, leading to an increase of Mn concentration in the columns or in the matrix  For samples grown at 100$^\circ$C, nanocolumns are always fully coherent with the surrounding matrix (Fig  4a)  

Increasing the Mn content in the samples grown at 150$^\circ$C from 1 3 \% to 11 3 \% leads to a decrease of the columns density (fig 3b)  Moreover, their average diameter increases significantly and size distributions become very broad (see Fig  3a)  For the highest Mn concentration (11 3 \%) we observe the coexistence of very small columns with a diameter of 2 5 nm and very large columns with a diameter of 9 nm  In samples grown at 150$^\circ$C containing 11 3 \% of Mn, the crystalline structure of nanocolumns is also highly modified  In plane view TEM micrographs, one can see columns exhibiting several different crystalline structures  We still observe some columns which are fully coherent with the Ge matrix like in the samples grown at lower temperature  Nevertheless, observations performed on these samples grown at 150$^\circ$C and with 11 3\% Mn reveal some uniaxially \cite{Jame06} or fully relaxed columns exhibiting a misfit of 4 \% between the matrix and the columns and leading to misfit dislocations at the interface between the column and the matrix (see fig  4b)  Thus we can conclude that coherent columns are probably in strong compression and the surrounding matrix in tension  On the same samples (T$_g$=150$^{\circ}$C, 11 3\% Mn), we also observe a large number of highly disordered nanocolumns leading to an amorphous like TEM contrast(fig  4c) 

\begin{figure}[htb]
    \center
   \includegraphics[width= 31\linewidth]{ /fig4a eps}
	\includegraphics[width= 31\linewidth]{ /fig4b eps}
	\includegraphics[width= 31\linewidth]{ /fig4c eps}
    \caption{Plane view high resolution transmission electron micrographs of different types of nanocolumns : (a) typical structure of a column grown at 100$^{\circ}$C  The crystal structure is exactly the same as germanium   (b) Partially relaxed nanocolumn  One can see dislocations at the interface between the columns and the matrix leading to stress relaxation  (c) Amorphous nanocolumn  These columns are typical in samples grown at 150$^{\circ}$C with high Mn contents }
 \label{fig4}
\end{figure}

In conclusion, we have evidenced a complex mechanism of Mn incorporation in Mn doped Ge films grown at low temperature  In particular Mn incorporation is highly inhomogeneous  For very low growth temperatures (below 120$^\circ$C) the diffusion of Mn atoms leads to the formation of Mn rich, vertical nanocolumns  Their density mostly depends on Mn concentration and their mean diameter is about 2 nm  These results can be compared with the theoretical predictions of Fukushima \textit{et al } \cite{Fuku06}: they proposed a model of spinodal decomposition in (Ga,Mn)N and (Zn,Cr)Te based on layer by layer growth conditions and a strong pair attraction between Mn atoms which leads to the formation of nanocolumns  This model may also properly describe the formation of Mn rich nanocolumns in our samples  Layer by layer growth conditions can be deduced from RHEED pattern evolution during growth  For all the samples grown at low temperature, RHEED observations clearly indicate two-dimensional growth  Moreover, Ge/Ge$_{1-x}$Mn$_{x}$/Ge heterostructures have been grown and observed by TEM (see Fig  5)  Ge$_{1-x}$Mn$_{x}$/Ge (as well as Ge/Ge$_{1-x}$Mn$_{x}$) interfaces are very flat and sharp thus confirming a two-dimensional, layer by layer growth mode  Therefore we can assume that the formation of Mn rich nanocolumns is a consequence of 2D-spinodal decomposition 

\begin{figure}[htb]
    \center
	\includegraphics[width= 7\linewidth]{ /fig5 eps}
    \caption{Cross section high resolution micrograph of a Ge/Ge$_{1-x}$Mn$_{x}$/Ge/Ge$_{1-x}$Mn$_{x}$/Ge heterostructure  This sample has been grown at 130 $^{\circ}$C with 6\% Mn  Ge$_{1-x}$Mn$_{x}$ layers are 15 nm thick and Ge spacers 5 nm thick  We clearly see the sharpness of both Ge$_{1-x}$Mn$_{x}$/Ge and Ge/Ge$_{1-x}$Mn$_{x}$ interfaces  Mn segregation leading to the columns formation already takes place in very thin Ge$_{1-x}$Mn$_{x}$ films }
\label{fig5}
\end{figure}

For growth temperatures higher than 160$^\circ$C, cross section TEM and EFTEM observations (not shown here) reveal the coexistence of two Mn-rich phases: nanocolumns and Ge$_{3}$Mn$_{5}$ nanoclusters embedded in the germanium matrix  A typical high resolution TEM image is shown in figure 6  
Ge$_{3}$Mn$_{5}$ clusters are not visible in RHEED patterns for temperatures below 180$^\circ$C  To investigate the nature of these clusters, we performed x-ray diffraction in $\theta-2\theta$ mode  Diffraction scans were acquired on a high resolution diffractometer using the copper K$_\alpha$ radiation and on the GMT station of the BM32 beamline at the European Synchrotron Radiation Facility (ESRF)  Three samples grown at different temperatures and/or annealed at high temperature were investigated  The two first samples are Ge$_{1-x}$Mn$_{x}$ films grown at 130$^\circ$C and 170$^\circ$C respectively  The third one has been grown at 130$^\circ$C and post-growth annealed at 650$^\circ$C  By analysing x-ray diffraction spectra, we can evidence two different crystalline structures  For the sample grown at 130$^\circ$C, the $\theta-2\theta$ scan only reveals the (004) Bragg peak of the germanium crystal, confirming the good epitaxial relationship between the layer and the substrate, and the absence of secondary phases in the film in spite of a high dynamics of the order of 10$^7$  For both samples grown at 170$^\circ$C and annealed at 650$^\circ$C, $\theta-2\theta$ spectra are identical  In addition to the (004) peak of germanium, we observe three additional weak peaks  The first one corresponds to the (002) germanium forbidden peak which probably comes from a small distortion of the germanium crystal, and the two other peaks are respectively attributed to the (002) and (004) Bragg peaks of a secondary phase  The $c$ lattice parameter of Ge$_3$Mn$_5$ hexagonal crystal is 5 053 \AA \ \cite{Fort90} which is in very good agreement with the values obtained from diffraction data for both (002) and (004) lines assuming that the $c$ axis of Ge$_3$Mn$_5$ is along the [001] direction of the Ge substrate 

\begin{figure}[htb]
    \center
	\includegraphics[width= 7\linewidth]{ /fig6 eps}
	\caption{Cross section high resolution transmission electron micrograph of a sample grown at 170$^{\circ}$C  We observe the coexistence of two different Mn-rich phases: Ge$_{1-x}$Mn$_{x}$ nanocolumns and Ge$_3$Mn$_5$ clusters }
\label{fig6}
\end{figure}

In summary, in a wide range of growth temperatures and Mn concentrations, we have evidenced a two-dimensional spinodal decomposition leading to the formation of Mn-rich nanocolumns in Ge$_{1-x}$Mn$_{x}$ films  This decomposition is probably the consequence of: $(i)$ a strong pair attraction between Mn atoms, $(ii)$ a strong surface diffusion of Mn atoms in germanium even at low growth temperatures and $(iii)$ layer by layer growth conditions  We have also investigated the influence of growth parameters on the spinodal decomposition: at low growth temperatures (100$^{\circ}$C), increasing the Mn content leads to higher columns densities while at higher growth temperatures (150$^{\circ}$C), the columns density remains nearly constant whereas their size increases drastically  By plotting the nanocolumns density as a function of Mn content, we have shown that the mechanism of Mn incorporation in Ge changes above 5 \% of Mn  Finally, using TEM observations and x-ray diffraction, we have shown that Ge$_3$Mn$_5$ nanoclusters start to form at growth temperatures higher than 160$^\circ$C 

\section{Magnetic properties \label{magnetic}}

We have thoroughly investigated the magnetic properties of thin Ge$_{1-x}$Mn$_{x}$ films for different growth temperatures and Mn concentrations  In this section, we focus on Mn concentrations between 2 \% and 11 \%  We could clearly identify four different magnetic phases in Ge$_{1-x}$Mn$_{x}$ films : diluted Mn atoms in the germanium matrix, low $T_{C}$ nanocolumns ($T_{C}$ $\leq$ 170 K), high $T_{C}$ nanocolumns ($T_{C}$ $\geq$ 400 K) and Ge$_{3}$Mn$_{5}$ clusters ($T_{C}$ $\thickapprox$ 300 K)  The relative weight of each phase clearly depends on the growth temperature and to a lesser extend on Mn concentration  For low growth temperature ($<$ 120$^{\circ}$C), we show that nanocolumns are actually made of four uncorrelated superparamagnetic nanostructures  Increasing T$_{g}$ above 120$^{\circ}$C, we first obtain continuous columns exhibiting low $T_{C}$ ($<$ 170 K) and high $T_{C}$ ($>$ 400 K) for $T_{g}\approx$130$^{\circ}$C  The larger columns become ferromagnetic \textit{i e } $T_{B}>T_{C}$  Meanwhile Ge$_{3}$Mn$_{5}$ clusters start to form  Finally for higher $T_{g}$, the magnetic contribution from Ge$_{3}$Mn$_{5}$ clusters keeps increasing while the nanocolumns signal progressively disappears 

\begin{figure}[htb]
\center
   \includegraphics[width= 6\linewidth]{ /fig7a eps}
   \includegraphics[width= 3\linewidth]{ /fig7b eps}
\caption{(a) Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{0 93}$Mn$_{0 07}$ samples for different growth temperatures  The magnetic field is applied in the film plane  The inset shows the temperature dependence of a sample grown at 130$^{\circ}$C and annealed at 650$^{\circ}$C for 15 minutes  After annealing, the magnetic signal mostly arises from Ge$_{3}$Mn$_{5}$ clusters  (b) ZFC-FC measurements performed on Ge$_{0 93}$Mn$_{0 07}$ samples for different growth temperatures  The in-plane applied field is 0 015 T  The ZFC peak at low temperature ($\leq$150 K) can be attributed to the superparamagnetic nanocolumns  This peak widens and shifts towards high blocking temperatures when increasing growth temperature  The second peak above 150 K in the ZFC curve which increases with increasing growth temperature is attributed to superparamagnetic Ge$_{3}$Mn$_{5}$ clusters  The increasing ZFC-FC irreversibility at $\approx$ 300 K is due to the increasing contribution from large ferromagnetic Ge$_{3}$Mn$_{5}$ clusters  The nanocolumns signal completely vanishes after annealing at 650$^{\circ}$C for 15 minutes }
\label{fig7}
\end{figure}

In Fig  7a, the saturation magnetization at 2 Tesla in $\mu_{B}$/Mn of Ge$_{1-x}$Mn$_{x}$ films with 7 \% of Mn is plotted as a function of temperature for different growth temperatures ranging from $T_{g}$=90$^{\circ}$C up to 160$^{\circ}$C  The inset shows the temperature dependence of the magnetization at 2 Tesla after annealing at 650$^{\circ}$C during 15 minutes  Figure 7b displays the corresponding Zero Field Cooled - Field Cooled (ZFC-FC) curves recorded at 0 015 Tesla  In the ZFC-FC procedure, the sample is first cooled down to 5 K in zero magnetic field and the susceptibility is subsequently recorded at 0 015 Tesla while increasing the temperature up to 400 K (ZFC curve)  Then, the susceptibility is recorded under the same magnetic field while decreasing the temperature down to 5 K (FC curve)  Three different regimes can be clearly distinguished  \\
For $T_{g}\leq$120$^{\circ}$C, the temperature dependence of the saturation magnetization remains nearly the same while increasing growth temperature  The overall magnetic signal vanishing above 200 K is attributed to the nanocolumns whereas the increasing signal below 50 K originates from diluted Mn atoms in the surrounding matrix  The Mn concentration dependence of the saturation magnetization is displayed in figure 8  For the lowest Mn concentration (4 \%), the contribution from diluted Mn atoms is very high and drops sharply for higher Mn concentrations (7 \%, 9 \% and 11 3 \%)  Therefore the fraction of Mn atoms in the diluted matrix decreases with Mn concentration probably because Mn atoms are more and more incorporated in the nanocolumns  In parallel, the Curie temperature of nanocolumns increases with the Mn concentration reaching 170 K for 11 3 \% of Mn  This behavior may be related to different Mn compositions and to the increasing diameter of nanocolumns (from 1 8 nm to 2 8 nm) as discussed in section \ref{structural} 

\begin{figure}[htb]
\center
   \includegraphics[width= 7\linewidth]{ /fig8 eps}
    \caption{Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{1-x}$Mn$_{x}$ films grown at 100$^{\circ}$C plotted for different Mn concentrations: 4 1 \%; 7 \%; 8 9 \% and 11 3 \% }
\label{fig8}
\end{figure}

ZFC-FC measurements show that the nanocolumns are superparamagnetic  The magnetic signal from the diluted Mn atoms in the matrix is too weak to be detected in susceptibility measurements at low temperature  In samples containing 4 \% of Mn, ZFC and FC curves superimpose down to low temperatures  As we do not observe hysteresis loops at low temperature, we believe that at this Mn concentration nanocolumns are  superparamagnetic in the whole temperature range and the blocking temperature cannot be measured  For higher Mn contents, the ZFC curve exhibits a very narrow peak with a maximum at the blocking temperature of 15 K whatever the Mn concentration and growth temperature (see Fig  7b)  Therefore the anisotropy barrier distribution is narrow and assuming that nanocolumns have the same magnetic anisotropy, this is a consequence of the very narrow size distribution of the nanocolumns as observed by TEM  To probe the anisotropy barrier distribution, we have performed ZFC-FC measurements but instead of warming the sample up to 400 K, we stopped at a lower temperature $T_{0}$  

\begin{figure}[htb]
\center
   \includegraphics[width= 6\linewidth]{ /fig9 eps}
\caption{Schematic drawing of the anisotropy barrier distribution n($E_{B}$) of superparamagnetic nanostructures  If magnetic anisotropy does not depend on the particle size, this distribution exactly reflects their magnetic size distribution  In this drawing the blocking temperature ($T_{B}$) corresponds to the distribution maximum  At a given temperature $T_{0}$ such that 25$k_{B}T_{0}$ falls into the anisotropy barrier distribution, the largest nanostructures with an anisotropy energy larger than 25$k_{B}T_{0}$ are blocked whereas the others are superparamagnetic }
\label{fig9}
\end{figure}

If this temperature falls into the anisotropy barrier distribution as depicted in Fig  9, the FC curve deviates from the ZFC curve  Indeed the smallest nanostructures have become superparamagnetic at $T_{0}$ and when decreasing again the temperature, their magnetization freezes along a direction close to the magnetic field and the FC susceptibility is higher than the ZFC susceptibility  Therefore any irreversibility in this procedure points at the presence of superparamagnetic nanostructures  The results are given in Fig  10a  ZFC and FC curves clearly superimpose up to $T_{0}$=250 K thus the nanocolumns are superparamagnetic up to their Curie temperature and no Ge$_{3}$Mn$_{5}$ clusters could be detected  Moreover for low $T_{0}$ values, a peak appears at low temperature in FC curves which evidences strong antiferromagnetic interactions between the nanocolumns \cite{Chan00} 

\begin{figure}[htb]
\center
    \includegraphics[width= 35\linewidth]{ /fig10a eps}
    \includegraphics[width= 63\linewidth]{ /fig10b eps}
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 115$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 30 K, 50 K, 100 K, 150 K, 200 K and 250 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0 015 T) }
\label{fig10}
\end{figure}

In order to derive the magnetic size and anisotropy of the Mn-rich nanocolumns embedded in the Ge matrix, we have fitted the inverse normalized in-plane (resp  out-of-plane) susceptibility: $\chi_{\parallel}^{-1}$ (resp  $\chi_{\perp}^{-1}$)  The corresponding experimental ZFC-FC curves are reported in Fig  10b  Since susceptibility measurements are performed at low field (0 015 T), the matrix magnetic signal remains negligible  In order to normalize susceptibility data, we need to divide the magnetic moment by the saturated magnetic moment recorded at 5 T  However the matrix magnetic signal becomes very strong at 5 T and low temperature so that we need to subtract it from the saturated magnetic moment using a simple Curie function  From Fig  10b, we can conclude that nanocolumns are isotropic  Therefore to fit experimental data we use the following expression well suited for isotropic systems or cubic anisotropy: $\chi_{\parallel}^{-1}= \chi_{\perp}^{-1}\approx 3k_{B}T/M(T)+\mu_{0}H_{eff}(T)$  $k_{B}$ is the Boltzmann constant, $M=M_{s}v$ is the magnetic moment of a single-domain nanostructure (macrospin approximation) where $M_{s}$ is its magnetization and $v$ its volume  The in-plane magnetic field is applied along $[110]$ or $[-110]$ crystal axes  Since the nanostructures Curie temperature does not exceed 170 K, the temperature dependence of the saturation magnetization is also accounted for by writting $M(T)$  Antiferromagnetic interactions between nanostructures are also considered by adding an effective field estimated in the mean field approximation \cite{Fruc02}: $\mu_{0}H_{eff}(T)$ 
The only fitting parameters are the maximum magnetic moment (\textit{i e } at low temperature) per nanostructure: $M$ (in Bohr magnetons $\mu_{B}$) and the maximum interaction field (\textit{i e } at low temperature): $\mu_{0}H_{eff}$ 

\begin{figure}[htb]
\center
   \includegraphics[width= 7\linewidth]{ /fig11 eps}
\caption{Temperature dependence of the inverse in-plane (open circles) and out-of-plane (open squares) normalized susceptibilities of a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 115$^{\circ}$C  Fits were performed assuming isotropic nanostructures or cubic anisotropy  Dashed line is for in-plane susceptibility and solid line for out-of-plane susceptibility }
\label{fig11}
\end{figure}

In Fig  11, the best fits lead to $M\approx$1250 $\mu_{B}$ and $\mu_{0}H_{eff}\approx$102 mT for in-plane susceptibility and $M\approx$1600 $\mu_{B}$ and $\mu_{0}H_{eff}\approx$98 mT for out-of-plane susceptibility  It gives an average magnetic moment of 1425 $\mu_{B}$ per column and an effective interaction field of 100 mT  Using this magnetic moment and its temperature dependence, magnetization curves could be fitted using a Langevin function and $M(H/T)$ curves superimpose for $T<$100 K  However, from the saturated magnetic moment of the columns and their density (35000 $\rm{\mu m}^{-2}$), we find almost 6000 $\mu_{B}$ per column  Therefore, for low growth temperatures, we need to assume that nanocolumns are actually made of almost four independent elongated magnetic nanostructures  The effective field for antiferromagnetic interactions between nanostructures estimated from the susceptibility fits is at least one order of magnitude larger than what is expected from pure magnetostatic coupling  This difference may be due to either an additional antiferromagnetic coupling through the matrix which origin remains unexplained or to the mean field approximation which is no more valid in this strong coupling regime  As for magnetic anisotropy, the nanostructures behave as isotropic magnetic systems or exhibit a cubic magnetic anisotropy  First we can confirm that nanostructures are not amorphous otherwise shape anisotropy would dominate leading to out-of-plane anisotropy  We can also rule out a random distribution of magnetic easy axes since the nanostructures are clearly crystallized in the diamond structure and would exhibit at least a cubic anisotropy (except if the random distribution of Mn atoms within the nanostructures can yield random easy axes)  Since the nanostructures are in strong in-plane compression (their lattice parameter is larger than the matrix one), the cubic symmetry of the diamond structure is broken and magnetic cubic anisotropy is thus unlikely  We rather believe that out-of-plane shape anisotropy is nearly compensated by in-plane magnetoelastic anisotropy due to compression leading to a \textit{pseudo} cubic anisotropy  From the blocking temperature (15 K) and the magnetic volume of the nanostructures , we can derive their magnetic anisotropy constant using $Kv=25k_{B}T_{B}$: K$\approx$10 kJ m$^{-3}$ which is of the same order of magnitude as shape anisotropy 

\begin{figure}[htb]
\center
    \includegraphics[width= 35\linewidth]{ /fig12a eps}
    \includegraphics[width= 63\linewidth]{ /fig12b eps} 
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 93}$Mn$_{0 07}$ sample grown at 122$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 50 K, 100 K, 150 K, 200 K and 250 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields  (0 015 T) }
\label{fig12}
\end{figure}

For growth temperatures $T_{g}\geq$120$^{\circ}$C and Mn concentrations $\geq$ 7 \%, samples exhibit a magnetic signal above 200 K corresponding to Ge$_{3}$Mn$_{5}$ clusters (see Fig  7a)  As we can see, SQUID measurements are much more sensitive to the presence of Ge$_{3}$Mn$_{5}$ clusters, even at low concentration, than TEM and x-ray diffraction used in section \ref{structural}  We also observe a sharp transition in the ZFC curve (see Fig  7b, Fig  12a and 12b): the peak becomes very large and is shifted towards high blocking temperatures (the signal is maximum at $T=$23 K)  This can be easily understood as a magnetic percolation of the four independent nanostructures obtained at low growth temperatures into a single magnetic nanocolumn  Therefore the magnetic volume increases sharply as well as blocking temperatures  At the same time, the size distribution widens as observed in TEM  In Fig  12a, we have performed ZFC-FC measurements at different $T_{0}$ temperatures  The ZFC-FC irreversibility is observed up to the Curie temperature of $\approx$120 K meaning that a fraction of nanocolumns is ferromagnetic (\textit{i e } $T_{B}\geq T_{C}$) 
In Fig  12b, in-plane and out-of-plane ZFC curves nearly superimpose for $T\leq$150 K due to the isotropic magnetic behavior of the nanocolumns: in-plane magnetoelastic anisotropy is still compensating out-of-plane shape anisotropy  Moreover the magnetic signal above 150 K corresponding to Ge$_{3}$Mn$_{5}$ clusters that start to form in this growth temperature range is strongly anisotropic  This perpendicular anisotropy confirms the epitaxial relation: (0002) Ge$_{3}$Mn$_{5}$ $\parallel$ (002) Ge discussed in Ref \cite{Bihl06}  The magnetic easy axis of the clusters lies along the hexagonal $c$-axis which is perpendicular to the film plane 

\begin{figure}[ht]
\center
   \includegraphics[width= 35\linewidth]{ /fig13a eps}
   \includegraphics[width= 63\linewidth]{ /fig13b eps} 
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 145$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 50 K, 100 K, 150 K, 200 K, 250 K and 300 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0 015 T) }
\label{fig13}
\end{figure}

For growth temperatures $T_{g}\geq$145$^{\circ}$C the cluster magnetic signal dominates (Fig  13b)  Superparamagnetic nanostructures are investigated performing ZFC-FC measurements at different $T_{0}$ temperatures (Fig  13a)  The first ZFC peak at low temperature \textit{i e } $\leq$ 150 K is attributed to  low-$T_{C}$ nanocolumns ($T_{C}\approx$130 K)  This peak is wider than for lower growth temperatures and its maximum is further shifted up to 30 K  These results are in agreement with TEM observations: increasing $T_{g}$ leads to larger nanocolumns (\textit{i e } higher blocking temperatures) and wider size distributions  ZFC-FC irreversibility is observed up to the Curie temperature due to the presence of ferromagnetic columns  The second peak above 180 K in the ZFC curve is attributed to Ge$_{3}$Mn$_{5}$ clusters and the corresponding ZFC-FC irreversibility persisting up to 300 K means that some clusters are ferromagnetic  We clearly evidence the out-of-plane anisotropy of Ge$_{3}$Mn$_{5}$ clusters and the isotropic magnetic behavior of nanocolumns (Fig  13b)  In this growth temperature range, we have also investigated the Mn concentration dependence of magnetic properties  

\begin{figure}[ht]
\center
    \includegraphics[width= 49\linewidth]{ /fig14a eps}
    \includegraphics[width= 49\linewidth]{ /fig14b eps} 
\caption{Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{1-x}$Mn$_{x}$ films grown at 150$^{\circ}$C plotted for different Mn concentrations: 2 3 \%; 4 \%; 7 \%; 9 \%; 11 3 \%  (b) ZFC-FC measurements performed on Ge$_{1-x}$Mn$_{x}$ films grown at 150$^{\circ}$C  The in-plane applied field is 0 025 T for 2 3 \% and 4 \% and 0 015 T for 8 \% and 11 3 \%  }
\label{fig14}
\end{figure}

In Fig  14a, for low Mn concentrations (2 3 \% and 4 \%) the contribution from diluted Mn atoms in the germanium matrix to the saturation magnetization is very high and nearly vanishes for higher Mn concentrations (7 \%, 9 \% and 13 \%) as observed for low growth temperatures  Above 7 \%, the magnetic signal mainly comes from nanocolumns and Ge$_{3}$Mn$_{5}$ clusters  We can derive more information from ZFC-FC measurements (Fig  14b)  Indeed, for 2 3 \% of Mn, ZFC and FC curves nearly superimpose down to low temperature meaning that nanocolumns are superparamagnetic in the whole temperature range  Moreover the weak irreversibility arising at 300 K  means that some Ge$_{3}$Mn$_{5}$ clusters have already formed in the samples even at very low Mn concentrations  For 4 \% of Mn, we can observe a peak with a maximum at the blocking temperature (12 K) in the ZFC curve  We can also derive the Curie temperature of nanocolumns: $\approx$45 K  The irresversibility arising at 300 K still comes from Ge$_{3}$Mn$_{5}$ clusters  Increasing the Mn concentration above 7 \% leads to: higher blocking temperatures (20 K and 30 K) due to larger nanocolumns and wider ZFC peaks due to wider size distributions in agreement with TEM observations (see Fig  3a)  Curie temperatures also increase (110 K and 130 K) as well as the contribution from Ge$_{3}$Mn$_{5}$ clusters \\
Finally when increasing $T_{g}$ above 160$^{\circ}$C, the nanocolumns magnetic signal vanishes and only Ge$_{3}$Mn$_{5}$ clusters and diluted Mn atoms coexist  The overall magnetic signal becomes comparable to the one measured on annealed samples in which only Ge$_{3}$Mn$_{5}$ clusters are observed by TEM (see Fig  7a) \\
The magnetic properties of high-$T_{C}$ nanocolumns obtained for $T_{g}$ close to 130$^{\circ}$C are discussed in detail in Ref \cite{Jame06} \\
In conclusion, at low growth temperatures ($T_{g}\leq$120$^{\circ}$C), nanocolumns are made of almost 4 independent elongated magnetic nanostructures  For $T_{g}\geq$120$^{\circ}$C, these independent nanostructures percolate into a single nanocolumn sharply leading to higher blocking temperatures  Increasing $T_{g}$ leads to larger columns with a wider size distribution as evidenced by ZFC-FC measurements and given by TEM observations  In parallel, some Ge$_{3}$Mn$_{5}$ clusters start to form and their contribution increases when increasing $T_{g}$  Results on magnetic anisotropy seems counter-intuitive  Indeed Ge$_{3}$Mn$_{5}$ clusters exhibit strong out-of-plane anisotropy whereas nanocolumns which are highly elongated magnetic structures are almost isotropic  This effect is probably due to compensating in-plane magnetoelastic coupling (due to the columns compression) and out-of-plane shape anisotropy  

\section{Conclusion}

In this paper, we have investigated the structural and magnetic properties of thin Ge$_{1-x}$Mn$_{x}$ films grown by low temperature molecular beam epitaxy  A wide range of growth temperatures and Mn concentrations have been explored  All the samples contain Mn-rich nanocolumns as a consequence of 2D-spinodal decomposition  However their size, crystalline structure and magnetic properties depend on growth temperature and Mn concentration  For low growth temperatures, nanocolumns are very small (their diameter ranges between 1 8 nm for 1 3 \% of Mn and 2 8 nm for 11 3 \% of Mn), their Curie temperature is rather low ($<$ 170 K) and they behave as almost four uncorrelated superparamagnetic nanostructures  Increasing Mn concentration leads to higher columns densities while diameters remain nearly unchanged  For higher growth temperatures, the nanocolumns mean diameter increases and their size distribution widens  Moreover the 4 independent magnetic nanostructures percolate into a single magnetic nanocolumn  Some columns are ferromagnetic even if Curie temperatures remain quite low  In this regime, increasing Mn concentration leads to larger columns while their density remains nearly the same  In parallel, Ge$_{3}$Mn$_{5}$ nanoclusters start to form in the film with their $c$-axis perpendicular to the film plane  In both temperature regimes, the Mn incorporation mechanism in the nanocolumns and/or in the matrix changes above 5 \% of Mn and nanocolumns exhibit an isotropic magnetic behaviour due to the competing effects of out-of-plane shape anisotropy and in-plane magnetoelastic coupling  Finally for a narrow range of growth temperatures around 130$^{\circ}$C, nanocolumns exhibit Curie temperatures higher than 400 K  Our goal is now to investigate the crystalline structure inside the nanocolumns, in particular the position of Mn atoms in the distorted diamond structure, which is essential to understand magnetic and future transport properties in Ge$_{1-x}$Mn$_{x}$ films 

\section{Aknowledgements}
The authors would like to thank Dr  F  Rieutord for grazing incidence x-ray diffraction measurements performed on the GMT station of BM32 beamline at the European Synchrotron Radiation Facility 





ground_truth: Goodwin became a Naval aviator in January 1929.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the average magnetic moment per column in Ge$_{1-x}$Mn$_{x}$ films?

Read the following text and answer briefly.

Context 4: 
,   ) in eq  ( ), which involves solving an optimization problem (such as a shortest path problem)  In section IV, we propose a specific encoding of preference θ for resolving eq  ( ), while ensuring the number of computations of the cost C g,θ ( ,  ) per update does not grow exponentially with the number of obstacles 

Decision Making

We consider a navigation problem where the robot receives reward according to the model R(s t , g, θ, a t )  We wish to find the optimal policy π that maximizes the expected discounted sum of future rewards, with discount factor γ  The above problem is a Partially Observable Markov Decision Process (POMDP)  
In this section, we propose an encoding of human's path preference θ for computing the posterior in eq  ( )  Devifrom the concept of homotopy classes, we define the preference according to a partitioning of the environment into polytopes, as shown in fig  , creating a hyperplane arrangement of the space 
Hyperplane arrangements have been used by Vincent and Schwager in the context of Neural Network verification  In our setting, we leverage this representation to define path preferences as preferred transitions between adjacent regions of the space 

Hyperplane Arrangement

We assume a two-dimensional environment composed of m polytopic obstacles, each defined by their half-space representation (H-representation) where A i ∈ R di×2 and b i ∈ R di , and where d i is the number of edges (hyperplanes) composing polytope i  Let n = i d i be the total number of hyperplanes  We leverage each obstacle's H-representation to construct a hyperplane arrangement of the environment as shown in fig 
 e  a partitioning of the space into polytopes  More specifically, each location in space belongs to a polytope j for which we can write an H-representation of the form where α j i ∈ {−1, 1} di is a vector specific to polytope j and obstacle i corresponding to the relative position of any point in the set with respect to each hyperplane in O i  
Fig  : Intent inference model in a hyperplane arrangement of the obstacle free space  We spatially decompose the preference θ into a set of preferred neighboring polytopes per region of the space  Within each polytope j, the human preference pj is a discrete distribution over the preferred neighbor in N (j) 
We assume that for a location st belonging to polytope j, and given goal g and preference pj, the observation ot and any other preference p i,i =j are conditionally independent  Concatenating elements from each obstacle's Hrepresentation, we can write polytope j's H-representation as where Some of the constraints in eq  ( ) (corresponding to rows of A, b and α j ) are redundant, i e  the set P j does not change upon their removal 
We can further reduce the Hrepresentation of a polytope to include only non-redundant constraints  By removing the rows corresponding to redundant constraints, we obtain new matrices A j e , b j e and α j e such that we can write the polytope's reduced H-representation as The non-redundant constraints correspond to edges of the polytope 
In other words, as the robot continually moves in space, the first hyperplane that it will cross upon exiting the polytope will correspond to one of the polytope's nonredundant constraints  Vincent and Schwager outline an iterative method for removing redundant constraints by solving n linear programs 
We use this method in practice for computing α j e for each polytope  We can now characterize each polytope by a vector α j e ∈ {−1, 1} n j e , where n j e ≤ n is the number of essential constraints of the polytope  The polytopes P j partition the environment into a hyperplane arrangement 

Path Preference

In this section, we provide a definition of preference θ according to a graphical representation of the environment based on the hyperplane arrangement  Under this representation, a path preference corresponds to a set of preferred transitions  In other words, for each polytope in the space, the human will have a preference to which neighboring polytope they wish to transition 
Let G := (V, E) be an undirected graph, where vertices are obstacle-free polytopes, and edges connect two adjacent polytopes  Each polytope is described by a unique vector α j as defined in eq  ( )  Two polytopes are adjacent if they share non-redundant constraints (rows in eq  ( )) corresponding to the same hyperplane (i e  they are on opposite sides of the hyperplane) 
Let N (v) be the set of neighbors of a vertex v  For each vertex, we denote p v the discrete-valued random variable describing which edge in N (v) the human intends to transition to  Using this formalism, we define a path preference as the set of preferred transitions over all nodes in the graph, Let m θ = v∈V |N (v)| be the cardinality of Θ, and m g = |Ω g | the number of possible goals 
A priori, the number of Bayesian updates required to update the belief at every iteration should be m θ × m g   Now, let us assume the conditional independence relationships described by the new problem diagram in fig    More specifically, we introduce the assumption that conditioned on a robot location s t , the goal g, and the preference for the corresponding vertex p v in the graph, the observation o t and the preference for any other vertex are conditionally independent 
In other words, the observations the human provides can be defined conditioned only on the robot location, the goal, and the human's preference for its current vertex p v   By introducing this assumption, each update step only requires updating the joint (p v , g), reducing the number of cost computations to |N (v)| × m g  
We can notice that by introducing this assumption, we removed the direct relationship between the number of polytopes in the environment and the complexity of the Bayesian update in eq  ( )  In practice, components of θ are not mutually independent  For example, if the human preference at a vertex v 1 is
, it is unlikely that the human will also prefer p v2 = (v 2 , v 1 ) (turning back)  We can improve our model by assuming a dependent relationship between preferences for adjacent edges, which does not significantly increase the complexity of the inference problem  An interesting property of our encoding is that any two paths that belong to different homotopy classes will cross different sequences of polytopes, i e  they correspond to a different sequence of edges on G 
This can be proved by contradiction  Let us suppose that two continuous trajectories ξ 1 and ξ 2 , with the same start and end points and that do not intersect any obstacle, traverse the same regions in G in the same order  From the construction of the hyperplane arrangement, each polytope that the paths traverse through is obstacle-free 
Therefore, within each polytope, there is no obstacle in the area located in between the portions of ξ 1 and ξ 2 that belong to the region  A smooth transformation of ξ 1 into ξ 2 can be obtained by transforming each portion of ξ 1 belonging to the polytopes it intersects into the corresponding portion of ξ 2 for the same polytopes, where the extremities of the trajectory portions are connected to one another along the polytope's edges (where the same edge is crossed by both paths) 
Along this transformation, the paths do not intersect any obstacle, and therefore ξ 1 and ξ 2 belong to the same homotopy class 

EXPERIMENTS

We evaluate our model on a simulated navigation task where the robot must reach a goal that is unknown a priori while respecting the path preferences indicated by a human  The robot navigates in a grid world containing obstacles  The transition model is deterministic: the robot selects an adjacent location on the grid to reach at the next time step 
The robot is also allowed to take diagonal actions  Each location s t in the map can be mapped to a vertex v t ∈ G  Therefore, the actions leading to locations mapped to different vertices correspond to edges on the graph  We note f (s t , a t ) the edge crossed by taking action a t from location s t  
The robot is given a mission time limit T max for reaching the goal  In this problem, we assume that the human selects actions to noisily minimize a cost function C g,θ , where θ is defined as per eq  ( ), corresponding to the length of the shortest path to the goal constrained by the preference (where the robot is only allowed to make transitions on G along preferred edges) 
More specifically, where δ(s t , g | o t , p vt ) designates the length of the shortest path from s t to g passing by o t and constrained by preference p vt   This is a slight variant of the cost function proposed by Best and Fitch , where we add in a conditioning on the path preference  We compute costs by running the A path planning algorithm on the environment maps (grid worlds with diagonal actions) and impose preference constraints by pruning invalid transitions from the search tree 
Reward model  At each step in time, the robot receives a reward which is a sum of three components: a goal-specific reward a preference-specific reward or penalty We compute solutions to the POMDP defined in section III-B with the online solver POMCP , and with the particularity that within the rollouts, the robot does not expect to collect human inputs 
Each time a solution is computed, the robot takes an action and may receive an observation  If it does, it updates its belief distribution over the unknown problem variables and resolves the POMDP over a receding horizon 

Baselines

• Goal only  The robot solves the POMDP while ignoring the effects of path preference  Similarly to , we assume the human is taking action to minimize a goaldependent cost C g (s t , o t ) = δ(s t , g | o t ), where the conditioning on the preference is removed  We also omit the path preference's contribution to the reward R pref  
• Compliant  The robot complies with the human input, but does not take an initiative  If the user stops providing information, the robot continues in the last direction indicated for 5 time steps (conserving its momentum), then stops  • Blended  We designed an arbitration function to decide between our proposed policy (accounting for path preferences) and the user's recommendation when the robot receives inputs 
Our metric to evaluate confidence in the robot's prediction for the purpose of arbitration is the entropy of the intention distribution H(g, p i ), where p i denotes the preferred neighbor for the current region  Because our representation of the world is discrete, the arbitration is given by a step function 
Denoted by U , the action corresponding to the human's input, and P , the robot's prediction for the optimal action, we write the policy where we chose h = 1 6 as the confidence threshold 

Results

When evaluating the algorithm, we consider that a run is successful if the robot reached the goal within its allocated mission time T max and only made transitions between graph vertices corresponding to the human's preferences  We vary the time delay between human inputs, from constant guidance (∆ T = 1) to only a single observation (∆ T ≥ T max ) 
Success rates  Table I reports the success rates for experiments conducted over six randomly sampled problem instances and 50 runs per instance in Map 1 (fig  )  When the human provides inputs at every iteration, the compliant policy shows the highest success rates  However, as ∆ T increases, the compliant robot is not able to accomplish the task within the allotted time as it does not receive sufficient inputs to do so, and performance decreases compared to the autonomous baselines 
We find that in these runs, accounting for path preference consistently improves performance compared with the goal-only baseline  Results also show that blending the user's input with the robot's policy (Path Preference + Blend) when the human provides information leads to improved performance  Belief entropy 
Figure shows a challenging problem instance where the directions the human provides do not align directly with the shortest path to the goal  By ignoring the effects of preferences in the problem model (goal only), the robot quickly infers from observations that the upper left goal is less likely than others (P (g) drops) 
The strong decrease in entropy shows that the robot becomes overconfident in this prediction  Overconfidence in an incorrect goal will prevent the agent from finding the correct goal once the human's indications directly align with it, as it needs to correct for the wrong predictions, as shown in the path realization (fig 
)  In this realization, the goal-only method (green robot) fails to search the upper left area within the allotted time  By accounting for path preferences in its model, the blue robot's entropy over the goal distribution decreases more steadily, allowing for it to leverage the human's latest observations and reach the goal successfully 
shows an over-confident prediction (shown by the strong reduction in belief entropy) that the correct goal is less likely, making it more difficult to reach the correct goal compared to a method that accounts for path preference  Computation time  In table II we provide the time required to solve the POMDP, and the time required to update the robot's belief as it receives new observations 
We compute solutions on three maps: a simple 10 × 10 grid world with 8 polytopes (fig  ), a 10 × 10 grid world with 56 polytopes (fig  ), and a 20×20 grid world with 73 polytopes (fig  )  The latter environment being larger, we increase the mission time and the depth of the search tree in POMCP from T max = 30 (Map 1 and Map 2) to T max = 60 (Map 3) 
We do not notice an increase in the time required to update the robot's belief with an increase in problem complexity, which is consistent with our observation that the complexity of the Bayesian update should not increase with the number of obstacles or polytopes  On the contrary, the belief update time on Map 2 and Map 3, containing more obstacles, is reduced compared to the first map 
More obstacles result in fewer iterations when solving the constrained shortest path problem with A   Adding constraints due to the obstacles and polytopes reduces the size of the A search tree  C  Limitations Simulation environments  In our simulations, we hardcoded the preference policy over the maps (e g  in Map 1, go around the table counter-clockwise) 
We randomly sampled problem instances (start and goal locations, and goal options) to reduce the bias introduced by these preference choices  To best evaluate and compare the different approaches, it would be best to sample preferences among a distribution of preferences chosen by a human (for example, from benchmarks resulting from a collection of data) 
Creating such a benchmark is an interesting direction for future work  Hyperplane arrangement construction  The main limitation of our approach is that the size and geometry of each polytope depends strongly on the geometry of the obstacles, as seen in fig    Because of this, the robot can make predictions over preferences that are too refined compared with the topology of the environment 
A direct consequence is that when the size of the polytopes is small, the information provided by the human can be incorrectly interpreted as a preference on the robot's immediate action  Our method can be improved by changing the structure of the hyperplane arrangement so that it relies on the topology of the environment, but does not vary strongly with the geometry of the features in the environment 
For this purpose, topometric maps and region construction algorithms are promising directions  We presented an approach for encoding and inferring a human's path preference in an environment with obstacles  By leveraging a partitioning of the space into polytopes and a stochastic observation model, our method allows for joint inference over the goal and path preference even when both are unknown a-priori 
Our experiments on an unknown-goal navigation problem with sparse human interventions demonstrate the effectiveness of our approach and its suitability for online applications  The time required to update the robot's belief does not increase with the complexity of the environment, which further highlights the practicality of our method 



ground_truth: 1425 $\mu_{B}$.